Geometry Study Guide

11 Units · Beginner-Friendly Notes · Step-by-Step Worked Examples · Interactive Quiz + Flashcards · Full-Length Practice Exam · Saved Progress

Unit 1: Foundations: Points, Lines & Angles4 concepts▾

The building blocks

Geometry starts with three undefined terms — point, line, and plane. You can't define them with simpler words; you just describe them.

Point: an exact location, no size. Named with a capital letter (point A).
Line: straight, goes forever in both directions, no thickness. Named by two points with a double arrow, like ↔AB, or a lowercase letter.
Plane: a flat surface that extends forever in all directions (like an endless tabletop).
Collinear points lie on the same line. Coplanar points lie on the same plane.
Line segment AB (written ̅AB): a piece of a line with two endpoints. It HAS length.
Ray AB (→AB): starts at endpoint A and goes forever through B. Order matters: ray AB ≠ ray BA.

Measuring segments

A segment has a definite length. If a point is between two others, the small pieces add up to the whole.

Segment Addition Postulate: if B is between A and C, then AB + BC = AC.
Midpoint: the point that splits a segment into two equal halves (AM = MB).
A segment bisector is any line, ray, or segment that passes through the midpoint.
Congruent segments (≅) have equal length. Equal numbers describe measures; congruent describes the figures.

Types of angles

An angle is formed by two rays sharing an endpoint (the vertex). Measured in degrees (°).

Acute: between 0° and 90°. Right: exactly 90° (square corner). Obtuse: between 90° and 180°. Straight: exactly 180°.
Angle Addition Postulate: if ray BD is inside ∠ABC, then m∠ABD + m∠DBC = m∠ABC.
An angle bisector cuts an angle into two equal angles.
Congruent angles have equal measure.

Angle pair relationships

These pairs show up constantly on exams. Learn the words and what equation each gives you.

Complementary angles: two angles that add to 90°.
Supplementary angles: two angles that add to 180°.
Linear pair: two adjacent angles that form a straight line → they are supplementary (add to 180°).
Vertical angles: the opposite angles formed by two crossing lines → they are ALWAYS congruent (equal).
Adjacent angles share a vertex and a side but no interior points.

“Complementary” = 90° (Corner). “Supplementary” = 180° (Straight). Mixing these up is the #1 beginner error.

Vertical angles are EQUAL, not supplementary. Don't set them equal to 180°.

Ray AB and ray BA are different rays — the first letter is always the starting point.

Complementary: x + y = 90°
Supplementary / Linear pair: x + y = 180°
Vertical angles: ∠1 = ∠3 (congruent)
Segment Addition: AB + BC = AC

Diagram

Two intersecting lines: ∠1 & ∠3 are vertical (equal); ∠1 & ∠2 form a linear pair (sum 180°).

Unit 2: Parallel Lines & Transversals4 concepts▾

The setup

When one line (a transversal) crosses two parallel lines, it makes 8 angles. Knowing which pairs are equal and which add to 180° lets you solve for almost anything.

Parallel lines (∥) never meet and have the same slope.
A transversal is a line that crosses two or more other lines.
“Interior” angles are between the two parallel lines; “exterior” angles are outside them.

Equal angle pairs (congruent)

When the two lines are parallel, these pairs are CONGRUENT (set them equal to each other).

Corresponding angles: same position at each intersection (e.g., both top-right) → congruent.
Alternate interior angles: between the lines, on opposite sides of the transversal → congruent.
Alternate exterior angles: outside the lines, on opposite sides → congruent.

Supplementary angle pairs

Some pairs add up to 180° instead of being equal.

Co-interior (same-side interior / consecutive interior) angles: between the lines, same side of the transversal → supplementary (add to 180°).
Co-exterior (same-side exterior) angles: outside the lines, same side → supplementary.
Tip: if two angles look about the same size, they're probably equal; if one looks big and one small, they probably add to 180°.

Working backward

You can also use these rules to PROVE two lines are parallel.

If corresponding angles are congruent, the lines are parallel.
If alternate interior angles are congruent, the lines are parallel.
If same-side interior angles are supplementary, the lines are parallel.

Only use these rules when the lines are PARALLEL. If they're not parallel, none of the pairs are equal.

Same-side interior angles are SUPPLEMENTARY (180°), not congruent — a very common slip.

“Big angle = small angle” is impossible; equal pairs are both big or both small.

Corresponding ≅, Alternate interior ≅, Alternate exterior ≅
Same-side interior: x + y = 180°
Parallel lines have equal slopes

Diagram

A transversal crossing parallel lines. Alternate interior angles (3 & 6) are equal; same-side interior (3 & 5) sum to 180°.

Unit 3: Triangles: Angles & Inequalities5 concepts▾

Classifying triangles

Triangles are named two ways: by their sides and by their angles.

By sides: scalene (no equal sides), isosceles (2 equal sides), equilateral (all 3 equal).
By angles: acute (all < 90°), right (one 90°), obtuse (one > 90°), equiangular (all 60°).
Equilateral triangles are also equiangular — every angle is 60°.

Angle sum & exterior angle

Two rules unlock most triangle problems.

Triangle Angle-Sum: the three interior angles always add to 180°.
Exterior Angle Theorem: an exterior angle equals the SUM of the two non-adjacent (remote) interior angles.
A right triangle's two acute angles are complementary (they add to 90°).

Isosceles triangle theorem

Isosceles triangles have a built-in symmetry that exams love to test.

Base Angles Theorem: if two sides are congruent, the angles opposite them are congruent.
The converse is also true: equal base angles → equal sides.
The altitude from the apex of an isosceles triangle bisects the base and the apex angle.

Triangle inequalities

These decide whether a triangle can even exist and which side/angle is biggest.

Triangle Inequality: the sum of any two sides must be GREATER than the third side.
The largest angle is opposite the longest side; the smallest angle is opposite the shortest side.
Quick test for three lengths: add the two smallest — if that sum beats the largest, a triangle is possible.

Special segments & centers

Four special lines create four special centers (good vocabulary to know).

Median: connects a vertex to the midpoint of the opposite side. The 3 medians meet at the centroid.
The centroid divides each median 2:1 (vertex-to-centroid is twice centroid-to-midpoint). It is the balance point.
Altitude: perpendicular from a vertex to the opposite side. The 3 altitudes meet at the orthocenter.
Perpendicular bisectors meet at the circumcenter (equidistant from the 3 vertices).
Angle bisectors meet at the incenter (equidistant from the 3 sides).

Exterior angle = SUM of the two far angles, NOT 180 minus one angle (though that also works via the linear pair).

Three lengths like 3, 4, 8 cannot form a triangle because 3 + 4 = 7 < 8.

Median goes to a midpoint; altitude makes a right angle — don't confuse them.

Interior angles: A + B + C = 180°
Exterior angle = sum of 2 remote interior angles
Triangle Inequality: a + b > c (for all three)
Centroid divides median 2 : 1

Diagram

Triangle interior angles always add to 180°. The exterior angle (d) equals a + b.

Unit 4: Congruent Triangles & Proofs4 concepts▾

What congruent means

Two triangles are congruent if they are exactly the same size and shape — you could slide/flip/turn one onto the other perfectly.

Congruent triangles have all corresponding sides equal and all corresponding angles equal (CPCTC).
Order matters in naming: △ABC ≅ △DEF means A↔D, B↔E, C↔F.
CPCTC = “Corresponding Parts of Congruent Triangles are Congruent” — used AFTER you prove triangles congruent to get one more equal part.

The five shortcuts

You don't need all six parts. Any one of these five combinations guarantees congruence.

SSS — three pairs of sides.
SAS — two sides and the angle BETWEEN them (included angle).
ASA — two angles and the side BETWEEN them (included side).
AAS — two angles and a non-included side.
HL — (right triangles only) hypotenuse and one leg.
NOT valid: SSA and AAA. AAA only proves SIMILAR, not congruent.

Reasons used in proofs

Proofs are just a chain of true statements, each with a reason. These reasons appear over and over.

Reflexive Property: a side or angle shared by both triangles is congruent to itself.
Vertical angles are congruent.
Alternate interior angles congruent (when lines are parallel).
Midpoint gives two congruent segments; bisector gives two congruent angles.
Definition of perpendicular gives right angles, which are all congruent.

Strategy for a proof

A reliable game plan for two-column or paragraph proofs.

1. Mark the diagram with everything the ‘Given’ tells you.
2. Look for free information: shared sides (reflexive), vertical angles, parallel-line angle pairs.
3. Decide which shortcut (SSS/SAS/ASA/AAS/HL) you can reach.
4. State the triangles congruent, then use CPCTC if the question asks for a specific side or angle.

SSA and AAA are NOT congruence shortcuts. Watch for the angle being in the wrong place (not included).

You can only use CPCTC AFTER proving the triangles congruent — never before.

Name corresponding vertices in matching order or your CPCTC pairs will be wrong.

Valid: SSS, SAS, ASA, AAS, HL
Invalid: SSA, AAA (AAA → similar only)
CPCTC used only AFTER congruence is proven

Diagram

Matching tick marks show corresponding congruent parts. Here two sides match (SAS needs the angle BETWEEN them; the shared side gives Reflexive).

Unit 5: Similarity & Proportions4 concepts▾

What similar means

Similar figures (~) have the same shape but not necessarily the same size — like a photo and its enlargement.

Corresponding angles are EQUAL; corresponding sides are PROPORTIONAL (same ratio).
That common ratio is the scale factor (k).
Congruent is the special case of similar where k = 1.

Proving triangles similar

Fewer parts are needed than for congruence.

AA — two pairs of equal angles (most common method used).
SAS~ — two pairs of proportional sides with the included angle equal.
SSS~ — all three pairs of sides proportional.
A line parallel to one side of a triangle cuts the other two sides proportionally (Side-Splitter Theorem).

Scale factor effects

When a figure is scaled by k, length, area, and volume change by different powers of k. This is heavily tested.

Lengths (sides, perimeter) scale by k.
Areas (and surface areas) scale by k².
Volumes scale by k³.
Example: if a model is 3× bigger in length, its area is 9× and its volume is 27×.

Right-triangle similarity

Dropping an altitude to the hypotenuse of a right triangle creates two smaller triangles similar to the original — this gives the ‘geometric mean’ relationships.

The altitude to the hypotenuse is the geometric mean of the two hypotenuse pieces: h = √(p·q).
Each leg is the geometric mean of the whole hypotenuse and the segment next to it.
Set up these as proportions: short/long = long/whole.

Areas scale by k² and volumes by k³ — a frequent trap is using k for all three.

In a proportion, keep corresponding parts in the same order (big-over-small on both sides).

AA needs two angles; one equal angle is not enough.

Similar: angles equal, sides in ratio k
Perimeter ratio = k, Area ratio = k², Volume ratio = k³
Geometric mean: altitude = √(p·q)
a/b = c/d ↔ ad = bc (cross-multiply)

Diagram

DE ∥ AC creates a smaller similar triangle: every length in △BDE is k times △BAC. Lengths scale by k, areas by k², volumes by k³.

Unit 6: Right Triangles & Trigonometry4 concepts▾

Pythagorean Theorem

In any RIGHT triangle, the square of the hypotenuse equals the sum of the squares of the legs.

a² + b² = c², where c is the hypotenuse (the side opposite the right angle — always the longest side).
Use it to find a missing side when you know the other two.
Pythagorean triples to recognize: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples like 6-8-10).

The trig ratios (SOH-CAH-TOA)

In a right triangle, the ratios of the sides depend only on the acute angle. These three ratios are your tools.

sin(θ) = Opposite / Hypotenuse (SOH).
cos(θ) = Adjacent / Hypotenuse (CAH).
tan(θ) = Opposite / Adjacent (TOA).
‘Opposite’ and ‘adjacent’ are relative to the angle you're using; the hypotenuse never changes.

Finding sides vs. angles

Pick your move based on what's missing.

Missing a SIDE and you know an angle: set up sin/cos/tan and solve.
Missing an ANGLE and you know two sides: use the inverse (sin⁻¹, cos⁻¹, tan⁻¹).
Make sure your calculator is in DEGREE mode for these problems.

Co-function & special right triangles

Two extra facts save time and appear on the exam.

Co-function relationship: sin(x) = cos(90° − x). The sine of an angle equals the cosine of its complement.
45-45-90 triangle: legs are equal; hypotenuse = leg · √2.
30-60-90 triangle: sides are in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).
Angle of elevation/depression: the angle up to (or down to) an object, measured from the horizontal.

Pythagorean Theorem ONLY works in right triangles.

The hypotenuse is opposite the right angle and is always the longest side — don't plug a leg in as c.

To find an angle you need the INVERSE trig button (sin⁻¹), not sin. Calculator must be in degree mode.

a² + b² = c²
sin = O/H, cos = A/H, tan = O/A
sin(x) = cos(90° − x)
45-45-90: 1 : 1 : √2 · 30-60-90: 1 : √3 : 2

Diagram

Right triangle: a² + b² = c². For angle θ: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Unit 7: Polygons & Quadrilaterals3 concepts▾

Polygon angle sums

A polygon is a closed figure made of straight sides. The number of sides (n) controls its angle totals.

Sum of INTERIOR angles = (n − 2) · 180°. (Triangle 180°, quadrilateral 360°, pentagon 540°…)
Sum of EXTERIOR angles of ANY polygon = 360° (always, no matter how many sides).
Regular polygon (all sides and angles equal): each interior angle = (n−2)·180 ⁄ n; each exterior angle = 360 ⁄ n.

The quadrilateral family

Quadrilaterals form a ‘family tree.’ Each special type inherits the properties above it and adds its own.

Parallelogram: both pairs of opposite sides parallel. Opposite sides ≅, opposite angles ≅, consecutive angles supplementary, diagonals BISECT each other.
Rectangle: a parallelogram with 4 right angles. Adds: diagonals are CONGRUENT.
Rhombus: a parallelogram with 4 ≅ sides. Adds: diagonals are PERPENDICULAR and bisect the angles.
Square: both a rectangle and a rhombus — has ALL of their properties.
Trapezoid: exactly one pair of parallel sides (the bases). Isosceles trapezoid: legs ≅, base angles ≅, diagonals ≅.

Proving a quadrilateral's type (coordinate)

You'll often need to PROVE what kind of quadrilateral a set of points makes, using coordinate tools.

Show sides parallel → equal SLOPES.
Show sides/diagonals congruent → equal DISTANCE (length).
Show right angles or a rhombus → slopes are NEGATIVE RECIPROCALS (perpendicular).
Parallelogram: opposite sides equal slope (or diagonals share a midpoint).
Rectangle: parallelogram + one right angle (perpendicular adjacent sides).
Rhombus: parallelogram + perpendicular diagonals (or 4 equal sides).

A square is a special rectangle AND a special rhombus — it satisfies every parallelogram property.

Exterior angles of any polygon always sum to 360°, not (n−2)·180°.

Trapezoid diagonals do NOT bisect each other (only parallelograms do).

Interior sum = (n − 2) · 180°
Exterior sum = 360° (always)
Regular interior angle = (n−2)·180 ⁄ n
Parallelogram diagonals bisect each other

Diagram

The quadrilateral family: each arrow adds properties. A square inherits EVERYTHING from both the rectangle and the rhombus.

Unit 8: Circles4 concepts▾

Parts of a circle

Learn the vocabulary first — most circle questions are about naming and relating these parts.

Radius: center to edge. Diameter: edge to edge through center (= 2 · radius). Chord: any segment with both endpoints on the circle.
Tangent: a line touching the circle at exactly one point (it is perpendicular to the radius at that point).
Secant: a line that cuts through the circle at two points.
Arc: part of the circle's edge. A central angle equals the measure of its intercepted arc.

Angle theorems

Where the vertex of an angle sits (center, on the circle, inside, or outside) decides the formula.

Central angle (vertex at center) = its intercepted arc.
Inscribed angle (vertex ON the circle) = HALF its intercepted arc.
An inscribed angle in a semicircle is a right angle (90°).
Inscribed angles that intercept the SAME arc are congruent.
Two chords crossing INSIDE: angle = half the SUM of the two intercepted arcs.
Two secants/tangents meeting OUTSIDE: angle = half the DIFFERENCE of the intercepted arcs.

Segment length relationships

When chords, secants, or tangents meet, the pieces multiply in predictable ways.

Two chords intersecting inside: (part)(part) = (part)(part).
Two secants from an outside point: (whole)(outside) = (whole)(outside).
Tangent-secant: tangent² = (whole secant)(outside part).

Arc length, sectors & equations

Circles connect to algebra through area, arc length, and the circle equation.

Circumference C = 2πr = πd. Area A = πr².
Arc length = (central angle ⁄ 360) · 2πr. Sector area = (central angle ⁄ 360) · πr².
Radian measure: arc length = r · θ (θ in radians). A full circle = 2π radians = 360°.
Equation of a circle: (x − h)² + (y − k)² = r², with center (h, k) and radius r.
To find the center/radius from a messy equation, COMPLETE THE SQUARE.

Inscribed angle = HALF the arc; central angle = the WHOLE arc. Mixing these is extremely common.

In the circle equation the center is (h, k) with MINUS signs: (x−h)²+(y−k)² means a +3 inside is center −3.

Inside vertex → SUM of arcs; outside vertex → DIFFERENCE of arcs.

C = 2πr · A = πr²
Inscribed angle = ½ arc · Central angle = arc
Arc length = (n ⁄ 360)·2πr · Sector = (n ⁄ 360)·πr²
Circle: (x − h)² + (y − k)² = r²

Diagram

Inscribed angle = ½ its arc. Central angle = the full arc. An angle in a semicircle is 90°.

Unit 9: Coordinate Geometry4 concepts▾

The three core formulas

Almost every coordinate problem uses distance, midpoint, or slope. Memorize all three.

Distance (length of a segment): d = √[(x₂−x₁)² + (y₂−y₁)²] — it's just the Pythagorean Theorem.
Midpoint (the middle point): M = ( (x₁+x₂)⁄2 , (y₁+y₂)⁄2 ) — average the x's and the y's.
Slope (steepness): m = (y₂−y₁) ⁄ (x₂−x₁) = rise ⁄ run.

Slope and parallel/perpendicular

Slope is how you test whether lines are parallel or perpendicular on a grid.

Parallel lines: SAME slope.
Perpendicular lines: slopes are NEGATIVE RECIPROCALS (multiply to −1). Example: 2 and −½.
Horizontal line: slope 0. Vertical line: slope undefined.

Equations of lines

Two main forms; pick whichever fits the given information.

Slope-intercept: y = mx + b (m = slope, b = y-intercept).
Point-slope: y − y₁ = m(x − x₁) (great when you have a point and a slope).
To write a perpendicular line through a point: flip-and-negate the slope, then use point-slope.

Partitioning a segment

Finding a point that divides a segment in a given ratio is a core skill to know.

To divide segment from A to B in ratio a:b, move a ⁄ (a+b) of the way from A to B.
Point = ( x₁ + (a ⁄ (a+b))(x₂−x₁) , y₁ + (a ⁄ (a+b))(y₂−y₁) ).
The midpoint is just the special 1:1 case.
Direction matters — partition from the FIRST named point toward the second.

Distance needs the square root — don't stop at the squared value.

Perpendicular slope is the negative RECIPROCAL: for 3⁄4 it's −4⁄3, not −3⁄4.

For partition ratio a:b, use the fraction a⁄(a+b), and start at the correct endpoint.

Distance = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint = ((x₁+x₂)⁄2 , (y₁+y₂)⁄2)
Slope m = (y₂−y₁)⁄(x₂−x₁)
⊥ slopes multiply to −1 · ∥ slopes equal

Diagram

On the coordinate plane: distance = √[(Δx)²+(Δy)²], slope = rise/run, midpoint = average of coordinates.

Unit 10: Transformations & Constructions4 concepts▾

Rigid motions (isometries)

A rigid motion slides, flips, or turns a figure WITHOUT changing its size or shape. The image is congruent to the original.

Translation: a slide. (x, y) → (x + a, y + b). Every point moves the same distance and direction.
Reflection: a flip over a line. Over x-axis: (x, y)→(x, −y). Over y-axis: (x, y)→(−x, y). Over y = x: (x, y)→(y, x).
Rotation: a turn about a center point. 90° CCW about origin: (x, y)→(−y, x). 180°: (x, y)→(−x, −y). 270° CCW: (x, y)→(y, −x).
Rigid motions PRESERVE distance, angle measure, parallelism, and area (everything but position).

Dilations

A dilation is the ONLY common transformation that changes size — it stretches or shrinks from a center point.

Centered at origin with scale factor k: (x, y) → (kx, ky).
k > 1 enlarges; 0 < k < 1 shrinks. The image is SIMILAR (same shape) but not congruent (unless k = 1).
A dilation preserves angle measure and maps a line to a parallel line (unless the line passes through the center).
Lengths multiply by k; areas multiply by k².

Symmetry & composition

Extra ideas often tested around transformations.

Line symmetry: a figure maps onto itself across a line of reflection.
Rotational symmetry: a figure maps onto itself after a turn of less than 360°.
A composition is doing one transformation then another; a glide reflection = translation + reflection.
Two figures are congruent if a sequence of rigid motions maps one to the other; similar if rigid motions + a dilation do.

Constructions (compass & straightedge)

Constructions use only a compass and straightedge — no measuring. Know what each construction produces.

Copy a segment / copy an angle: reproduce a length or angle exactly with arcs.
Perpendicular bisector: equal arcs from both endpoints; the crossing points define a line that is perpendicular AND passes through the midpoint.
Angle bisector: an arc from the vertex, then equal arcs from the two crossings, to split an angle in half.
Equilateral triangle: two arcs of the same radius from each endpoint of a segment.
Constructions rely on the fact that all radii drawn with the same compass setting are EQUAL.

Only DILATIONS change size; translations, reflections, and rotations keep the figure congruent.

Memorize rotation rules by quadrant signs: 90° CCW (x,y)→(−y,x), 180° (x,y)→(−x,−y).

A dilation with k=2 doubles lengths but quadruples (×4) the area.

Translation: (x,y)→(x+a, y+b)
Reflect x-axis: (x,−y) · y-axis: (−x,y) · y=x: (y,x)
Rotate 90°CCW: (−y,x) · 180°: (−x,−y) · 270°CCW: (y,−x)
Dilation k about origin: (kx, ky)

Diagram

One point, three images: reflections keep one coordinate, 180° negates both. Memorize by the quadrant the image lands in.

Unit 11: 3D Solids: Surface Area, Volume & Density4 concepts▾

Volume formulas

Volume is the space inside a 3-D solid, measured in cubic units. ‘B’ means the area of the base.

Prism or cylinder (same all the way up): V = B · h (base area × height). Cylinder base = πr², so V = πr²h.
Pyramid or cone (comes to a point): V = ⅓ · B · h. Cone: V = ⅓πr²h.
Sphere: V = (4 ⁄ 3)πr³.
A pyramid/cone is exactly ⅓ of the prism/cylinder with the same base and height.

Surface area

Surface area is the total area of all the outside faces (the ‘wrapping paper’), in square units.

Add up the area of every face. For a prism: 2 bases + the lateral (side) area.
Cylinder: 2πr² (two circles) + 2πrh (the wrapped-around rectangle).
Sphere surface area = 4πr².
Lateral area excludes the base(s); total surface area includes them.

Cross-sections & rotations

You may be asked what 2-D shape you get when you slice a solid, or spin a flat shape.

A cross-section is the 2-D shape formed when a plane slices through a solid.
Slicing a cylinder parallel to its base → a circle; perpendicular (vertical) → a rectangle.
Rotating a 2-D shape around an axis sweeps out a 3-D solid: a rectangle → cylinder; a right triangle → cone; a semicircle → sphere.

Density & modeling

Density problems combine volume with a rate — they are common ‘real-world’ questions.

Density = mass ⁄ volume. Rearranged: mass = density × volume.
Population density = people ⁄ area. Cost problems: cost = volume × price-per-unit-volume.
Step 1: find the volume (or area). Step 2: multiply or divide by the given rate.
Watch units — convert feet/inches or grams/kilograms so they match before computing.

Pyramids and cones need the ⅓ — forgetting it is the most common volume error.

Volume uses CUBIC units (cm³); surface area uses SQUARE units (cm²).

In a cone/pyramid, the slant height and the vertical height are different — surface area uses slant height, volume uses vertical height.

Prism/Cylinder: V = Bh (cyl = πr²h)
Pyramid/Cone: V = ⅓Bh (cone = ⅓πr²h)
Sphere: V = 4⁄3 πr³ · SA = 4πr²
Density = mass ⁄ volume

Diagram

A cone (or pyramid) is exactly ⅓ of the cylinder (or prism) with the same base and height.

Practice Question Bank — 440 questions

Unit 1: Foundations: Points, Lines & Angles (40)

Two lines intersect. One angle measures 72°. What is the measure of its vertical angle?
- 18°
- 72°
- 108°
- 144°
Vertical angles are always congruent (equal). The vertical angle is also 72°.
Two complementary angles are such that one is twice the other. What is the larger angle?
- 30°
- 45°
- 60°
- 120°
x + 2x = 90 → 3x = 90 → x = 30. The larger angle is 2x = 60°.
Two angles are supplementary. One is 3 times the other. The smaller angle is
- 30°
- 45°
- 60°
- 135°
x + 3x = 180 → 4x = 180 → x = 45°.
An inscribed angle in a semicircle (subtending a diameter) measures
- 45°
- 60°
- 90°
- 180°
An angle inscribed in a semicircle is always a right angle, 90°.
Two complementary angles are such that one is 4 times the other. The larger angle is
- 18°
- 36°
- 72°
- 54°
x + 4x = 90 → 5x = 90 → x = 18; larger = 4(18) = 72°.
∠1 and ∠2 are vertical angles. If ∠1 = (2x + 10)° and ∠2 = 50°, then x =
- 15
- 20
- 25
- 30
Vertical angles are equal: 2x + 10 = 50 → 2x = 40 → x = 20.
The supplement of an angle is 3 times the angle. The angle is
- 30°
- 45°
- 60°
- 135°
x + 3x = 180 → 4x = 180 → x = 45°.
M is the midpoint of ̅AB. If AM = 2x + 1 and MB = 11, then x =
- 4
- 5
- 6
- 10
AM = MB → 2x + 1 = 11 → 2x = 10 → x = 5.
Ray BD is in the interior of ∠ABC. If ∠ABD = 35° and ∠DBC = 55°, then m∠ABC =
- 20°
- 55°
- 90°
- 110°
Angle Addition: 35 + 55 = 90°.
If an angle is congruent to its own complement, the angle measures
- 30°
- 45°
- 60°
- 90°
x = 90 − x → 2x = 90 → x = 45°.
Adjacent angles ∠AOB = (3x)° and ∠BOC = (2x)° together form a right angle. Then x =
- 12
- 15
- 18
- 20
3x + 2x = 90 → 5x = 90 → x = 18.
An angle is 40° more than its complement. The angle measures
- 25°
- 50°
- 65°
- 115°
x = (90 − x) + 40 → 2x = 130 → x = 65°.
Two angles form a linear pair. One angle is 24° more than twice the other. What is the measure of the larger angle?
- 52°
- 76°
- 128°
- 104°
x + (2x + 24) = 180 → 3x = 156 → x = 52. Larger = 2(52) + 24 = 128°.
∠A and ∠B are vertical angles with m∠A = (5x − 17)° and m∠B = (3x + 25)°. What is m∠A?
- 21°
- 46°
- 88°
- 92°
Vertical angles are equal: 5x − 17 = 3x + 25 → x = 21 → m∠A = 5(21) − 17 = 88°.
B is between A and C. AB = 2x + 3, BC = 3x − 1, and AC = 42. What is the length of AB?
- 8
- 19
- 23
- 34
(2x+3) + (3x−1) = 42 → 5x + 2 = 42 → x = 8 → AB = 2(8) + 3 = 19.
M is the midpoint of segment AB. AM = 5x − 4 and MB = 3x + 10. What is the length of AB?
- 7
- 31
- 62
- 44
5x − 4 = 3x + 10 → x = 7 → AM = 31 → AB = 2(31) = 62.
Ray OC bisects ∠AOB. If m∠AOC = (3x + 5)° and m∠COB = (5x − 13)°, find m∠AOB.
- 32°
- 41°
- 64°
- 96°
Bisected halves are equal: 3x + 5 = 5x − 13 → x = 9 → each half = 32° → whole = 64°.
The complement of an angle is one-fourth of its supplement. Find the angle.
- 30°
- 45°
- 60°
- 72°
90 − x = ¼(180 − x) → 360 − 4x = 180 − x → 180 = 3x → x = 60°.
Three angles measuring (2x)°, (3x)°, and (4x)° together form a straight line. The largest of the three angles is
- 40°
- 60°
- 80°
- 100°
2x + 3x + 4x = 180 → x = 20 → largest = 4(20) = 80°.
An angle exceeds its supplement by 36°. The angle measures
- 72°
- 108°
- 126°
- 144°
x − (180 − x) = 36 → 2x = 216 → x = 108°.
In the diagram, two lines intersect. Find x.
- 10
- 20
- 30
- 60
The labeled angles are vertical, so they are equal: 3x + 10 = 70 → x = 20.
Ray OC lies inside right angle AOB. If m∠COB = 25° and m∠AOC = (2x + 5)°, find x.
- 25
- 30
- 32.5
- 60
Angle Addition: (2x + 5) + 25 = 90 → 2x = 60 → x = 30.
An angle is 3° more than twice its complement. The angle measures
- 29°
- 58°
- 61°
- 73°
x = 2(90 − x) + 3 → x = 183 − 2x → 3x = 183 → x = 61°.
Two angles in a linear pair are in the ratio 4 : 5. The smaller angle measures
- 40°
- 80°
- 90°
- 100°
4x + 5x = 180 → x = 20 → smaller = 80°.
B is the midpoint of AC. If AB = 3x − 7 and AC = 4x + 6, then AC =
- 10
- 23
- 46
- 40
AC = 2·AB: 4x + 6 = 2(3x − 7) → 4x + 6 = 6x − 14 → x = 10 → AC = 46.
Vertical angles measure (7x − 9)° and (4x + 27)°. What is the SUPPLEMENT of either angle?
- 75°
- 105°
- 93°
- 87°
7x − 9 = 4x + 27 → x = 12 → angle = 75° → supplement = 105°.
Three angles around a point measure (2x)°, (3x)°, and (4x)°. The largest angle is
- 40°
- 80°
- 120°
- 160°
Angles around a point total 360°: 9x = 360 → x = 40 → largest = 160°.
An angle exceeds its complement by 22°. The angle measures
- 34°
- 44°
- 56°
- 68°
x − (90 − x) = 22 → 2x = 112 → x = 56°.
Two angles are complementary. One measures 17°. The other measures
- 163°
- 73°
- 107°
- 17°
Complementary → 90 − 17 = 73°.
Two angles are complementary. One measures 41°. The other measures
- 139°
- 41°
- 131°
- 49°
Complementary → 90 − 41 = 49°.
Angle A and angle B form a linear pair. If m∠A = 38°, then m∠B =
- 38°
- 52°
- 142°
- 322°
Linear pair → 180 − 38 = 142°.
Two angles are complementary. One measures 23°. The other measures
- 23°
- 157°
- 67°
- 113°
Complementary → 90 − 23 = 67°.
Angle A and angle B form a linear pair. If m∠A = 66°, then m∠B =
- 294°
- 114°
- 66°
- 24°
Linear pair → 180 − 66 = 114°.
Angle A and angle B form a linear pair. If m∠A = 95°, then m∠B =
- 95°
- 5°
- 85°
- 265°
Linear pair → 180 − 95 = 85°.
Two angles are complementary. One measures 29°. The other measures
- 151°
- 29°
- 119°
- 61°
Complementary → 90 − 29 = 61°.
Angle A and angle B form a linear pair. If m∠A = 81°, then m∠B =
- 9°
- 81°
- 99°
- 279°
Linear pair → 180 − 81 = 99°.
Two angles are complementary. One measures 47°. The other measures
- 133°
- 43°
- 137°
- 47°
Complementary → 90 − 47 = 43°.
Angle A and angle B form a linear pair. If m∠A = 54°, then m∠B =
- 54°
- 306°
- 126°
- 36°
Linear pair → 180 − 54 = 126°.
Two angles are complementary. One measures 34°. The other measures
- 56°
- 146°
- 124°
- 34°
Complementary → 90 − 34 = 56°.
Angle A and angle B form a linear pair. If m∠A = 72°, then m∠B =
- 108°
- 72°
- 288°
- 18°
Linear pair → 180 − 72 = 108°.

Unit 2: Parallel Lines & Transversals (40)

Alternate interior angles measure (2x + 10)° and 50°. Find x.
- 20
- 30
- 25
- 15
Alternate interior angles are congruent: 2x + 10 = 50 → 2x = 40 → x = 20.
Same-side interior angles measure 120° and y°. Find y.
- 120
- 60
- 180
- 30
Same-side interior angles are supplementary: 120 + y = 180 → y = 60.
If corresponding angles formed by a transversal are congruent, then the two lines are:
- perpendicular
- parallel
- intersecting at 60°
- skew
Congruent corresponding angles prove the lines are parallel (the converse rule).
Co-interior (same-side interior) angles are (3x)° and (x + 20)°. Find x.
- 40
- 35
- 45
- 30
They are supplementary: 3x + x + 20 = 180 → 4x = 160 → x = 40.
Corresponding angles are (5x)° and (2x + 45)°. Find x.
- 9
- 15
- 12
- 45
5x = 2x + 45 → 3x = 45 → x = 15.
If two lines are cut by a transversal and same-side interior angles are supplementary, the lines are
- perpendicular
- skew
- parallel
- intersecting
Supplementary same-side interior angles prove the lines are parallel.
Parallel lines are cut by a transversal. Corresponding angles measure (4x)° and (x + 60)°. Find x.
- 12
- 15
- 20
- 24
Corresponding angles are equal: 4x = x + 60 → 3x = 60 → x = 20.
Alternate exterior angles measure 110° and (5x)°. Find x.
- 20
- 22
- 24
- 26
Alternate exterior angles are congruent: 5x = 110 → x = 22.
Same-side exterior angles measure x° and (x + 50)°. Find x.
- 55
- 65
- 70
- 75
Same-side exterior angles are supplementary: 2x + 50 = 180 → x = 65.
If a pair of alternate interior angles each measure 75°, then a same-side interior angle measures
- 75°
- 95°
- 105°
- 15°
Same-side interior is the supplement: 180 − 75 = 105°.
A transversal makes a 130° angle with one of two parallel lines. The same-side interior angle on the other line is
- 40°
- 50°
- 130°
- 230°
Same-side interior angles are supplementary: 180 − 130 = 50°.
Alternate interior angles measure (2x − 5)° and (x + 25)°. The measure of each angle is
- 30°
- 55°
- 60°
- 65°
2x − 5 = x + 25 → x = 30; angle = 30 + 25 = 55°.
Two parallel lines are cut by a transversal. Same-side interior angles measure (3x + 20)° and (2x − 15)°. The smaller angle measures
- 35°
- 55°
- 125°
- 145°
Supplementary: 5x + 5 = 180 → x = 35 → angles are 125° and 55°. Smaller = 55°.
Alternate interior angles measure (7x − 14)° and (4x + 22)°. Each angle measures
- 12°
- 36°
- 70°
- 110°
Equal: 7x − 14 = 4x + 22 → x = 12 → angle = 7(12) − 14 = 70°.
Corresponding angles measure (5x − 22)° and (3x + 14)°. What is the measure of the same-side interior angle paired with them?
- 68°
- 112°
- 18°
- 158°
5x − 22 = 3x + 14 → x = 18 → angle = 68°. Its same-side partner = 180 − 68 = 112°.
Alternate exterior angles measure (2x + 30)° and (4x − 10)°. Each measures
- 20°
- 50°
- 70°
- 110°
Equal: 2x + 30 = 4x − 10 → x = 20 → angle = 70°.
Lines ℓ and m are parallel. A third line is perpendicular to ℓ. The acute/right angle it makes with m is
- 45°
- 60°
- 90°
- cannot be determined
A line perpendicular to one of two parallel lines is perpendicular to the other → 90°.
Two same-side interior angles are in the ratio 1 : 4. The smaller angle measures
- 36°
- 45°
- 30°
- 144°
x + 4x = 180 → x = 36°.
Which is an equation of the line through (2, 3) parallel to the line 3x + 2y = 8?
- y − 3 = −3/2(x − 2)
- y − 3 = 2/3(x − 2)
- y − 3 = 3/2(x − 2)
- y − 3 = −2/3(x − 2)
3x + 2y = 8 → y = −3/2 x + 4, slope −3/2. Parallel keeps the slope: y − 3 = −3/2(x − 2).
Parallel lines are cut by a transversal. One angle measures 115°. An alternate interior angle's LINEAR-PAIR partner measures
- 115°
- 75°
- 65°
- 25°
The alternate interior angle is 115°; its linear pair = 180 − 115 = 65°.
Lines ℓ and m are parallel. The two labeled angles are same-side interior angles. Find x.
- 40
- 70
- 80
- 110
Same-side interior angles are supplementary: (x + 30) + 70 = 180 → x = 80.
The two labeled angles are corresponding angles formed by a transversal crossing parallel lines. Find x.
- 26
- 32
- 38
- 52
Corresponding angles are congruent: 2x + 20 = 84 → 2x = 64 → x = 32.
Corresponding angles measure (5x + 15)° and (7x − 25)°. Each measures
- 20°
- 95°
- 115°
- 65°
5x + 15 = 7x − 25 → x = 20 → angle = 115°.
Alternate exterior angles measure (6x − 12)° and (4x + 20)°. Each measures
- 16°
- 84°
- 96°
- 104°
6x − 12 = 4x + 20 → x = 16 → angle = 84°.
Same-side interior angles measure (2x + 40)° and (3x + 30)°. The smaller one is
- 22°
- 84°
- 96°
- 110°
Sum 180: 5x + 70 = 180 → x = 22 → angles 84° and 96° → smaller 84°.
Which is an equation of the line through (1, 2) parallel to 2x − 3y = 6?
- y − 2 = ⅔(x − 1)
- y − 2 = −⅔(x − 1)
- y − 2 = 3/2(x − 1)
- y − 2 = −3/2(x − 1)
2x − 3y = 6 → y = ⅔x − 2, slope ⅔. Parallel keeps it: y − 2 = ⅔(x − 1).
Alternate interior angles measure (3x − 20)° and 70°. Find x.
- 16.7
- 30
- 50
- 90
Congruent: 3x − 20 = 70 → 3x = 90 → x = 30.
Two parallel lines are cut by a transversal. One angle measures 112°. A same-side EXTERIOR angle paired with it measures
- 112°
- 68°
- 22°
- 158°
Same-side exterior angles are supplementary: 180 − 112 = 68°.
A line has slope 3. Any line parallel to it has slope
- -3
- -0.333333
- 0.333333
- 3
Parallel lines have equal slopes.
Parallel lines are cut by a transversal. An angle of 109° and angle x are same-side interior angles. x =
- 19°
- 109°
- 251°
- 71°
Same-side interior → supplementary: 180 − 109 = 71°.
Parallel lines are cut by a transversal. An angle of 71° and angle x are same-side interior angles. x =
- 289°
- 19°
- 71°
- 109°
Same-side interior → supplementary: 180 − 71 = 109°.
A line has slope 2. Any line parallel to it has slope
- -0.5
- 2
- 0.5
- -2
Parallel lines have equal slopes.
Parallel lines are cut by a transversal. An angle of 117° and angle x are same-side interior angles. x =
- 27°
- 117°
- 63°
- 243°
Same-side interior → supplementary: 180 − 117 = 63°.
A line has slope 5. Any line parallel to it has slope
- 5
- -0.2
- 0.2
- -5
Parallel lines have equal slopes.
A line has slope -3. Any line parallel to it has slope
- -0.333333
- -3
- 0.333333
- 3
Parallel lines have equal slopes.
A line has slope -2. Any line parallel to it has slope
- -2
- 2
- -0.5
- 0.5
Parallel lines have equal slopes.
Parallel lines are cut by a transversal. An angle of 84° and angle x are same-side interior angles. x =
- 96°
- 276°
- 84°
- 6°
Same-side interior → supplementary: 180 − 84 = 96°.
Parallel lines are cut by a transversal. An angle of 63° and angle x are same-side interior angles. x =
- 297°
- 117°
- 63°
- 27°
Same-side interior → supplementary: 180 − 63 = 117°.
Parallel lines are cut by a transversal. An angle of 98° and angle x are same-side interior angles. x =
- 262°
- 8°
- 98°
- 82°
Same-side interior → supplementary: 180 − 98 = 82°.
A line has slope 4. Any line parallel to it has slope
- 0.25
- -0.25
- 4
- -4
Parallel lines have equal slopes.

Unit 3: Triangles: Angles & Inequalities (40)

An exterior angle of a triangle equals the sum of the two remote interior angles. If those are 50° and 70°, the exterior angle is:
- 60°
- 110°
- 120°
- 130°
Exterior Angle Theorem: 50 + 70 = 120°.
In an isosceles triangle the vertex angle is 40°. Each base angle measures:
- 40°
- 70°
- 100°
- 140°
Base angles are equal: (180 − 40) ÷ 2 = 70°.
Which set of lengths CANNOT form a triangle?
- 5, 6, 10
- 3, 4, 8
- 7, 8, 9
- 6, 8, 10
Triangle Inequality fails: 3 + 4 = 7, which is not greater than 8.
A triangle has sides 5, 12, and 13. It is a
- right triangle
- acute triangle
- obtuse triangle
- not a triangle
5² + 12² = 13² (25 + 144 = 169), so it's a right triangle.
The centroid of a triangle divides each median in the ratio
- 1 : 1
- 2 : 1
- 3 : 1
- 1 : 3
The centroid divides each median 2:1 (vertex side is twice the midpoint side).
A median of a triangle is 18 long. The distance from the vertex to the centroid is
- 6
- 9
- 12
- 15
The centroid is 2/3 of the way from the vertex: (2/3)(18) = 12.
The angles of a triangle are 3x, 4x, and 5x. The largest angle is
- 45°
- 60°
- 75°
- 90°
12x = 180 → x = 15; largest = 5(15) = 75°.
An exterior angle of a triangle is 120°. If one remote interior angle is 45°, the other is
- 55°
- 60°
- 75°
- 85°
Exterior = sum of remote interiors: 120 = 45 + x → x = 75°.
Which length CANNOT be the third side of a triangle with sides 7 and 10?
- 4
- 9
- 16
- 18
Third side must be < 17 and > 3. 18 fails (7 + 10 = 17, not > 18).
In △ABC, ∠A = 50°, ∠B = 60°, ∠C = 70°. The longest side is
- BC
- AC
- AB
- all equal
The longest side is opposite the largest angle (∠C = 70°), which is AB.
The acute angles of a right triangle are 2x and 3x. The larger acute angle is
- 36°
- 45°
- 54°
- 72°
2x + 3x = 90 → x = 18; larger = 3(18) = 54°.
On a median, the distance from the centroid to the midpoint is 5. The distance from the vertex to the centroid is
- 2.5
- 5
- 10
- 15
Centroid ratio is 2:1, so vertex-to-centroid = 2 × 5 = 10.
The angles of a triangle measure (2x + 10)°, (3x − 5)°, and (x + 25)°. The largest angle is
- 50°
- 60°
- 70°
- 80°
Sum: 6x + 30 = 180 → x = 25 → angles 60°, 70°, 50°. Largest = 70°.
An exterior angle measures (7x − 4)°; the remote interior angles are (3x + 8)° and (2x + 12)°. The exterior angle measures
- 12°
- 44°
- 80°
- 100°
7x − 4 = (3x+8) + (2x+12) → 7x − 4 = 5x + 20 → x = 12 → exterior = 80°.
In an isosceles triangle the vertex angle is (2x)° and each base angle is (4x − 10)°. The vertex angle measures
- 20°
- 40°
- 70°
- 80°
2x + 2(4x − 10) = 180 → 10x − 20 = 180 → x = 20 → vertex = 40°.
A triangle has sides 8 and 15. Which could NOT be the length of the third side?
- 8
- 14
- 22
- 7
Third side must satisfy 7 < s < 23 strictly. 7 fails because 8 + 7 = 15 is not greater than 15.
A median of a triangle is 24 units long. How far is the centroid from the MIDPOINT of the opposite side?
- 8
- 12
- 16
- 24
The centroid is 2:1 from the vertex, so the short piece = ⅓(24) = 8.
A triangle has sides 7, 24, and 25. This triangle is
- right
- acute
- obtuse
- equiangular
7² + 24² = 49 + 576 = 625 = 25² → right triangle by the Pythagorean converse.
In △ABC, m∠A = 2·m∠B and m∠C = m∠B + 20°. Find m∠A.
- 40°
- 60°
- 80°
- 100°
2b + b + (b + 20) = 180 → 4b = 160 → b = 40 → m∠A = 80°.
An isosceles triangle has perimeter 50 and base 12. Each leg measures
- 19
- 25
- 38
- 31
Legs: (50 − 12) ÷ 2 = 19.
Find the value of x in the triangle.
- 50
- 60
- 70
- 120
Angles of a triangle sum to 180: x = 180 − 55 − 65 = 60.
A side of the triangle is extended. Find the measure of exterior angle x.
- 80°
- 100°
- 120°
- 140°
Exterior angle = sum of the two remote interior angles = 40 + 60 = 100°.
The angles of a triangle are x°, (x + 15)°, and (2x − 35)°. The triangle is
- equilateral
- isosceles
- right
- obtuse scalene
4x − 20 = 180 → x = 50 → angles 50°, 65°, 65° → two equal angles → isosceles.
An exterior angle measures (2x + 30)°; its remote interior angles are (x + 40)° and 30°. The exterior angle is
- 40°
- 70°
- 110°
- 150°
2x + 30 = x + 70 → x = 40 → exterior = 110°.
A triangle has sides 9, 40, and 41. It is
- acute
- right
- obtuse
- impossible
9² + 40² = 81 + 1600 = 1681 = 41² → right triangle.
MN joins the midpoints of two sides of a triangle. If MN = 3x − 2 and the parallel side BC = 5x + 6, then BC =
- 10
- 28
- 56
- 62
Midsegment = half the side: 2(3x − 2) = 5x + 6 → 6x − 4 = 5x + 6 → x = 10 → BC = 56.
A median of a triangle is 30 units. The distance from the vertex to the centroid is
- 10
- 15
- 20
- 25
Vertex-to-centroid = ⅔ of the median = 20.
An isosceles triangle has vertex angle (3x + 12)° and base angles (x + 24)° each. The vertex angle is
- 24°
- 48°
- 84°
- 96°
(3x+12) + 2(x+24) = 180 → 5x + 60 = 180 → x = 24 → vertex = 84°.
Two angles of a triangle measure 77° and 33°. The third angle measures
- 110°
- 57°
- 103°
- 70°
180 − 77 − 33 = 70°.
The remote interior angles of a triangle measure 31° and 88°. The exterior angle measures
- 57°
- 61°
- 119°
- 149°
Exterior = sum of remote interiors = 119°.
The remote interior angles of a triangle measure 48° and 73°. The exterior angle measures
- 121°
- 132°
- 59°
- 25°
Exterior = sum of remote interiors = 121°.
Two angles of a triangle measure 42° and 66°. The third angle measures
- 108°
- 72°
- 138°
- 48°
180 − 42 − 66 = 72°.
The remote interior angles of a triangle measure 40° and 70°. The exterior angle measures
- 110°
- 140°
- 30°
- 70°
Exterior = sum of remote interiors = 110°.
Two angles of a triangle measure 35° and 75°. The third angle measures
- 70°
- 55°
- 110°
- 145°
180 − 35 − 75 = 70°.
The remote interior angles of a triangle measure 55° and 45°. The exterior angle measures
- 10°
- 125°
- 80°
- 100°
Exterior = sum of remote interiors = 100°.
Two angles of a triangle measure 51° and 49°. The third angle measures
- 80°
- 41°
- 129°
- 100°
180 − 51 − 49 = 80°.
The remote interior angles of a triangle measure 62° and 38°. The exterior angle measures
- 100°
- 80°
- 24°
- 118°
Exterior = sum of remote interiors = 100°.
The remote interior angles of a triangle measure 66° and 52°. The exterior angle measures
- 62°
- 114°
- 14°
- 118°
Exterior = sum of remote interiors = 118°.
Two angles of a triangle measure 28° and 94°. The third angle measures
- 62°
- 122°
- 58°
- 152°
180 − 28 − 94 = 58°.
Two angles of a triangle measure 63° and 58°. The third angle measures
- 121°
- 32°
- 117°
- 59°
180 − 63 − 58 = 59°.

Unit 4: Congruent Triangles & Proofs (36)

In SAS, the angle used must be:
- any angle
- the included angle (between the two sides)
- a right angle
- the largest angle
SAS requires the angle BETWEEN the two given sides (the included angle).
CPCTC can be used:
- before proving triangles congruent
- only after proving triangles congruent
- to prove triangles similar
- instead of a congruence shortcut
CPCTC (corresponding parts...) is only used AFTER you prove the triangles congruent.
If △ABC ≅ △DEF, then angle B corresponds to:
- angle D
- angle E
- angle F
- side DE
Matching the naming order A↔D, B↔E, C↔F. So ∠B ↔ ∠E.
Two angles and the side between them are congruent. This is:
- SSS
- SAS
- ASA
- AAS
ASA = two angles with the INCLUDED side between them.
Which pair of given parts would let you use AAS?
- two sides and a non-included angle
- two angles and a non-included side
- three sides
- two sides and the included angle
AAS = two angles and a side that is NOT between them.
Vertical angles in two triangles can be used in a proof because they are
- supplementary
- congruent
- complementary
- right angles
Vertical angles are always congruent — useful as an angle pair in proofs.
To prove △MIE ≅ △LSE using a pair of vertical angles plus two angle pairs, you would cite
- SSS
- SAS
- ASA or AAS
- HL
Two angles (including the vertical-angle pair) plus a side give ASA or AAS.
Two angles and the side between them are congruent. This proves congruence by
- SSS
- SAS
- ASA
- AAS
ASA = two angles with the included side.
In △ABC ≅ △DEF, angle B corresponds to
- angle D
- angle E
- angle F
- side DE
Matching order A↔D, B↔E, C↔F → ∠B ↔ ∠E.
If △ABC ≅ △DEF and m∠A = 50°, then m∠D =
- 40°
- 50°
- 90°
- 130°
Corresponding angles of congruent triangles are equal: ∠D = 50°.
If △ABC ≅ △DEF and EF = 9, then BC =
- 3
- 6
- 9
- 18
Corresponding sides are congruent: BC ↔ EF = 9.
Two sides and the angle BETWEEN them are congruent. This is
- SSS
- SAS
- ASA
- AAS
SAS = two sides with the included angle.
Given AB ≅ DE and ∠A ≅ ∠D, which additional congruence lets you conclude △ABC ≅ △DEF by ASA?
- ∠B ≅ ∠E
- ∠C ≅ ∠F
- BC ≅ EF
- AC ≅ DF
For ASA the given side must be INCLUDED between the two angles: AB lies between ∠A and ∠B, so you need ∠B ≅ ∠E.
△ABC ≅ △XYZ, AB = 3x + 7 and XY = 5x − 9. The length of AB is
- 8
- 23
- 31
- 39
Corresponding sides equal: 3x + 7 = 5x − 9 → x = 8 → AB = 31.
Triangles ABC and ADE share ∠A, with AB ≅ AD and AC ≅ AE. The triangles are congruent by
- SSS
- SAS
- ASA
- HL
Two pairs of sides with the shared included angle ∠A → SAS.
In right triangles ABC and DEF, ∠C and ∠F are right angles, AC ≅ DF, and hypotenuses AB ≅ DE. The triangles are congruent by
- SSA
- ASA
- AAS
- HL
Right angle + congruent hypotenuse + congruent leg = HL.
Which piece of information is NOT sufficient (with nothing else) to help prove two triangles congruent?
- three pairs of congruent sides
- two sides and the included angle
- the triangles have equal areas
- two angles and an included side
Equal areas say nothing about shape — many non-congruent triangles share an area.
After proving △ABC ≅ △DEF, the statement BC ≅ EF is justified by
- the Reflexive Property
- CPCTC
- the definition of midpoint
- SAS
Once the triangles are congruent, corresponding parts are congruent (CPCTC).
Segments AC and BD bisect each other at M. △AMB ≅ △CMD by
- SAS
- ASA
- SSS
- HL
Bisecting gives AM ≅ MC and BM ≅ MD; vertical angles ∠AMB ≅ ∠CMD are included → SAS.
△ABC ≅ △DEF, m∠B = (4x + 6)° and m∠E = (6x − 18)°. Find m∠B.
- 12°
- 30°
- 54°
- 66°
4x + 6 = 6x − 18 → x = 12 → m∠B = 4(12) + 6 = 54°.
Segments AC and BD intersect at M, with AM ≅ MC (single ticks) and BM ≅ MD (double ticks). △AMB ≅ △CMD by
- SSS
- SAS
- ASA
- AAS
Two pairs of congruent sides plus the included VERTICAL angles at M → SAS.
Two right triangles have congruent hypotenuses (single ticks) and one pair of congruent legs (double ticks). They are congruent by
- SSA — not valid
- HL
- AAS
- SAS
Right angle + hypotenuse + leg is the HL theorem (valid for right triangles only).
Given ∠A ≅ ∠D, ∠B ≅ ∠E, and AB ≅ DE, the triangles are congruent by
- SAS
- ASA
- SSS
- HL
AB is the side INCLUDED between ∠A and ∠B → ASA.
Given AB ≅ DE, BC ≅ EF, and ∠B ≅ ∠E, the triangles are congruent by
- ASA
- AAS
- SAS
- SSS
∠B is included between sides AB and BC → SAS.
A sequence of rigid motions maps △ABC onto △DEF. The triangles must be congruent because rigid motions preserve
- orientation only
- distance and angle measure
- area only
- slope
Rigid motions preserve distance and angle measure, so all corresponding parts match.
△ABC ≅ △DEF with AB = 2x + 9 and DE = 4x − 13. AB =
- 11
- 22
- 31
- 35
2x + 9 = 4x − 13 → x = 11 → AB = 31.
Two triangles share vertical angles at M, and one pair of sides AM ≅ MC is marked. For SAS you also need
- ∠A ≅ ∠C
- BM ≅ MD
- AB ≅ CD
- ∠B ≅ ∠D
SAS needs the second pair of sides that FORM the vertical angle: BM ≅ MD.
After a reflection then a translation, △PQR lands exactly on △P'Q'R'. Which statement must be true?
- PQ ∥ P'Q'
- the triangles are congruent
- orientation is preserved
- the triangles are similar but not congruent
Reflections and translations are rigid motions → the image is congruent (orientation flipped by the reflection).
△ABC ≅ △DEF. AB = 5x + 4 and DE = 34. Then x =
- 7
- 34
- 5
- 6
Corresponding sides equal: 5x + 4 = 34 → x = 6.
△PQR ≅ △XYZ and m∠Q = 71°. Then m∠Y =
- 19°
- 71°
- 81°
- 109°
Corresponding angles of congruent triangles are equal.
△ABC ≅ △DEF. AB = 4x + 11 and DE = 43. Then x =
- 9
- 7
- 43
- 8
Corresponding sides equal: 4x + 11 = 43 → x = 8.
△PQR ≅ △XYZ and m∠Q = 58°. Then m∠Y =
- 122°
- 58°
- 68°
- 32°
Corresponding angles of congruent triangles are equal.
△ABC ≅ △DEF. AB = 2x + 9 and DE = 33. Then x =
- 33
- 11
- 12
- 13
Corresponding sides equal: 2x + 9 = 33 → x = 12.
△PQR ≅ △XYZ and m∠Q = 47°. Then m∠Y =
- 133°
- 57°
- 43°
- 47°
Corresponding angles of congruent triangles are equal.
△ABC ≅ △DEF. AB = 3x + 7 and DE = 34. Then x =
- 9
- 8
- 34
- 10
Corresponding sides equal: 3x + 7 = 34 → x = 9.
△PQR ≅ △XYZ and m∠Q = 34°. Then m∠Y =
- 146°
- 34°
- 56°
- 44°
Corresponding angles of congruent triangles are equal.

Unit 5: Similarity & Proportions (39)

A figure is dilated by a scale factor of 3. Its area is multiplied by:
- 3
- 6
- 9
- 27
Area scales by k²: 3² = 9.
A solid is enlarged by scale factor 2. Its volume is multiplied by:
- 2
- 4
- 6
- 8
Volume scales by k³: 2³ = 8.
The altitude to the hypotenuse splits it into pieces of 4 and 9. The altitude length is:
- 6
- 6.5
- 13
- 36
Geometric mean: altitude = √(4·9) = √36 = 6.
Two similar triangles have a side ratio of 2 : 5. The ratio of their areas is
- 2 : 5
- 4 : 10
- 4 : 25
- 8 : 125
Area ratio = (side ratio)² = 2² : 5² = 4 : 25.
In △ABC, DE ∥ AC with D on AB and E on CB. If BD = 6, DA = 4, and BE = 9, then EC =
- 6
- 8
- 12
- 13.5
Side-Splitter: BD/DA = BE/EC → 6/4 = 9/EC → EC = 6.
Two similar triangles have a side ratio of 3:4. If the smaller perimeter is 15, the larger perimeter is
- 18
- 20
- 24
- 45
Perimeter ratio = side ratio: 15/x = 3/4 → x = 20.
A solid is dilated by scale factor ½. Its volume is multiplied by
- 1/2
- 1/4
- 1/8
- 1/16
Volume scales by k³: (½)³ = 1/8.
In △ABC, DE ∥ AC with BD = 8, DA = 4, BE = 10. Then EC =
- 5
- 6
- 8
- 20
Side-Splitter: BD/DA = BE/EC → 8/4 = 10/EC → EC = 5.
The geometric mean (altitude) for hypotenuse segments 9 and 4 is
- 6
- 6.5
- 13
- 36
Altitude = √(9·4) = √36 = 6.
Two similar polygons have a scale factor of 5:2. The ratio of their areas is
- 5:2
- 10:4
- 25:4
- 125:8
Area ratio = (5:2)² = 25:4.
△ABC ∼ △DEF with ratio 2:3. If AB = 10, then DE =
- 6.7
- 12
- 15
- 20
10/DE = 2/3 → 2·DE = 30 → DE = 15.
In similar figures, corresponding angles are
- proportional
- congruent
- supplementary
- complementary
Similar figures have congruent (equal) corresponding angles.
△ABC ∼ △DEF with AB = 8 and DE = 12. If the perimeter of △ABC is 30, the perimeter of △DEF is
- 34
- 40
- 45
- 67.5
Scale factor 12/8 = 3/2. Perimeter scales by k: 30 × 3/2 = 45.
Two similar triangles have areas 36 and 81. If a side of the smaller is 10, the corresponding side of the larger is
- 12.5
- 15
- 22.5
- 25
Area ratio 36:81 = 4:9 → side ratio 2:3 → 10 × 3/2 = 15.
The altitude to the hypotenuse measures 12 and one hypotenuse segment measures 9. The other segment measures
- 3
- 15
- 16
- 21
altitude² = (seg₁)(seg₂): 144 = 9x → x = 16.
DE ∥ AC in △ABC. BD = x, DA = x − 4, BE = 15, EC = 10. Find x.
- 8
- 10
- 12
- 20
Side-Splitter: x/(x−4) = 15/10 → 10x = 15x − 60 → x = 12.
In a right triangle, a leg measures 10 and the hypotenuse segment adjacent to that leg measures 5. The whole hypotenuse measures
- 15
- 20
- 25
- 50
leg² = (adjacent segment)(whole hypotenuse): 100 = 5h → h = 20.
Two similar solids have volumes in the ratio 8 : 27. Their surface areas are in the ratio
- 2 : 3
- 4 : 9
- 8 : 27
- 64 : 729
Volume ratio 8:27 → side ratio 2:3 → area ratio 2²:3² = 4:9.
DE ∥ AC creates △BDE ∼ △BAC. If BD = 6, BA = 10, and DE = 9, then AC =
- 5.4
- 12
- 15
- 54
BD/BA = DE/AC → 6/10 = 9/AC → AC = 15.
A dilation with scale factor 2.5 maps segment PQ of length 6 onto P'Q'. The length of P'Q' is
- 2.4
- 8.5
- 15
- 20
Lengths multiply by k: 6 × 2.5 = 15.
In △ABC, DE ∥ AC. BD = 4, DA = 2, BE = 6, EC = x. Find x.
- 2
- 3
- 4
- 12
Side-Splitter: BD/DA = BE/EC → 4/2 = 6/x → x = 3.
The two right triangles are similar, with corresponding sides as labeled. Find x.
- 10.5
- 12
- 13.5
- 17
6/9 = 8/x → 6x = 72 → x = 12.
Two similar triangles have areas in the ratio 9 : 25. If the smaller perimeter is 27, the larger perimeter is
- 45
- 75
- 33
- 135
Side ratio = √(9:25) = 3:5 → perimeter 27 × 5/3 = 45.
The altitude to the hypotenuse splits it into segments of 8 and 18. The altitude measures
- 12
- 13
- √26
- 144
h = √(8·18) = √144 = 12.
DE ∥ AC with BD = 5, DA = 7, BE = 10. Then EC =
- 12
- 14
- 17.5
- 24
5/7 = 10/EC → 5·EC = 70 → EC = 14.
Two similar triangles have shortest sides 6 and 15. If the longer triangle's longest side is 20, the smaller triangle's longest side is
- 8
- 9
- 10
- 12
k = 15/6 = 2.5 → 20 ÷ 2.5 = 8.
Two similar solids have volumes in the ratio 27 : 64. Their surface areas are in the ratio
- 3 : 4
- 9 : 16
- 27 : 64
- 81 : 256
Side ratio = ∛(27:64) = 3:4 → area ratio = 9:16.
In a right triangle, a leg measures 12 and the whole hypotenuse measures 16. The hypotenuse segment adjacent to that leg is
- 8
- 9
- 10
- 12
leg² = (adjacent segment)(hypotenuse): 144 = 16s → s = 9.
A figure is dilated by scale factor 5. Its area is multiplied by
- 5
- 125
- 10
- 25
Area scales by k² = 25.
△ABC ∼ △DEF. AB = 4, DE = 10, BC = 6. Then EF =
- 2.4
- 15
- 17
- 12
4/10 = 6/EF → EF = 15.
△ABC ∼ △DEF. AB = 8, DE = 12, BC = 10. Then EF =
- 15
- 14
- 17
- 6.66667
8/12 = 10/EF → EF = 15.
△ABC ∼ △DEF. AB = 6, DE = 8, BC = 9. Then EF =
- 11
- 6.75
- 12
- 14
6/8 = 9/EF → EF = 12.
△ABC ∼ △DEF. AB = 3, DE = 9, BC = 5. Then EF =
- 11
- 1.66667
- 17
- 15
3/9 = 5/EF → EF = 15.
A figure is dilated by scale factor 6. Its area is multiplied by
- 36
- 12
- 216
- 6
Area scales by k² = 36.
△ABC ∼ △DEF. AB = 5, DE = 10, BC = 7. Then EF =
- 12
- 3.5
- 16
- 14
5/10 = 7/EF → EF = 14.
A figure is dilated by scale factor 3. Its area is multiplied by
- 3
- 6
- 27
- 9
Area scales by k² = 9.
△ABC ∼ △DEF. AB = 4, DE = 6, BC = 10. Then EF =
- 6.66667
- 17
- 15
- 12
4/6 = 10/EF → EF = 15.
A figure is dilated by scale factor 4. Its area is multiplied by
- 16
- 64
- 8
- 4
Area scales by k² = 16.
A figure is dilated by scale factor 10. Its area is multiplied by
- 1000
- 20
- 100
- 10
Area scales by k² = 100.

Unit 6: Right Triangles & Trigonometry (48)

A right triangle has a hypotenuse of 10 and an angle of 30°. The side OPPOSITE the 30° angle is:
- 5
- 5√3
- 10
- 20
sin30° = opp/10 → opp = 10 × 0.5 = 5.
sin(30°) is equal to:
- sin(60°)
- cos(30°)
- cos(60°)
- tan(60°)
Co-function: sin(x) = cos(90 − x), so sin30° = cos60°.
In a 45-45-90 triangle with leg length 5, the hypotenuse is:
- 5
- 10
- 5√2
- 5√3
In a 45-45-90 triangle, hypotenuse = leg · √2 = 5√2.
In a 45-45-90 triangle, the hypotenuse is 10. Each leg is
- 5
- 10
- 5√2
- 10√2
hypotenuse = leg·√2 → leg = 10/√2 = 5√2.
cos(40°) is equal to
- sin(40°)
- sin(50°)
- cos(50°)
- tan(50°)
Co-function: cos(x) = sin(90 − x), so cos40° = sin50°.
If sin(2x)° = cos(40)°, then x =
- 20
- 25
- 30
- 50
Cofunctions: 2x + 40 = 90 → 2x = 50 → x = 25.
In a right triangle, an acute angle is 40° and the adjacent leg is 10. The opposite leg is about
- 6.4
- 7.7
- 8.4
- 11.9
tan(40°) = opp/10 → opp = 10·tan40° ≈ 8.4.
A triangle has sides 10 and 12 with a 30° included angle. Its area is
- 30
- 60
- 30√3
- 120
Area = ½·a·b·sin(C) = ½·10·12·sin30° = ½·120·0.5 = 30.
In a 30-60-90 triangle the hypotenuse is 12. The longer leg is
- 6
- 6√2
- 6√3
- 12√3
Short leg = ½·12 = 6; longer leg = 6√3 (ratio 1 : √3 : 2).
Which expression equals cos(40°)?
- sin(40°)
- sin(50°)
- cos(50°)
- tan(50°)
Cofunction: cos(x) = sin(90 − x), so cos40° = sin50°.
In a 30-60-90 triangle, the shorter leg is 5. The hypotenuse is
- 5√3
- 10
- 5√2
- 15
Hypotenuse = 2 × shorter leg = 10.
A right triangle has the side opposite θ = 3 and the adjacent side = 4. Then θ ≈
- 30°
- 37°
- 45°
- 53°
tan θ = 3/4 → θ = tan⁻¹(0.75) ≈ 37°.
A 20-foot ladder makes a 65° angle with the ground. How high up the wall does it reach, to the nearest tenth?
- 8.5 ft
- 18.1 ft
- 9.3 ft
- 21.9 ft
Height is opposite the 65° angle: h = 20·sin65° ≈ 20(0.9063) ≈ 18.1 ft.
From a point 120 ft from a building's base, the angle of elevation to the top is 32°. The building's height, to the nearest foot, is
- 64 ft
- 75 ft
- 102 ft
- 192 ft
tan32° = h/120 → h = 120·tan32° ≈ 75 ft.
A right triangle has the side opposite θ = 7 and the side adjacent = 10. Find θ to the nearest degree.
- 32°
- 35°
- 41°
- 55°
θ = tan⁻¹(7/10) = tan⁻¹(0.7) ≈ 35°.
In a 45-45-90 triangle the hypotenuse measures 14. Each leg measures
- 7
- 7√2
- 14√2
- 7√3
leg = hyp ÷ √2 = 14/√2 = 7√2.
In a 30-60-90 triangle the LONGER leg measures 9. The hypotenuse measures
- 18
- 6√3
- 9√3
- 4.5
Short leg = 9/√3 = 3√3; hypotenuse = 2(3√3) = 6√3.
If sin(3x + 10)° = cos(2x + 20)°, then x =
- 6
- 12
- 14
- 30
Cofunctions sum to 90: (3x+10) + (2x+20) = 90 → 5x = 60 → x = 12.
A right triangle has hypotenuse 25 and one leg 15. The sine of the angle opposite the OTHER leg is
- 3/5
- 4/5
- 3/4
- 5/4
Other leg = √(625−225) = 20. sin = opposite/hypotenuse = 20/25 = 4/5.
From the top of a 90-ft lighthouse, the angle of depression to a boat is 28°. The horizontal distance to the boat, to the nearest foot, is
- 48 ft
- 79 ft
- 169 ft
- 192 ft
tan28° = 90/d → d = 90/tan28° ≈ 90/0.5317 ≈ 169 ft.
Find the missing leg x of the right triangle.
- 8
- 10
- 12
- √194
x² + 5² = 13² → x² = 169 − 25 = 144 → x = 12. (5-12-13 triple.)
In the right triangle, the hypotenuse is 20 and the marked angle is 35°. Find x to the nearest tenth.
- 11.5
- 14.0
- 16.4
- 28.6
x is opposite the 35° angle: x = 20·sin35° ≈ 20(0.5736) ≈ 11.5.
If sin(4x − 2)° = cos(3x + 8)°, then x =
- 10
- 12
- 14
- 18
Cofunctions: (4x−2)+(3x+8) = 90 → 7x + 6 = 90 → x = 12.
A kite string is 100 m long and makes a 42° angle with the ground. The kite's height, to the nearest tenth, is
- 66.9 m
- 74.3 m
- 90.1 m
- 111.1 m
h = 100·sin42° ≈ 100(0.6691) ≈ 66.9 m.
A wheelchair ramp rises 3 ft over a horizontal run of 20 ft. The angle it makes with the ground, to the nearest tenth of a degree, is
- 8.5°
- 9.6°
- 81.5°
- 6.1°
θ = tan⁻¹(3/20) = tan⁻¹(0.15) ≈ 8.5°.
In a 30-60-90 triangle the hypotenuse is 18. The longer leg measures
- 9
- 9√2
- 9√3
- 18√3
Short leg = 9; longer leg = 9√3.
A right triangle has a leg of 9 and hypotenuse 15. The cosine of the angle opposite the 9-side is
- 3/5
- 4/5
- 3/4
- 5/3
Other leg = 12. The angle opposite 9 has adjacent 12 → cos = 12/15 = 4/5.
From the top of a 150-ft tower, the angle of depression to a car is 22°. The horizontal distance to the car, to the nearest foot, is
- 56 ft
- 162 ft
- 371 ft
- 405 ft
d = 150/tan22° ≈ 150/0.4040 ≈ 371 ft.
A right triangle has legs 20 and 21. The hypotenuse is
- 41
- 6.4
- 30
- 29
√(20²+21²) = 29.
In a right triangle the side opposite θ is 11 and the adjacent side is 6. θ to the nearest degree is
- 61°
- 29°
- 52°
- 69°
θ = tan⁻¹(11/6) ≈ 61°.
A right triangle has legs 3 and 4. The hypotenuse is
- 2.65
- 7
- 6
- 5
√(3²+4²) = 5.
A right triangle has hypotenuse 13 and one leg 5. The other leg is
- 8
- 14
- 13.93
- 12
√(13²−5²) = 12.
A right triangle has hypotenuse 15 and one leg 9. The other leg is
- 6
- 12
- 14
- 17.49
√(15²−9²) = 12.
A right triangle has legs 5 and 12. The hypotenuse is
- 17
- 14
- 13
- 10.91
√(5²+12²) = 13.
A right triangle has legs 8 and 15. The hypotenuse is
- 12.69
- 23
- 18
- 17
√(8²+15²) = 17.
A right triangle has legs 7 and 24. The hypotenuse is
- 25
- 31
- 26
- 22.96
√(7²+24²) = 25.
In a right triangle the side opposite θ is 7 and the adjacent side is 10. θ to the nearest degree is
- 55°
- 35°
- 26°
- 43°
θ = tan⁻¹(7/10) ≈ 35°.
A right triangle has hypotenuse 17 and one leg 8. The other leg is
- 18.79
- 17
- 9
- 15
√(17²−8²) = 15.
In a right triangle the side opposite θ is 9 and the adjacent side is 12. θ to the nearest degree is
- 28°
- 37°
- 53°
- 45°
θ = tan⁻¹(9/12) ≈ 37°.
A right triangle has hypotenuse 25 and one leg 7. The other leg is
- 24
- 18
- 25.96
- 26
√(25²−7²) = 24.
In a right triangle the side opposite θ is 3 and the adjacent side is 8. θ to the nearest degree is
- 21°
- 69°
- 12°
- 29°
θ = tan⁻¹(3/8) ≈ 21°.
A right triangle has hypotenuse 29 and one leg 20. The other leg is
- 23
- 9
- 35.23
- 21
√(29²−20²) = 21.
A right triangle has legs 9 and 12. The hypotenuse is
- 7.94
- 16
- 21
- 15
√(9²+12²) = 15.
A right triangle has hypotenuse 5 and one leg 3. The other leg is
- 5.83
- 4
- 6
- 2
√(5²−3²) = 4.
A right triangle has legs 6 and 8. The hypotenuse is
- 5.29
- 10
- 14
- 11
√(6²+8²) = 10.
In a right triangle the side opposite θ is 8 and the adjacent side is 5. θ to the nearest degree is
- 49°
- 58°
- 32°
- 66°
θ = tan⁻¹(8/5) ≈ 58°.
A right triangle has hypotenuse 10 and one leg 6. The other leg is
- 4
- 8
- 11.66
- 10
√(10²−6²) = 8.
In a right triangle the side opposite θ is 5 and the adjacent side is 12. θ to the nearest degree is
- 31°
- 14°
- 67°
- 23°
θ = tan⁻¹(5/12) ≈ 23°.

Unit 7: Polygons & Quadrilaterals (39)

The sum of the interior angles of a pentagon (5 sides) is:
- 360°
- 540°
- 720°
- 900°
(n − 2)·180 = (5 − 2)·180 = 540°.
The sum of the exterior angles of ANY polygon is:
- 180°
- 360°
- depends on sides
- 540°
Exterior angles of any polygon always total 360°.
Each interior angle of a regular hexagon (6 sides) measures:
- 108°
- 120°
- 135°
- 144°
(6−2)·180 ÷ 6 = 720 ÷ 6 = 120°.
Which quadrilateral always has perpendicular diagonals?
- rectangle
- rhombus
- trapezoid
- isosceles trapezoid
A rhombus has diagonals that are perpendicular (and bisect the angles).
A regular polygon has each exterior angle equal to 40°. How many sides does it have?
- 6
- 8
- 9
- 10
Number of sides = 360 ÷ exterior angle = 360 ÷ 40 = 9.
The sum of the interior angles of a decagon (10 sides) is
- 1260°
- 1440°
- 1620°
- 1800°
(10 − 2)·180 = 8·180 = 1440°.
Which is always true of a rectangle but NOT always of a rhombus?
- diagonals bisect each other
- opposite sides parallel
- diagonals are congruent
- it is a parallelogram
Rectangles have congruent diagonals; rhombus diagonals are usually unequal.
A trapezoid has bases of length 8 and 14. The length of its midsegment (median) is
- 6
- 11
- 22
- 21
Midsegment = ½(b₁ + b₂) = ½(8 + 14) = 11.
The sum of the interior angles of a heptagon (7 sides) is
- 720°
- 900°
- 1080°
- 1260°
(7 − 2)·180 = 900°.
A regular polygon has each exterior angle equal to 36°. It has
- 8 sides
- 9 sides
- 10 sides
- 12 sides
360 ÷ 36 = 10 sides.
Each interior angle of a regular pentagon measures
- 72°
- 108°
- 120°
- 135°
(5−2)·180 ÷ 5 = 540 ÷ 5 = 108°.
Consecutive angles of a parallelogram are always
- congruent
- complementary
- supplementary
- right angles
Consecutive angles of a parallelogram are supplementary.
Each interior angle of a regular polygon measures 156°. The number of sides is
- 12
- 13
- 15
- 18
Exterior = 180 − 156 = 24° → n = 360/24 = 15.
In parallelogram ABCD, m∠A = (3x + 12)° and m∠C = (5x − 26)°. Find m∠A.
- 19°
- 57°
- 69°
- 111°
Opposite angles equal: 3x + 12 = 5x − 26 → x = 19 → m∠A = 69°.
Consecutive angles of a parallelogram measure (2x + 40)° and (3x − 10)°. The larger angle measures
- 80°
- 90°
- 100°
- 110°
Supplementary: 5x + 30 = 180 → x = 30 → angles 100° and 80°. Larger = 100°.
The diagonals of a rhombus measure 12 and 16. The length of one side of the rhombus is
- 10
- 14
- 20
- 28
Diagonals bisect at right angles: side = √(6² + 8²) = 10.
In an isosceles trapezoid, a lower base angle measures 70°. An upper base angle measures
- 70°
- 90°
- 110°
- 140°
Same-side angles between the parallel bases are supplementary: 180 − 70 = 110°.
The interior angles of a polygon sum to 1620°. The polygon has how many sides?
- 9
- 10
- 11
- 12
(n − 2)·180 = 1620 → n − 2 = 9 → n = 11.
In rectangle ABCD, diagonal AC = 3x + 8 and diagonal BD = 5x − 12. The length of each diagonal is
- 10
- 28
- 38
- 46
Rectangle diagonals are congruent: 3x + 8 = 5x − 12 → x = 10 → AC = 38.
A trapezoid's midsegment measures 17 and one base measures 12. The other base measures
- 5
- 22
- 29
- 34
17 = ½(12 + b) → 34 = 12 + b → b = 22.
In the parallelogram, find x.
- 65
- 105
- 115
- 125
Consecutive angles of a parallelogram are supplementary: x = 180 − 65 = 115.
The figure is a regular hexagon. Find the measure of interior angle x.
- 108
- 120
- 135
- 144
Each interior angle of a regular hexagon = (6−2)·180 ÷ 6 = 120°.
Each interior angle of a regular 20-gon measures
- 144°
- 150°
- 162°
- 168°
(20−2)·180/20 = 3240/20 = 162°.
Parallelogram diagonals meet at E. If AE = 3x − 4 and EC = x + 12, diagonal AC =
- 8
- 20
- 40
- 32
Diagonals bisect: 3x − 4 = x + 12 → x = 8 → AE = 20 → AC = 40.
A rhombus has perimeter 52 and one diagonal of length 24. The other diagonal measures
- 5
- 10
- 13
- 26
Side = 13; half-diagonals 12 and √(13²−12²) = 5 → other diagonal = 10.
In an isosceles trapezoid, the diagonals measure (5x − 9) and (3x + 7). Each diagonal is
- 8
- 16
- 31
- 23
Diagonals congruent: 5x − 9 = 3x + 7 → x = 8 → 31.
Each exterior angle of a regular polygon measures 18°. The polygon has
- 10 sides
- 18 sides
- 20 sides
- 24 sides
n = 360/18 = 20 sides.
A square has a diagonal of length 8√2. Its area is
- 32
- 64
- 128
- 16√2
Diagonal = s√2 → s = 8 → area = 64.
Each exterior angle of a regular polygon measures 40°. The polygon has
- 18 sides
- 9 sides
- 8 sides
- 11 sides
n = 360/40 = 9.
The interior angles of a regular 6-gon each measure
- 150°
- 720°
- 120°
- 60°
(n−2)·180/n = 120°.
The interior angles of a regular 5-gon each measure
- 540°
- 162°
- 108°
- 72°
(n−2)·180/n = 108°.
Each exterior angle of a regular polygon measures 30°. The polygon has
- 11 sides
- 12 sides
- 24 sides
- 14 sides
n = 360/30 = 12.
The interior angles of a regular 12-gon each measure
- 150°
- 120°
- 1800°
- 30°
(n−2)·180/n = 150°.
Each exterior angle of a regular polygon measures 72°. The polygon has
- 4 sides
- 5 sides
- 10 sides
- 7 sides
n = 360/72 = 5.
The interior angles of a regular 9-gon each measure
- 40°
- 1260°
- 130°
- 140°
(n−2)·180/n = 140°.
Each exterior angle of a regular polygon measures 60°. The polygon has
- 12 sides
- 6 sides
- 8 sides
- 5 sides
n = 360/60 = 6.
Each exterior angle of a regular polygon measures 36°. The polygon has
- 10 sides
- 12 sides
- 20 sides
- 9 sides
n = 360/36 = 10.
The interior angles of a regular 10-gon each measure
- 126°
- 36°
- 144°
- 1440°
(n−2)·180/n = 144°.
Each exterior angle of a regular polygon measures 45°. The polygon has
- 16 sides
- 8 sides
- 7 sides
- 10 sides
n = 360/45 = 8.

Unit 8: Circles (39)

The equation (x − 2)² + (y + 3)² = 16 describes a circle with center and radius:
- center (2, −3), r = 16
- center (−2, 3), r = 4
- center (2, −3), r = 4
- center (2, 3), r = 4
Center is (h, k) = (2, −3) (signs flip), and r = √16 = 4.
Two chords intersect inside a circle, cutting arcs of 60° and 80°. The angle formed is:
- 20°
- 70°
- 140°
- 10°
Inside angle = ½(sum of arcs) = ½(60 + 80) = 70°.
A 90° central angle in a circle of radius 6 cuts off an arc of length:
- 3π
- 6π
- 9π
- 12π
Arc length = (90/360)·2π(6) = ¼·12π = 3π.
A tangent and a radius drawn to the point of tangency form an angle of
- 45°
- 60°
- 90°
- 180°
A tangent is perpendicular to the radius at the point of tangency → 90°.
The circle (x + 1)² + (y − 5)² = 9 has center and radius
- (1, −5), r = 9
- (−1, 5), r = 3
- (1, 5), r = 3
- (−1, −5), r = 9
Center (h, k) = (−1, 5) (signs flip); r = √9 = 3.
A circle has radius 9. The length of an arc with a central angle of 80° is
- 2π
- 4π
- 8π
- 18π
(80/360)·2π(9) = (2/9)(18π) = 4π.
The center of the circle x² + y² + 6x − 4y − 12 = 0 is
- (3, −2)
- (−3, 2)
- (6, −4)
- (−6, 4)
Complete the square: (x+3)² + (y−2)² = 25 → center (−3, 2), radius 5.
A quadrilateral is inscribed in a circle. If one angle is 95°, its opposite angle is
- 85°
- 95°
- 105°
- 185°
Opposite angles of a cyclic quadrilateral are supplementary: 180 − 95 = 85°.
From an external point, a tangent of length 6 and a secant are drawn. If the external part of the secant is 4, the whole secant length is
- 6
- 8
- 9
- 12
Tangent² = (whole)(external): 6² = 4·whole → whole = 36/4 = 9.
Two secants from an external point have (whole)(external) products. If one secant gives 8 and 3, and the other has whole length 6 and external part x, then x =
- 2
- 4
- 6
- 9
(8)(3) = (6)(x) → 24 = 6x → x = 4.
The circle (x − 5)² + (y + 2)² = 49 has center and radius
- (−5, 2), r = 49
- (5, −2), r = 7
- (−5, 2), r = 7
- (5, −2), r = 49
Center (5, −2); r = √49 = 7.
In a circle of radius 12, an arc with a 30° central angle has length
- π
- 2π
- 4π
- 6π
(30/360)·2π(12) = (1/12)(24π) = 2π.
Two chords intersect inside a circle. One chord is split into 6 and 8; the other into 4 and x. Find x.
- 10
- 12
- 14
- 48
(6)(8) = (4)(x) → 48 = 4x → x = 12.
An inscribed angle measures 48°. A CENTRAL angle intercepting the same arc measures
- 24°
- 48°
- 96°
- 132°
The arc = 2(48) = 96°, and the central angle equals its arc → 96°.
A sector of a circle with radius 9 has a central angle of 80°. The area of the sector is
- 9π
- 18π
- 36π
- 81π
(80/360)·π(9²) = (2/9)(81π) = 18π.
A tangent of length 12 and a secant are drawn from the same external point. The secant's external segment is 8. The part of the secant INSIDE the circle measures
- 6
- 10
- 18
- 144
tangent² = (whole)(external): 144 = 8w → w = 18 → inside part = 18 − 8 = 10.
The circle x² + y² − 8x + 4y + 4 = 0 has radius
- 2
- 4
- 16
- √20
(x−4)² + (y+2)² = −4 + 16 + 4 = 16 → r = 4.
Two tangents from an external point form a 40° angle. If the NEAR arc measures 140°, the FAR arc measures
- 100°
- 180°
- 220°
- 320°
The two arcs total 360°: far = 360 − 140 = 220°. Check: ½(220 − 140) = 40° ✓.
A quadrilateral is inscribed in a circle. If ∠A = (2x + 10)° and its opposite angle ∠C = (3x + 20)°, find m∠A.
- 30°
- 70°
- 110°
- 150°
Opposite angles are supplementary: 5x + 30 = 180 → x = 30 → ∠A = 70°.
A chord of length 24 is drawn in a circle of radius 13. The distance from the center to the chord is
- 1
- 5
- 12
- √313
Half-chord = 12; distance = √(13² − 12²) = √25 = 5.
The inscribed angle x intercepts a 100° arc. Find x.
- 50
- 100
- 200
- 80
Inscribed angle = ½ its intercepted arc = ½(100) = 50°.
Two chords intersect inside the circle with segments as labeled. Find x.
- 6
- 7
- 9
- 24
(3)(8) = (4)(x) → 24 = 4x → x = 6.
A circle has radius 9. An arc intercepted by a 120° central angle has length
- 3π
- 6π
- 27π
- 12π
(120/360)·2π(9) = ⅓·18π = 6π.
A tangent and a chord meet at the point of tangency forming a 62° angle. The intercepted arc measures
- 31°
- 62°
- 124°
- 118°
Tangent-chord angle = ½ the arc → arc = 2(62) = 124°.
Which is the equation of a circle with center (−4, 3) and radius 6?
- (x−4)²+(y+3)²=6
- (x+4)²+(y−3)²=36
- (x+4)²+(y−3)²=6
- (x−4)²+(y+3)²=36
(x−h)²+(y−k)²=r² with h=−4, k=3, r²=36.
The circle x² + y² − 10x + 6y + 18 = 0 has center and radius
- (5,−3), r=4
- (−5,3), r=4
- (5,−3), r=16
- (−5,3), r=√18
(x−5)² + (y+3)² = −18+25+9 = 16 → center (5,−3), r = 4.
Two secants from an external point intercept arcs of 200° and 80°. The angle between the secants is
- 40°
- 60°
- 120°
- 140°
½(200 − 80) = 60°.
A chord of length 30 is 8 units from the center of a circle. The radius is
- 15
- 16
- 17
- 23
r = √(15² + 8²) = √289 = 17.
An inscribed angle intercepts an arc of 126°. The angle measures
- 252°
- 63°
- 54°
- 126°
Inscribed = ½ arc = 63°.
An inscribed angle intercepts an arc of 88°. The angle measures
- 176°
- 44°
- 92°
- 88°
Inscribed = ½ arc = 44°.
An inscribed angle intercepts an arc of 148°. The angle measures
- 296°
- 32°
- 148°
- 74°
Inscribed = ½ arc = 74°.
The circle (x − 2)² + (y + 3)² = 16 has radius
- 3
- 6
- 16
- 4
r = √16 = 4.
An inscribed angle intercepts an arc of 104°. The angle measures
- 208°
- 52°
- 104°
- 76°
Inscribed = ½ arc = 52°.
An inscribed angle intercepts an arc of 72°. The angle measures
- 72°
- 36°
- 108°
- 144°
Inscribed = ½ arc = 36°.
The circle (x + 5)² + (y + 2)² = 49 has radius
- 9
- 7
- 49
- 6
r = √49 = 7.
The circle (x − 0)² + (y − 6)² = 9 has radius
- 2
- 9
- 5
- 3
r = √9 = 3.
The circle (x − 4)² + (y + 1)² = 81 has radius
- 9
- 11
- 81
- 8
r = √81 = 9.
The circle (x + 1)² + (y − 5)² = 36 has radius
- 36
- 5
- 8
- 6
r = √36 = 6.
An inscribed angle intercepts an arc of 56°. The angle measures
- 56°
- 112°
- 28°
- 124°
Inscribed = ½ arc = 28°.

Unit 9: Coordinate Geometry (40)

The midpoint of the segment from (2, 4) to (6, 8) is:
- (4, 6)
- (8, 12)
- (2, 2)
- (4, 4)
Average the coordinates: ((2+6)/2, (4+8)/2) = (4, 6).
The slope of the line through (1, 2) and (3, 8) is:
- 2
- 3
- 1/3
- 6
m = (8 − 2)/(3 − 1) = 6/2 = 3.
A line has slope 2. A line PERPENDICULAR to it has slope:
- 2
- −2
- 1/2
- −1/2
Perpendicular slope is the negative reciprocal of 2, which is −1/2.
The distance between (1, 1) and (4, 5) is:
- 4
- 5
- 7
- √7
d = √(3² + 4²) = √25 = 5.
The midpoint of (−2, 3) and (4, 7) is:
- (1, 5)
- (2, 10)
- (3, 2)
- (6, 4)
((−2+4)/2, (3+7)/2) = (1, 5).
A line perpendicular to a line with slope −2/3 has slope
- −2/3
- 2/3
- 3/2
- −3/2
Negative reciprocal of −2/3 is +3/2.
The midpoint of A(−4, 2) and B(6, −8) is
- (1, −3)
- (2, −6)
- (1, 3)
- (5, −5)
((−4+6)/2, (2−8)/2) = (1, −3).
Point P divides the segment from A(0, 0) to B(10, 5) in the ratio 3:2. P is
- (4, 2)
- (6, 3)
- (5, 2.5)
- (3, 2)
Move 3/5 of the way: (0.6·10, 0.6·5) = (6, 3).
A line perpendicular to a line of slope 5 has slope
- 5
- −5
- 1/5
- −1/5
Negative reciprocal of 5 is −1/5.
Which is the equation of the line through (2, 3) perpendicular to y = −2x + 1?
- y − 3 = ½(x − 2)
- y − 3 = −2(x − 2)
- y − 3 = 2(x − 2)
- y − 3 = −½(x − 2)
Perpendicular slope = ½ (negative reciprocal of −2); point-slope through (2,3): y − 3 = ½(x − 2).
The slope of the line through (2, 1) and (5, 10) is
- 1/3
- 3
- 9
- 6
(10 − 1)/(5 − 2) = 9/3 = 3.
A line perpendicular to a line with slope 1/4 has slope
- 1/4
- −1/4
- 4
- −4
Negative reciprocal of 1/4 is −4.
Point P divides the segment from A(−4, 2) to B(6, 7) in the ratio 2 : 3. The coordinates of P are
- (0, 4)
- (2, 5)
- (1, 4.5)
- (−2, 3)
Move 2/5 of the way: x = −4 + 0.4(10) = 0; y = 2 + 0.4(5) = 4 → (0, 4).
Which is an equation of the perpendicular bisector of the segment with endpoints (2, 4) and (6, 8)?
- y − 6 = −(x − 4)
- y − 6 = x − 4
- y − 4 = −(x − 2)
- y = x + 2
Midpoint (4, 6); segment slope = 1 → perpendicular slope = −1: y − 6 = −(x − 4).
A triangle has vertices (0, 0), (4, 0), and (2, 6). This triangle is
- scalene
- isosceles
- equilateral
- right
Sides: 4, √(4+36) = √40, √40. Two equal sides → isosceles.
Which is an equation of the line through (−2, 5) parallel to y = −3x + 1?
- y = −3x − 1
- y = −3x + 5
- y = ⅓x + 5
- y = 3x + 11
Keep slope −3: y − 5 = −3(x + 2) → y = −3x − 1.
The distance between (−3, 4) and (5, −2) is
- 8
- 10
- 12
- 14
√(8² + (−6)²) = √100 = 10.
To prove a quadrilateral is a parallelogram using slopes, you must show
- all four slopes are equal
- both pairs of opposite sides have equal slopes
- the diagonals have equal slopes
- adjacent sides have opposite slopes
Equal slopes on both pairs of opposite sides prove both pairs parallel → parallelogram.
A circle has center (3, −1) and passes through (7, 2). Its equation is
- (x−3)² + (y+1)² = 5
- (x−3)² + (y+1)² = 25
- (x+3)² + (y−1)² = 25
- (x−7)² + (y−2)² = 25
r = √(4² + 3²) = 5 → (x−3)² + (y+1)² = 25.
M(1, 2) is the midpoint of AB with A(−3, 5). The coordinates of B are
- (−1, 3.5)
- (5, −1)
- (4, −3)
- (2, 4)
B = (2·1 − (−3), 2·2 − 5) = (5, −1).
Points A(1, 1) and B(7, 9) are graphed. What is the length of segment AB?
- 8
- 10
- 14
- √28
d = √((7−1)² + (9−1)²) = √(36 + 64) = √100 = 10.
A line passes through (0, 1) and (4, 4). What is its slope?
- 3/4
- 4/3
- 3
- −3/4
m = (4 − 1)/(4 − 0) = 3/4.
Point P divides the segment from R(−3, 5) to Z(9, −3) in the ratio 3 : 1. P is
- (0, 3)
- (3, 1)
- (6, −1)
- (−6, 1)
¾ of the way: x = −3 + ¾(12) = 6; y = 5 + ¾(−8) = −1 → (6, −1).
Which is an equation of the line through (4, −1) perpendicular to y = ½x + 3?
- y + 1 = 2(x − 4)
- y + 1 = −2(x − 4)
- y + 1 = ½(x − 4)
- y − 1 = −2(x + 4)
Perpendicular slope = −2: y + 1 = −2(x − 4).
To prove a quadrilateral is a trapezoid using coordinates, it is sufficient to show
- all sides congruent
- exactly one pair of opposite sides with equal slopes
- diagonals bisect each other
- two pairs of parallel sides
A trapezoid needs one pair of parallel sides — shown by equal slopes (and the other pair not parallel).
The distance from (−2, −1) to (10, 4) is
- 12
- 13
- 17
- √119
√(12² + 5²) = √169 = 13.
The line y = ½x + 2 is dilated by scale factor 4 centered at the origin. The image is
- y = 2x + 2
- y = ½x + 8
- y = ½x + 2
- y = 2x + 8
Slope unchanged; the y-intercept scales: 2 × 4 = 8 → y = ½x + 8.
The midpoint of the segment joining (−2, 7) and (6, −3) is
- (2, 2)
- (4, 5)
- (2, 5)
- (4, 2)
((−2+6)/2, (7−3)/2) = (2, 2).
The distance between (-3, 2) and (3, 10) is
- 12
- 14
- 5
- 10
√(6² + 8²) = 10.
The midpoint of (5, 0) and (9, 8) is
- (7, 4)
- (4, 7)
- (8, 4)
- (4, 8)
Average the x's and the y's.
The midpoint of (0, -5) and (6, 1) is
- (4, -2)
- (-2, 3)
- (6, 6)
- (3, -2)
Average the x's and the y's.
The midpoint of (-2, 3) and (4, 7) is
- (6, 4)
- (5, 1)
- (1, 5)
- (2, 5)
Average the x's and the y's.
The distance between (1, -2) and (9, 4) is
- 5
- 10
- 12
- 14
√(8² + 6²) = 10.
The distance between (0, 3) and (8, 9) is
- 14
- 12
- 5
- 10
√(8² + 6²) = 10.
The distance between (0, 0) and (6, 8) is
- 5
- 14
- 12
- 10
√(6² + 8²) = 10.
The midpoint of (3, 3) and (9, 11) is
- (7, 6)
- (7, 7)
- (6, 7)
- (6, 8)
Average the x's and the y's.
The distance between (2, 1) and (7, 13) is
- 17
- 6.5
- 15
- 13
√(5² + 12²) = 13.
The midpoint of (2, 4) and (6, 8) is
- (6, 4)
- (5, 6)
- (4, 6)
- (4, 4)
Average the x's and the y's.
The distance between (-2, -1) and (4, 7) is
- 12
- 14
- 5
- 10
√(6² + 8²) = 10.
The midpoint of (-4, 2) and (2, -6) is
- (-2, -1)
- (0, -2)
- (6, -8)
- (-1, -2)
Average the x's and the y's.

Unit 10: Transformations & Constructions (42)

Rotating (2, 5) 180° about the origin gives:
- (5, 2)
- (−2, −5)
- (−5, 2)
- (2, −5)
180° rotation: (x, y) → (−x, −y) = (−2, −5).
Rotating (3, 1) 90° counterclockwise about the origin gives:
- (1, 3)
- (−1, 3)
- (1, −3)
- (3, −1)
90° CCW: (x, y) → (−y, x) = (−1, 3).
Reflecting (2, 7) over the line y = x gives:
- (7, 2)
- (−2, 7)
- (2, −7)
- (−7, −2)
Reflection over y = x swaps coordinates: (x, y) → (y, x) = (7, 2).
A rigid motion preserves all of the following EXCEPT:
- distance
- angle measure
- position
- area
Rigid motions preserve distance, angle, and area — but change the figure's position.
Reflecting (−2, 7) over the line y = x gives
- (7, −2)
- (2, −7)
- (−7, 2)
- (−2, −7)
Reflection over y = x swaps coordinates: (−2, 7) → (7, −2).
A translation maps (1, 1) to (4, −3). The rule is
- (x+3, y−4)
- (x+4, y−3)
- (x−3, y+4)
- (x+3, y+4)
x: 1→4 is +3; y: 1→−3 is −4. Rule (x+3, y−4).
What is the smallest angle of rotation that maps a regular octagon onto itself?
- 30°
- 36°
- 45°
- 60°
360 ÷ 8 = 45°.
A regular pentagon carries onto itself after a rotation of
- 54°
- 60°
- 72°
- 108°
360 ÷ 5 = 72°.
The line y = 3x is dilated by scale factor 2 centered at the origin. The image is
- y = 6x
- y = 3x
- y = 3x + 2
- y = 6x + 2
The line passes through the center of dilation, so it maps onto itself: y = 3x.
Reflecting (3, −5) over the line y = x gives
- (−5, 3)
- (5, −3)
- (−3, 5)
- (−5, −3)
Reflection over y = x swaps coordinates: (3, −5) → (−5, 3).
Under the translation (x, y) → (x − 5, y + 2), the point (3, 3) maps to
- (−2, 5)
- (8, 1)
- (−2, 1)
- (8, 5)
(3 − 5, 3 + 2) = (−2, 5).
Rotating (5, 2) 90° counterclockwise about the origin gives
- (2, 5)
- (−2, 5)
- (2, −5)
- (−5, −2)
90° CCW: (x, y) → (−y, x) = (−2, 5).
The point (2, 3) is rotated 90° counterclockwise about the origin, then reflected over the x-axis. The final image is
- (−3, 2)
- (−3, −2)
- (3, −2)
- (−2, −3)
R90CCW: (2,3) → (−3, 2). Reflect over x-axis: (−3, 2) → (−3, −2).
The point (1, 4) is translated by (x + 3, y − 2), then rotated 180° about the origin. The final image is
- (4, 2)
- (−4, −2)
- (−4, 2)
- (4, −2)
Translate: (4, 2). Rotate 180°: (−4, −2).
The point (3, 1) is dilated by scale factor 2 about the origin, then translated by (x, y − 3). The final image is
- (6, 2)
- (6, −1)
- (5, −1)
- (6, 5)
Dilate: (6, 2). Translate down 3: (6, −1).
The point (4, 7) is reflected over the line y = x, then over the y-axis. The final image is
- (−4, 7)
- (7, −4)
- (−7, 4)
- (7, 4)
Over y = x: (7, 4). Over the y-axis: (−7, 4).
Rotating (−2, 5) by 270° counterclockwise about the origin gives
- (5, 2)
- (−5, −2)
- (2, −5)
- (−5, 2)
270° CCW: (x, y) → (y, −x) = (5, −(−2)) = (5, 2).
A dilation with scale factor ½ centered at the origin is applied to a segment of length 10. The image segment has length
- 5
- 10
- 20
- 2.5
Lengths multiply by k = ½: 10 × ½ = 5.
A glide reflection is the composition of a
- rotation and a dilation
- translation and a reflection
- reflection and a reflection
- translation and a rotation
A glide reflection = translation followed by (or with) a reflection over a parallel line.
Which transformation REVERSES the orientation of a figure's vertices?
- translation
- rotation
- reflection
- dilation
Reflections flip orientation (clockwise labels become counterclockwise). Translations, rotations, dilations preserve it.
△A'B'C' is the mirror image of △ABC across the vertical axis shown. Which single transformation maps △ABC onto △A'B'C'?
- rotation of 180°
- translation right
- reflection over the y-axis
- dilation
The image is flipped across the vertical (y) axis — a reflection over the y-axis.
Point P(3, 2) maps to P'(−2, 3). Which transformation occurred?
- reflection over y = x
- rotation 90° counterclockwise about the origin
- rotation 90° clockwise about the origin
- reflection over the y-axis
90° CCW rule: (x, y) → (−y, x): (3, 2) → (−2, 3) ✓.
The smallest rotation that carries a regular decagon onto itself is
- 10°
- 18°
- 36°
- 45°
360/10 = 36°.
The rule (x, y) → (y, −x) represents a rotation of
- 90° counterclockwise
- 90° clockwise
- 180°
- 270° clockwise
(y, −x) is 90° clockwise (equivalently 270° CCW).
P(5, −1) is reflected over the y-axis, then translated up 2. The final image is
- (−5, 1)
- (−5, −3)
- (5, 1)
- (−1, 5)
Reflect: (−5, −1). Up 2: (−5, 1).
Rotating (2, 3) 90° clockwise about the origin gives
- (−3, 2)
- (3, −2)
- (−2, −3)
- (3, 2)
90° CW: (x, y) → (y, −x) = (3, −2).
Which transformation preserves distance but NOT orientation?
- translation
- rotation
- reflection
- dilation
A reflection is rigid (distance-preserving) but flips orientation.
The line y = 2x + 4 is dilated by scale factor 3 centered at the origin. The image is
- y = 6x + 4
- y = 2x + 12
- y = 2x + 4
- y = 6x + 12
Slope stays 2; intercept triples: y = 2x + 12.
Reflecting (3, 2) over the line y = x gives
- (3, -2)
- (2, 3)
- (-2, -3)
- (-3, 2)
Over y = x: swap the coordinates.
Rotating (1, 7) 90° counterclockwise about the origin gives
- (7, 1)
- (-1, -7)
- (-7, 1)
- (7, -1)
90° CCW: (x, y) → (−y, x).
Reflecting (6, -4) over the line y = x gives
- (4, -6)
- (6, 4)
- (-4, 6)
- (-6, -4)
Over y = x: swap the coordinates.
Reflecting (-3, -2) over the line y = x gives
- (-3, 2)
- (-2, -3)
- (3, -2)
- (2, 3)
Over y = x: swap the coordinates.
Reflecting (5, 1) over the line y = x gives
- (-5, 1)
- (1, 5)
- (-1, -5)
- (5, -1)
Over y = x: swap the coordinates.
Rotating (-4, 5) 90° counterclockwise about the origin gives
- (5, 4)
- (5, -4)
- (4, -5)
- (-5, -4)
90° CCW: (x, y) → (−y, x).
Rotating (5, 1) 90° counterclockwise about the origin gives
- (1, -5)
- (-5, -1)
- (-1, 5)
- (1, 5)
90° CCW: (x, y) → (−y, x).
Rotating (6, -4) 90° counterclockwise about the origin gives
- (-6, 4)
- (-4, -6)
- (4, 6)
- (-4, 6)
90° CCW: (x, y) → (−y, x).
Reflecting (1, 7) over the line y = x gives
- (7, 1)
- (-1, 7)
- (-7, -1)
- (1, -7)
Over y = x: swap the coordinates.
Rotating (3, 2) 90° counterclockwise about the origin gives
- (-2, 3)
- (2, -3)
- (-3, -2)
- (2, 3)
90° CCW: (x, y) → (−y, x).
Reflecting (2, -6) over the line y = x gives
- (-2, -6)
- (2, 6)
- (-6, 2)
- (6, -2)
Over y = x: swap the coordinates.
Rotating (-3, -2) 90° counterclockwise about the origin gives
- (3, 2)
- (-2, 3)
- (2, -3)
- (-2, -3)
90° CCW: (x, y) → (−y, x).
Rotating (2, -6) 90° counterclockwise about the origin gives
- (-6, -2)
- (-6, 2)
- (-2, 6)
- (6, 2)
90° CCW: (x, y) → (−y, x).
Reflecting (-4, 5) over the line y = x gives
- (4, 5)
- (-5, 4)
- (-4, -5)
- (5, -4)
Over y = x: swap the coordinates.

Unit 11: 3D Solids: Surface Area, Volume & Density (37)

A cylinder has radius 3 and height 10. Its volume is:
- 30π
- 90π
- 60π
- 900π
V = πr²h = π(9)(10) = 90π.
A cone has radius 3 and height 10. Its volume is:
- 30π
- 90π
- 60π
- 10π
V = ⅓πr²h = ⅓π(9)(10) = 30π.
A sphere has radius 3. Its volume is:
- 9π
- 27π
- 36π
- 12π
V = 4⁄3 πr³ = 4⁄3 π(27) = 36π.
Rotating a rectangle about one of its sides sweeps out a:
- cone
- cylinder
- sphere
- pyramid
Spinning a rectangle around a side produces a cylinder.
A sphere has radius 3. Its surface area is
- 12π
- 27π
- 36π
- 9π
SA = 4πr² = 4π(9) = 36π.
Rotating a semicircle about its diameter produces a
- cone
- cylinder
- sphere
- hemisphere
Spinning a semicircle around its straight edge (diameter) sweeps out a full sphere.
A region has a population of 500,000 over 2,000 square miles. Its population density is
- 100 / mi²
- 250 / mi²
- 500 / mi²
- 2500 / mi²
500,000 ÷ 2,000 = 250 people per square mile.
A sphere has a volume of 36π. Its radius is
- 2
- 3
- 4
- 6
(4/3)πr³ = 36π → r³ = 27 → r = 3.
A cylinder has radius 2 and height 9. Its volume is
- 18π
- 36π
- 72π
- 144π
V = πr²h = π(4)(9) = 36π.
A cone has radius 6 and height 5. Its volume is
- 30π
- 60π
- 180π
- 90π
V = ⅓πr²h = ⅓π(36)(5) = 60π.
A sphere has radius 6. Its volume is
- 72π
- 144π
- 216π
- 288π
V = 4⁄3 πr³ = 4⁄3 π(216) = 288π.
A plane slices a sphere through any point. The cross-section is a
- square
- triangle
- circle
- oval
Every cross-section of a sphere is a circle.
A cylinder has volume 250π and radius 5. Its height is
- 5
- 10
- 25
- 50
250π = π(25)h → h = 10.
A cone has radius 9 and slant height 15. Its volume is
- 324π
- 405π
- 972π
- 108π
Height = √(15² − 9²) = 12. V = ⅓π(81)(12) = 324π.
A sphere has surface area 100π. Its volume is
- 500π/3
- 100π
- 125π
- 500π
4πr² = 100π → r = 5. V = 4/3 π(125) = 500π/3.
A circular pool has a 20-ft diameter and is 4 ft deep everywhere. Its volume, to the nearest cubic foot, is
- 251
- 1257
- 5027
- 80
V = π(10²)(4) = 400π ≈ 1257 ft³.
A metal block measures 8 cm × 5 cm × 4 cm and has a mass of 1248 g. Its density is
- 7.8 g/cm³
- 6.2 g/cm³
- 9.6 g/cm³
- 0.13 g/cm³
V = 160 cm³. Density = 1248/160 = 7.8 g/cm³.
A hemisphere has radius 6. Its volume is
- 72π
- 144π
- 288π
- 36π
Half a sphere: ½ · 4/3 π(216) = 144π.
A 5 × 3 rectangle is rotated about its LONGER side. The volume of the resulting solid is
- 15π
- 45π
- 75π
- 30π
Cylinder with height 5 (the axis) and radius 3: V = π(9)(5) = 45π.
A plane parallel to the base slices a square pyramid halfway up. The cross-section is a
- triangle
- trapezoid
- square
- rectangle larger than the base
Slices parallel to the base of a square pyramid are smaller squares.
Find the volume of the cylinder, in terms of π.
- 36π
- 72π
- 144π
- 288π
V = πr²h = π(16)(9) = 144π.
Find the volume of the cone, in terms of π.
- 48π
- 96π
- 288π
- 128π
V = ⅓πr²h = ⅓π(36)(8) = 96π.
A conical pile of gravel has radius and height both 4 ft. If a cart carries 6 ft³ per trip, the fewest trips to move the pile is
- 9
- 11
- 12
- 17
V = ⅓π(16)(4) ≈ 67.0 ft³ → 67.0/6 ≈ 11.2 → round UP to 12 trips.
A sphere has volume 288π. Its radius is
- 4
- 6
- 8
- 12
4/3 πr³ = 288π → r³ = 216 → r = 6.
A cylindrical rain barrel has diameter 4 ft and height 10 ft. Its volume, to the nearest tenth, is
- 40.0 ft³
- 125.7 ft³
- 251.3 ft³
- 502.7 ft³
r = 2 → V = π(4)(10) = 40π ≈ 125.7 ft³.
A solid is a 10×4×2 prism topped by a pyramid of height 6 on the same 10×4 base. Total volume ×2.7 g/cm³ density gives a mass of
- 216 g
- 432 g
- 648 g
- 160 g
Prism 80 + pyramid ⅓(40)(6)=80 → 160 cm³ × 2.7 = 432 g.
A plane slices a cylinder perpendicular to its bases, passing through the axis. The cross-section is a
- circle
- ellipse
- rectangle
- triangle
A vertical cut through the axis of a cylinder is a rectangle.
Gold has density 19.3 g/cm³. A bar of volume 25 cm³ has mass
- 44.3 g
- 482.5 g
- 772 g
- 0.77 g
mass = density × volume = 19.3 × 25 = 482.5 g.
A cylinder has radius 3 and height 7. Its volume is
- 42π
- 21π
- 63π
- 189π
V = πr²h = 63π.
A cone has radius 5 and height 3. Its volume is
- 15π
- 75π
- 25π
- 225π
V = ⅓πr²h = 25π.
A cone has radius 6 and height 5. Its volume is
- 30π
- 60π
- 180π
- 540π
V = ⅓πr²h = 60π.
A cylinder has radius 3 and height 11. Its volume is
- 99π
- 297π
- 33π
- 66π
V = πr²h = 99π.
A cylinder has radius 4 and height 5. Its volume is
- 240π
- 80π
- 20π
- 40π
V = πr²h = 80π.
A cylinder has radius 5 and height 6. Its volume is
- 150π
- 60π
- 450π
- 50π
V = πr²h = 150π.
A cone has radius 2 and height 12. Its volume is
- 16π
- 144π
- 24π
- 48π
V = ⅓πr²h = 16π.
A cone has radius 9 and height 4. Its volume is
- 324π
- 972π
- 36π
- 108π
V = ⅓πr²h = 108π.
A cone has radius 4 and height 9. Its volume is
- 432π
- 36π
- 144π
- 48π
V = ⅓πr²h = 48π.

Hard Mode Questions — 440 questions

Unit 1: Foundations: Points, Lines & Angles (40)

Two complementary angles are in the ratio 2:3. The larger angle measures
- 64°
- 74°
- 36°
- 54°
2x + 3x = 90 → x = 18 → angles 36° and 54° → larger = 54°.
Two complementary angles are in the ratio 4:5. The larger angle measures
- 70°
- 60°
- 40°
- 50°
4x + 5x = 90 → x = 10 → angles 40° and 50° → larger = 50°.
Two complementary angles are in the ratio 3:7. The larger angle measures
- 27°
- 73°
- 63°
- 83°
3x + 7x = 90 → x = 9 → angles 27° and 63° → larger = 63°.
Two complementary angles are in the ratio 1:2. The larger angle measures
- 30°
- 60°
- 80°
- 70°
1x + 2x = 90 → x = 30 → angles 30° and 60° → larger = 60°.
Two complementary angles are in the ratio 1:4. The larger angle measures
- 92°
- 82°
- 18°
- 72°
1x + 4x = 90 → x = 18 → angles 18° and 72° → larger = 72°.
Vertical angles measure (4x + 31)° and (6x + 7)°. What is the SUPPLEMENT of either angle?
- 91°
- 11°
- 79°
- 101°
Vertical angles are congruent: 4x + 31 = 6x + 7 → x = 12 → angle = 79° → supplement = 101°.
Vertical angles measure (7x + 18)° and (6x + 24)°. What is the SUPPLEMENT of either angle?
- 30°
- 120°
- 60°
- 110°
Vertical angles are congruent: 7x + 18 = 6x + 24 → x = 6 → angle = 60° → supplement = 120°.
Vertical angles measure (3x + 7)° and (2x + 31)°. What is the SUPPLEMENT of either angle?
- 101°
- 91°
- 79°
- 11°
Vertical angles are congruent: 3x + 7 = 2x + 31 → x = 24 → angle = 79° → supplement = 101°.
Vertical angles measure (6x + 1)° and (3x + 25)°. What is the SUPPLEMENT of either angle?
- 49°
- 41°
- 121°
- 131°
Vertical angles are congruent: 6x + 1 = 3x + 25 → x = 8 → angle = 49° → supplement = 131°.
Vertical angles measure (7x + 27)° and (5x + 47)°. What is the SUPPLEMENT of either angle?
- 112°
- 97°
- 83°
- 73°
Vertical angles are congruent: 7x + 27 = 5x + 47 → x = 10 → angle = 97° → supplement = 83°.
Three angles around a point measure (3x)°, (4x)°, and (5x)°. The LARGEST angle is
- 150°
- 30°
- 120°
- 90°
Angles around a point total 360°: 12x = 360 → x = 30 → largest = 150°.
Three angles around a point measure (8x)°, (3x)°, and (4x)°. The LARGEST angle is
- 96°
- 72°
- 24°
- 192°
Angles around a point total 360°: 15x = 360 → x = 24 → largest = 192°.
Three angles around a point measure (7x)°, (8x)°, and (3x)°. The LARGEST angle is
- 60°
- 20°
- 140°
- 160°
Angles around a point total 360°: 18x = 360 → x = 20 → largest = 160°.
Three angles around a point measure (8x)°, (5x)°, and (7x)°. The LARGEST angle is
- 144°
- 126°
- 18°
- 90°
Angles around a point total 360°: 20x = 360 → x = 18 → largest = 144°.
Three angles around a point measure (7x)°, (5x)°, and (8x)°. The LARGEST angle is
- 126°
- 18°
- 144°
- 90°
Angles around a point total 360°: 20x = 360 → x = 18 → largest = 144°.
B is between A and C. If AB = 2x + 8, BC = 3x + 18, and AC = 56, then AB =
- 6
- 36
- 20
- 33
AB + BC = AC: (2x+8) + (3x+18) = 56 → x = 6 → AB = 20.
B is between A and C. If AB = 2x + 4, BC = 3x + 8, and AC = 57, then AB =
- 36
- 9
- 22
- 35
AB + BC = AC: (2x+4) + (3x+8) = 57 → x = 9 → AB = 22.
B is between A and C. If AB = 2x + 3, BC = 2x + 7, and AC = 62, then AB =
- 33
- 35
- 13
- 29
AB + BC = AC: (2x+3) + (2x+7) = 62 → x = 13 → AB = 29.
B is between A and C. If AB = 2x + 9, BC = 2x + 21, and AC = 58, then AB =
- 7
- 23
- 35
- 31
AB + BC = AC: (2x+9) + (2x+21) = 58 → x = 7 → AB = 23.
B is between A and C. If AB = 2x + 10, BC = 4x + 20, and AC = 54, then AB =
- 31
- 36
- 18
- 4
AB + BC = AC: (2x+10) + (4x+20) = 54 → x = 4 → AB = 18.
B is the midpoint of AC. If AB = 2x + 2 and AC = 3x + 10, then AC =
- 6
- 14
- 28
- 17
AC = 2·AB: 3x + 10 = 2(2x + 2) → x = 6 → AC = 28.
B is the midpoint of AC. If AB = 4x + 12 and AC = 6x + 50, then AC =
- 13
- 67
- 64
- 128
AC = 2·AB: 6x + 50 = 2(4x + 12) → x = 13 → AC = 128.
B is the midpoint of AC. If AB = 3x + 7 and AC = 2x + 50, then AC =
- 37
- 9
- 68
- 34
AC = 2·AB: 2x + 50 = 2(3x + 7) → x = 9 → AC = 68.
B is the midpoint of AC. If AB = 2x + 9 and AC = 5x + 15, then AC =
- 15
- 3
- 30
- 18
AC = 2·AB: 5x + 15 = 2(2x + 9) → x = 3 → AC = 30.
B is the midpoint of AC. If AB = 2x + 15 and AC = 6x + 10, then AC =
- 38
- 70
- 10
- 35
AC = 2·AB: 6x + 10 = 2(2x + 15) → x = 10 → AC = 70.
Ray BD bisects ∠ABC. If m∠ABD = (2x + 1)° and m∠ABC = 58°, then x =
- 29
- 58
- 3
- 14
Bisector splits the angle evenly: m∠ABD = ½(58) = 29° → 2x + 1 = 29 → x = 14.
Ray BD bisects ∠ABC. If m∠ABD = (4x + 16)° and m∠ABC = 80°, then x =
- 40
- 6
- 20
- 80
Bisector splits the angle evenly: m∠ABD = ½(80) = 40° → 4x + 16 = 40 → x = 6.
Ray BD bisects ∠ABC. If m∠ABD = (2x + 4)° and m∠ABC = 60°, then x =
- 6
- 30
- 60
- 13
Bisector splits the angle evenly: m∠ABD = ½(60) = 30° → 2x + 4 = 30 → x = 13.
Ray BD bisects ∠ABC. If m∠ABD = (2x + 29)° and m∠ABC = 66°, then x =
- 2
- 31
- 33
- 66
Bisector splits the angle evenly: m∠ABD = ½(66) = 33° → 2x + 29 = 33 → x = 2.
Ray BD bisects ∠ABC. If m∠ABD = (4x + 6)° and m∠ABC = 36°, then x =
- 36
- 10
- 3
- 18
Bisector splits the angle evenly: m∠ABD = ½(36) = 18° → 4x + 6 = 18 → x = 3.
An angle measures 15°. The DIFFERENCE between its supplement and its complement is
- 165°
- 15°
- 75°
- 90°
Supplement = 180−15 = 165°. Complement = 90−15 = 75°. Difference = 165−75 = 90°.
An angle measures 11°. The DIFFERENCE between its supplement and its complement is
- 79°
- 11°
- 169°
- 90°
Supplement = 180−11 = 169°. Complement = 90−11 = 79°. Difference = 169−79 = 90°.
An angle measures 35°. The DIFFERENCE between its supplement and its complement is
- 90°
- 35°
- 145°
- 55°
Supplement = 180−35 = 145°. Complement = 90−35 = 55°. Difference = 145−55 = 90°.
An angle measures 52°. The DIFFERENCE between its supplement and its complement is
- 52°
- 128°
- 38°
- 90°
Supplement = 180−52 = 128°. Complement = 90−52 = 38°. Difference = 128−38 = 90°.
An angle measures 58°. The DIFFERENCE between its supplement and its complement is
- 122°
- 90°
- 32°
- 58°
Supplement = 180−58 = 122°. Complement = 90−58 = 32°. Difference = 122−32 = 90°.
Ray BD bisects ∠ABC. If m∠ABD = (6x + 1)° and m∠ABC = 110°, then x =
- 7
- 55
- 9
- 110
Bisector splits the angle evenly: m∠ABD = ½(110) = 55° → 6x + 1 = 55 → x = 9.
Ray BD bisects ∠ABC. If m∠ABD = (5x + 12)° and m∠ABC = 124°, then x =
- 124
- 17
- 10
- 62
Bisector splits the angle evenly: m∠ABD = ½(124) = 62° → 5x + 12 = 62 → x = 10.
B is between A and C. If AB = 2x + 6, BC = 5x + 16, and AC = 43, then AB =
- 30
- 31
- 12
- 3
AB + BC = AC: (2x+6) + (5x+16) = 43 → x = 3 → AB = 12.
An angle measures 49°. The DIFFERENCE between its supplement and its complement is
- 90°
- 41°
- 49°
- 131°
Supplement = 180−49 = 131°. Complement = 90−49 = 41°. Difference = 131−41 = 90°.
Three angles around a point measure (6x)°, (4x)°, and (8x)°. The LARGEST angle is
- 80°
- 120°
- 20°
- 160°
Angles around a point total 360°: 18x = 360 → x = 20 → largest = 160°.

Unit 2: Parallel Lines & Transversals (40)

Alternate interior angles measure (5x + 37)° and (6x + 7)°. Each measures
- 187°
- 30°
- -7°
- 202°
Alternate interior angles are congruent: 5x + 37 = 6x + 7 → x = 30 → each = 187°.
Alternate exterior angles measure (4x + 32)° and (2x + 58)°. Each measures
- 84°
- 13°
- 99°
- 96°
Alternate exterior angles are congruent: 4x + 32 = 2x + 58 → x = 13 → each = 84°.
Alternate exterior angles measure (4x + 24)° and (6x + 4)°. Each measures
- 116°
- 79°
- 10°
- 64°
Alternate exterior angles are congruent: 4x + 24 = 6x + 4 → x = 10 → each = 64°.
Alternate interior angles measure (5x + 12)° and (4x + 16)°. Each measures
- 32°
- 4°
- 47°
- 148°
Alternate interior angles are congruent: 5x + 12 = 4x + 16 → x = 4 → each = 32°.
Alternate exterior angles measure (2x + 26)° and (3x + 12)°. Each measures
- 69°
- 126°
- 14°
- 54°
Alternate exterior angles are congruent: 2x + 26 = 3x + 12 → x = 14 → each = 54°.
Same-side interior angles measure (6x + 27)° and (5x + 4)°. The smaller one is
- 123°
- 96°
- 84°
- 16°
Same-side interior angles are supplementary: (6x+27) + (5x+4) = 180 → x = 16 → angles 123° and 84° → smaller = 84°.
Same-side exterior angles measure (5x + 19)° and (6x + 37)°. The smaller one is
- 115°
- 84°
- 96°
- 13°
Same-side exterior angles are supplementary: (5x+19) + (6x+37) = 180 → x = 13 → angles 84° and 115° → smaller = 84°.
Same-side exterior angles measure (4x + 23)° and (2x + 42)°. The smaller one is
- 92°
- 115°
- 23°
- 88°
Same-side exterior angles are supplementary: (4x+23) + (2x+42) = 180 → x = 23 → angles 115° and 88° → smaller = 88°.
Same-side interior angles measure (5x + 27)° and (2x + 40)°. The smaller one is
- 80°
- 127°
- 20°
- 100°
Same-side interior angles are supplementary: (5x+27) + (2x+40) = 180 → x = 20 → angles 127° and 80° → smaller = 80°.
Same-side exterior angles measure (2x + 5)° and (6x + 20)°. The smaller one is
- 45°
- 135°
- 20°
- 140°
Same-side exterior angles are supplementary: (2x+5) + (6x+20) = 180 → x = 20 → angles 45° and 140° → smaller = 45°.
A line passes through (3, -4) and (6, -2). An equation of the line through (-5, 5) parallel to it is
- y − -5 = 2/3(x − 5)
- y + 5 = 2/3(x + -5)
- y − 5 = 2/3(x − -5)
- y − 5 = -3/2(x − -5)
Slope between the given points = (2)/(3) = 2/3. Parallel lines share slope, so through (-5,5): y − 5 = 2/3(x − -5).
A line passes through (-1, -6) and (3, -3). An equation of the line through (5, -4) parallel to it is
- y + -4 = 3/4(x + 5)
- y − -4 = 3/4(x − 5)
- y − -4 = -4/3(x − 5)
- y − 5 = 3/4(x − -4)
Slope between the given points = (3)/(4) = 3/4. Parallel lines share slope, so through (5,-4): y − -4 = 3/4(x − 5).
A line passes through (-5, 6) and (0, 8). An equation of the line through (3, -4) parallel to it is
- y + -4 = 2/5(x + 3)
- y − 3 = 2/5(x − -4)
- y − -4 = 2/5(x − 3)
- y − -4 = -5/2(x − 3)
Slope between the given points = (2)/(5) = 2/5. Parallel lines share slope, so through (3,-4): y − -4 = 2/5(x − 3).
A line passes through (6, 4) and (10, 7). An equation of the line through (1, -1) parallel to it is
- y − -1 = -4/3(x − 1)
- y + -1 = 3/4(x + 1)
- y − 1 = 3/4(x − -1)
- y − -1 = 3/4(x − 1)
Slope between the given points = (3)/(4) = 3/4. Parallel lines share slope, so through (1,-1): y − -1 = 3/4(x − 1).
A line passes through (5, 1) and (8, 3). An equation of the line through (-3, 1) parallel to it is
- y − -3 = 2/3(x − 1)
- y + 1 = 2/3(x + -3)
- y − 1 = 2/3(x − -3)
- y − 1 = -3/2(x − -3)
Slope between the given points = (2)/(3) = 2/3. Parallel lines share slope, so through (-3,1): y − 1 = 2/3(x − -3).
Line ℓ has slope 3/4. Line m is perpendicular to ℓ and has slope (2x)/3. Find x.
- 3
- 2
- -2
- 0
Perpendicular slopes multiply to −1. Solving (2x)/3 = −4/3 gives x = -2.
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (5x + 11)° and its corresponding angle at line r is 101°, then x =
- 5
- 101
- 18
- 90
Corresponding angles across parallel lines are equal: 5x + 11 = 101 → x = 18.
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (3x + 8)° and its corresponding angle at line r is 68°, then x =
- 60
- 3
- 68
- 20
Corresponding angles across parallel lines are equal: 3x + 8 = 68 → x = 20.
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (4x + 20)° and its corresponding angle at line r is 40°, then x =
- 20
- 5
- 40
- 4
Corresponding angles across parallel lines are equal: 4x + 20 = 40 → x = 5.
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (3x + 7)° and its corresponding angle at line r is 55°, then x =
- 16
- 3
- 55
- 48
Corresponding angles across parallel lines are equal: 3x + 7 = 55 → x = 16.
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (4x + 12)° and its corresponding angle at line r is 36°, then x =
- 6
- 4
- 24
- 36
Corresponding angles across parallel lines are equal: 4x + 12 = 36 → x = 6.
Two lines are cut by a transversal. Which condition guarantees the lines are parallel?
- the two lines are the same length
- a pair of vertical angles is congruent
- the transversal is perpendicular to one line
- a pair of alternate interior angles is congruent
The Converse of the Alternate Interior Angles Theorem: if alternate interior angles are congruent, the lines must be parallel.
A line has slope 9/4. A line parallel to it also has slope
- 9/4
- -4/9
- 4/9
- -9/4
Parallel lines have equal slopes.
A line has slope 8/7. A line parallel to it also has slope
- -8/7
- 7/8
- -7/8
- 8/7
Parallel lines have equal slopes.
A line has slope 7/3. A line parallel to it also has slope
- -3/7
- 3/7
- 7/3
- -7/3
Parallel lines have equal slopes.
A line has slope 6/7. A line parallel to it also has slope
- -6/7
- 6/7
- -7/6
- 7/6
Parallel lines have equal slopes.
A line has slope 3/5. A line parallel to it also has slope
- -5/3
- -3/5
- 3/5
- 5/3
Parallel lines have equal slopes.
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (4x + 20)° and its corresponding angle at line r is 44°, then x =
- 4
- 6
- 44
- 24
Corresponding angles across parallel lines are equal: 4x + 20 = 44 → x = 6.
A line passes through (-3, 4) and (2, 6). An equation of the line through (1, 4) parallel to it is
- y + 4 = 2/5(x + 1)
- y − 4 = 2/5(x − 1)
- y − 1 = 2/5(x − 4)
- y − 4 = -5/2(x − 1)
Slope between the given points = (2)/(5) = 2/5. Parallel lines share slope, so through (1,4): y − 4 = 2/5(x − 1).
Lines p, q, and r are all parallel, cut by transversal t. If one angle at line p is (3x + 6)° and its corresponding angle at line r is 54°, then x =
- 3
- 48
- 54
- 16
Corresponding angles across parallel lines are equal: 3x + 6 = 54 → x = 16.
A line has slope 5/8. A line parallel to it also has slope
- -5/8
- -8/5
- 5/8
- 8/5
Parallel lines have equal slopes.
Same-side interior angles measure (6x + 5)° and (5x + 37)°. The smaller one is
- 13°
- 102°
- 97°
- 83°
Same-side interior angles are supplementary: (6x+5) + (5x+37) = 180 → x = 13 → angles 83° and 102° → smaller = 83°.
Alternate exterior angles measure (6x + 37)° and (4x + 59)°. Each measures
- 77°
- 118°
- 103°
- 11°
Alternate exterior angles are congruent: 6x + 37 = 4x + 59 → x = 11 → each = 103°.
A line has slope 3/7. A line parallel to it also has slope
- -7/3
- 3/7
- -3/7
- 7/3
Parallel lines have equal slopes.
A line has slope 2/4. A line parallel to it also has slope
- -2/4
- -4/2
- 4/2
- 2/4
Parallel lines have equal slopes.
Alternate interior angles measure (6x + 16)° and (3x + 49)°. Each measures
- 98°
- 11°
- 97°
- 82°
Alternate interior angles are congruent: 6x + 16 = 3x + 49 → x = 11 → each = 82°.
A line has slope 9/2. A line parallel to it also has slope
- 2/9
- 9/2
- -9/2
- -2/9
Parallel lines have equal slopes.
Alternate interior angles measure (4x + 29)° and (5x + 12)°. Each measures
- 112°
- 83°
- 97°
- 17°
Alternate interior angles are congruent: 4x + 29 = 5x + 12 → x = 17 → each = 97°.
Same-side interior angles measure (3x + 3)° and (3x + 36)°. The smaller one is
- 105°
- 108°
- 24°
- 75°
Same-side interior angles are supplementary: (3x+3) + (3x+36) = 180 → x = 24 → angles 75° and 108° → smaller = 75°.
A line passes through (-2, -3) and (0, -4). An equation of the line through (4, 6) parallel to it is
- y + 6 = -1/2(x + 4)
- y − 6 = -1/2(x − 4)
- y − 6 = -2/-1(x − 4)
- y − 4 = -1/2(x − 6)
Slope between the given points = (-1)/(2) = -1/2. Parallel lines share slope, so through (4,6): y − 6 = -1/2(x − 4).

Unit 3: Triangles: Angles & Inequalities (40)

The angles of a triangle measure (2x+14)°, (3x+2)°, and (4x+38)°. The LARGEST angle is
- 44°
- 94°
- 14°
- 42°
Angles sum to 180: 9x + 54 = 180 → x = 14 → angles 42°, 44°, 94° → largest = 94°.
The angles of a triangle measure (2x+17)°, (1x+8)°, and (3x+23)°. The LARGEST angle is
- 61°
- 30°
- 89°
- 22°
Angles sum to 180: 6x + 48 = 180 → x = 22 → angles 61°, 30°, 89° → largest = 89°.
The angles of a triangle measure (2x+2)°, (4x+20)°, and (1x+18)°. The LARGEST angle is
- 100°
- 38°
- 42°
- 20°
Angles sum to 180: 7x + 40 = 180 → x = 20 → angles 42°, 100°, 38° → largest = 100°.
The angles of a triangle measure (3x+20)°, (2x+2)°, and (1x+32)°. The LARGEST angle is
- 53°
- 21°
- 44°
- 83°
Angles sum to 180: 6x + 54 = 180 → x = 21 → angles 83°, 44°, 53° → largest = 83°.
The angles of a triangle measure (4x+15)°, (3x+18)°, and (2x+21)°. The LARGEST angle is
- 14°
- 60°
- 49°
- 71°
Angles sum to 180: 9x + 54 = 180 → x = 14 → angles 71°, 60°, 49° → largest = 71°.
An exterior angle of a triangle has two remote interior angles measuring 66° and 74°. The exterior angle measures
- 74°
- 140°
- 40°
- 8°
Exterior Angle Theorem: exterior angle = sum of remote interior angles = 66 + 74 = 140°.
An exterior angle of a triangle has two remote interior angles measuring 76° and 45°. The exterior angle measures
- 31°
- 59°
- 76°
- 121°
Exterior Angle Theorem: exterior angle = sum of remote interior angles = 76 + 45 = 121°.
An exterior angle of a triangle has two remote interior angles measuring 31° and 30°. The exterior angle measures
- 1°
- 31°
- 61°
- 119°
Exterior Angle Theorem: exterior angle = sum of remote interior angles = 31 + 30 = 61°.
An exterior angle of a triangle has two remote interior angles measuring 49° and 68°. The exterior angle measures
- 19°
- 63°
- 117°
- 68°
Exterior Angle Theorem: exterior angle = sum of remote interior angles = 49 + 68 = 117°.
An exterior angle of a triangle has two remote interior angles measuring 22° and 60°. The exterior angle measures
- 98°
- 38°
- 60°
- 82°
Exterior Angle Theorem: exterior angle = sum of remote interior angles = 22 + 60 = 82°.
An exterior angle measures (5x + 22)°; its remote interior angles are (3x + 30)° and 28°. The exterior angle is
- 84°
- 28°
- 18°
- 112°
Exterior = sum of remote interiors: 5x+22 = (3x+30) + 28 → x = 18 → exterior = 112°.
An exterior angle measures (5x + 26)°; its remote interior angles are (3x + 37)° and 23°. The exterior angle is
- 23°
- 88°
- 17°
- 111°
Exterior = sum of remote interiors: 5x+26 = (3x+37) + 23 → x = 17 → exterior = 111°.
An exterior angle measures (3x + 29)°; its remote interior angles are (1x + 60)° and 15°. The exterior angle is
- 83°
- 23°
- 98°
- 15°
Exterior = sum of remote interiors: 3x+29 = (1x+60) + 15 → x = 23 → exterior = 98°.
An exterior angle measures (4x + 2)°; its remote interior angles are (2x + 46)° and 26°. The exterior angle is
- 142°
- 35°
- 26°
- 116°
Exterior = sum of remote interiors: 4x+2 = (2x+46) + 26 → x = 35 → exterior = 142°.
An exterior angle measures (4x + 27)°; its remote interior angles are (3x + 40)° and 20°. The exterior angle is
- 33°
- 20°
- 139°
- 159°
Exterior = sum of remote interiors: 4x+27 = (3x+40) + 20 → x = 33 → exterior = 159°.
In isosceles triangle ABC, AB ≅ AC, and the base angles each measure 56°. The exterior angle at a BASE vertex measures
- 112°
- 56°
- 68°
- 124°
The exterior angle at a base vertex is supplementary to the base angle: 180 − 56 = 124°.
In isosceles triangle ABC, AB ≅ AC, and the base angles each measure 44°. The exterior angle at a BASE vertex measures
- 44°
- 88°
- 92°
- 136°
The exterior angle at a base vertex is supplementary to the base angle: 180 − 44 = 136°.
In isosceles triangle ABC, AB ≅ AC, and the base angles each measure 58°. The exterior angle at a BASE vertex measures
- 58°
- 122°
- 116°
- 64°
The exterior angle at a base vertex is supplementary to the base angle: 180 − 58 = 122°.
In isosceles triangle ABC, AB ≅ AC, and the base angles each measure 79°. The exterior angle at a BASE vertex measures
- 158°
- 22°
- 101°
- 79°
The exterior angle at a base vertex is supplementary to the base angle: 180 − 79 = 101°.
In isosceles triangle ABC, AB ≅ AC, and the base angles each measure 63°. The exterior angle at a BASE vertex measures
- 117°
- 54°
- 63°
- 126°
The exterior angle at a base vertex is supplementary to the base angle: 180 − 63 = 117°.
Two sides of a triangle measure 15 and 15. The length of the third side could be any value between
- 0 and 35
- 0 and 30
- -1 and 31
- 15 and 15
Triangle Inequality: the third side is between |15−15| = 0 and 15+15 = 30 (exclusive).
Two sides of a triangle measure 20 and 10. The length of the third side could be any value between
- 10 and 35
- 9 and 31
- 10 and 30
- 20 and 10
Triangle Inequality: the third side is between |20−10| = 10 and 20+10 = 30 (exclusive).
Two sides of a triangle measure 19 and 6. The length of the third side could be any value between
- 13 and 30
- 12 and 26
- 19 and 6
- 13 and 25
Triangle Inequality: the third side is between |19−6| = 13 and 19+6 = 25 (exclusive).
Two sides of a triangle measure 19 and 8. The length of the third side could be any value between
- 10 and 28
- 19 and 8
- 11 and 32
- 11 and 27
Triangle Inequality: the third side is between |19−8| = 11 and 19+8 = 27 (exclusive).
Isosceles triangle ABC has AB ≅ AC. The base angles each measure (3x + 29)°, and the vertex angle measures 86°. Find x.
- 86
- 47
- 6
- 3
Base angles are equal and sum with vertex to 180: 2(3x+29) + 86 = 180 → x = 6.
Isosceles triangle ABC has AB ≅ AC. The base angles each measure (2x + 20)°, and the vertex angle measures 124°. Find x.
- 28
- 2
- 124
- 4
Base angles are equal and sum with vertex to 180: 2(2x+20) + 124 = 180 → x = 4.
Isosceles triangle ABC has AB ≅ AC. The base angles each measure (3x + 9)°, and the vertex angle measures 120°. Find x.
- 30
- 3
- 120
- 7
Base angles are equal and sum with vertex to 180: 2(3x+9) + 120 = 180 → x = 7.
Isosceles triangle ABC has AB ≅ AC. The base angles each measure (3x + 14)°, and the vertex angle measures 86°. Find x.
- 47
- 11
- 3
- 86
Base angles are equal and sum with vertex to 180: 2(3x+14) + 86 = 180 → x = 11.
A median of a triangle measures 30 units. The distance from the vertex to the centroid is
- 30
- 20
- 10
- 15
The centroid divides each median 2:1 from the vertex: 30 × ⅔ = 20.
A median of a triangle measures 27 units. The distance from the vertex to the centroid is
- 18
- 13
- 9
- 27
The centroid divides each median 2:1 from the vertex: 27 × ⅔ = 18.
A median of a triangle measures 33 units. The distance from the vertex to the centroid is
- 11
- 33
- 22
- 16
The centroid divides each median 2:1 from the vertex: 33 × ⅔ = 22.
A median of a triangle measures 15 units. The distance from the vertex to the centroid is
- 5
- 10
- 7
- 15
The centroid divides each median 2:1 from the vertex: 15 × ⅔ = 10.
A triangle has sides 10, 13, and 16. It is
- obtuse
- impossible
- right
- acute
Compare 16² = 256 to 10²+13² = 269: less → acute triangle.
A triangle has sides 9, 40, and 41. It is
- acute
- obtuse
- impossible
- right
Compare 41² = 1681 to 9²+40² = 1681: equal → right triangle.
A triangle has sides 7, 24, and 25. It is
- impossible
- acute
- obtuse
- right
Compare 25² = 625 to 7²+24² = 625: equal → right triangle.
A triangle has sides 10, 10, and 14. It is
- impossible
- right
- acute
- obtuse
Compare 14² = 196 to 10²+10² = 200: less → acute triangle.
A triangle has sides 5, 12, and 13. It is
- right
- impossible
- acute
- obtuse
Compare 13² = 169 to 5²+12² = 169: equal → right triangle.
Two sides of a triangle measure 15 and 14. The length of the third side could be any value between
- 1 and 29
- 1 and 34
- 0 and 30
- 15 and 14
Triangle Inequality: the third side is between |15−14| = 1 and 15+14 = 29 (exclusive).
Isosceles triangle ABC has AB ≅ AC. The base angles each measure (1x + 16)°, and the vertex angle measures 138°. Find x.
- 138
- 1
- 5
- 21
Base angles are equal and sum with vertex to 180: 2(1x+16) + 138 = 180 → x = 5.
An exterior angle of a triangle has two remote interior angles measuring 72° and 48°. The exterior angle measures
- 24°
- 60°
- 72°
- 120°
Exterior Angle Theorem: exterior angle = sum of remote interior angles = 72 + 48 = 120°.

Unit 4: Congruent Triangles & Proofs (40)

In △ABC and △DEF, AB ≅ DE, ∠B ≅ ∠E, BC ≅ EF, where ∠B is between AB and BC. The triangles are congruent by
- SSS
- SAS
- AAS
- HL
The given parts match the SAS congruence pattern.
In △ABC and △DEF, AB ≅ DE, BC ≅ EF, AC ≅ DF. The triangles are congruent by
- HL
- SSS
- AAS
- ASA
The given parts match the SSS congruence pattern.
In △ABC and △DEF, ∠B and ∠E are right angles, AB ≅ DE (hypotenuses), AC ≅ DF (a leg). The triangles are congruent by
- SSS
- HL
- SAS
- ASA
The given parts match the HL congruence pattern.
In △ABC and △DEF, ∠A ≅ ∠D, ∠B ≅ ∠E, BC ≅ EF, where BC is NOT between the two angle pairs. The triangles are congruent by
- SAS
- SSS
- ASA
- AAS
The given parts match the AAS congruence pattern.
In △ABC and △DEF, ∠A ≅ ∠D, AB ≅ DE, ∠B ≅ ∠E, where AB is between ∠A and ∠B. The triangles are congruent by
- HL
- SSS
- ASA
- AAS
The given parts match the ASA congruence pattern.
△ABC ≅ △DEF with AB = 4x + 20 and DE = 3x + 35. AB =
- 15
- 80
- 75
- 20
Corresponding sides are equal (CPCTC): 4x+20 = 3x+35 → x = 15 → AB = 80.
△ABC ≅ △DEF with AB = 4x + 9 and DE = 3x + 32. AB =
- 23
- 101
- 9
- 96
Corresponding sides are equal (CPCTC): 4x+9 = 3x+32 → x = 23 → AB = 101.
△ABC ≅ △DEF with AB = 5x + 23 and DE = 2x + 38. AB =
- 43
- 5
- 48
- 23
Corresponding sides are equal (CPCTC): 5x+23 = 2x+38 → x = 5 → AB = 48.
△ABC ≅ △DEF with AB = 5x + 2 and DE = 3x + 24. AB =
- 11
- 52
- 57
- 2
Corresponding sides are equal (CPCTC): 5x+2 = 3x+24 → x = 11 → AB = 57.
△ABC ≅ △DEF with AB = 3x + 25 and DE = 5x + 17. AB =
- 25
- 4
- 37
- 32
Corresponding sides are equal (CPCTC): 3x+25 = 5x+17 → x = 4 → AB = 37.
△PQR ≅ △XYZ and m∠Q = 43°. Then m∠Y =
- 53°
- 137°
- 43°
- 47°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 56°. Then m∠Y =
- 34°
- 124°
- 56°
- 66°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 57°. Then m∠Y =
- 57°
- 67°
- 33°
- 123°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 61°. Then m∠Y =
- 29°
- 119°
- 71°
- 61°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 79°. Then m∠Y =
- 79°
- 11°
- 89°
- 101°
Corresponding angles of congruent triangles are equal (CPCTC).
Two triangles share side BD, with ∠A ≅ ∠C and AB ≅ CD marked. To prove congruence by SAS, the missing justification for BD ≅ BD is
- Reflexive Property
- Alternate Interior Angles Theorem
- CPCTC
- Vertical Angles Theorem
A side shared by both triangles is congruent to itself — the Reflexive Property.
Two triangles overlap at point M, sharing vertical angles ∠AMB and ∠CMD, with AM ≅ CM marked. For SAS you also need
- AB ≅ CD
- ∠A ≅ ∠C
- BM ≅ DM
- ∠B ≅ ∠D
SAS needs the second side forming the vertical angle: BM ≅ DM, since ∠AMB ≅ ∠CMD by the Vertical Angles Theorem.
△ABC is mapped onto △A'B'C' by a rotation of 90° about the origin. Which statement must be true?
- △ABC ∼ △A'B'C' but not congruent
- △ABC ≅ △A'B'C'
- the triangles have different areas
- only one pair of sides is congruent
A rotation of 90° about the origin is a rigid motion — it preserves distance and angle measure, so the image is congruent to the original.
△ABC is mapped onto △A'B'C' by a reflection over the y-axis. Which statement must be true?
- the triangles have different areas
- △ABC ≅ △A'B'C'
- only one pair of sides is congruent
- △ABC ∼ △A'B'C' but not congruent
A reflection over the y-axis is a rigid motion — it preserves distance and angle measure, so the image is congruent to the original.
△ABC is mapped onto △A'B'C' by a translation. Which statement must be true?
- △ABC ∼ △A'B'C' but not congruent
- △ABC ≅ △A'B'C'
- the triangles have different areas
- only one pair of sides is congruent
A translation is a rigid motion — it preserves distance and angle measure, so the image is congruent to the original.
In △ABC, ∠B ≅ ∠C. Which conclusion is justified by the Converse of the Isosceles Triangle Theorem?
- BC ≅ AC
- ∠A ≅ ∠B
- AB ≅ AC
- AB ⊥ BC
If two angles of a triangle are congruent, the sides opposite them are congruent: AB ≅ AC.
△ABD ≅ △CBD share side BD. If AD = 3x + 10 and CD = 4x + 6, then AD =
- 22
- 6
- 4
- 25
CPCTC: AD = CD → 3x+10 = 4x+6 → x = 4 → AD = 22.
△ABD ≅ △CBD share side BD. If AD = 3x + 13 and CD = 2x + 31, then AD =
- 70
- 67
- 18
- 31
CPCTC: AD = CD → 3x+13 = 2x+31 → x = 18 → AD = 67.
△ABD ≅ △CBD share side BD. If AD = 3x + 12 and CD = 2x + 25, then AD =
- 13
- 54
- 25
- 51
CPCTC: AD = CD → 3x+12 = 2x+25 → x = 13 → AD = 51.
△ABD ≅ △CBD share side BD. If AD = 2x + 20 and CD = 3x + 4, then AD =
- 16
- 55
- 52
- 4
CPCTC: AD = CD → 2x+20 = 3x+4 → x = 16 → AD = 52.
△ABD ≅ △CBD share side BD. If AD = 2x + 16 and CD = 3x + 7, then AD =
- 9
- 7
- 37
- 34
CPCTC: AD = CD → 2x+16 = 3x+7 → x = 9 → AD = 34.
△ABD ≅ △CBD share side BD. If AD = 2x + 19 and CD = 3x + 16, then AD =
- 16
- 25
- 28
- 3
CPCTC: AD = CD → 2x+19 = 3x+16 → x = 3 → AD = 25.
△ABD ≅ △CBD share side BD. If AD = 4x + 20 and CD = 3x + 25, then AD =
- 25
- 43
- 40
- 5
CPCTC: AD = CD → 4x+20 = 3x+25 → x = 5 → AD = 40.
△ABD ≅ △CBD share side BD. If AD = 3x + 5 and CD = 2x + 11, then AD =
- 26
- 6
- 11
- 23
CPCTC: AD = CD → 3x+5 = 2x+11 → x = 6 → AD = 23.
△PQR ≅ △XYZ and m∠Q = 49°. Then m∠Y =
- 131°
- 41°
- 49°
- 59°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 75°. Then m∠Y =
- 105°
- 85°
- 15°
- 75°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 72°. Then m∠Y =
- 18°
- 82°
- 108°
- 72°
Corresponding angles of congruent triangles are equal (CPCTC).
△PQR ≅ △XYZ and m∠Q = 22°. Then m∠Y =
- 158°
- 22°
- 32°
- 68°
Corresponding angles of congruent triangles are equal (CPCTC).
△ABC ≅ △DEF with AB = 6x + 4 and DE = 4x + 22. AB =
- 53
- 9
- 4
- 58
Corresponding sides are equal (CPCTC): 6x+4 = 4x+22 → x = 9 → AB = 58.
△ABD ≅ △CBD share side BD. If AD = 4x + 9 and CD = 3x + 28, then AD =
- 85
- 88
- 28
- 19
CPCTC: AD = CD → 4x+9 = 3x+28 → x = 19 → AD = 85.
△ABD ≅ △CBD share side BD. If AD = 4x + 12 and CD = 3x + 31, then AD =
- 88
- 19
- 31
- 91
CPCTC: AD = CD → 4x+12 = 3x+31 → x = 19 → AD = 88.
△ABC ≅ △DEF with AB = 4x + 2 and DE = 2x + 14. AB =
- 2
- 21
- 26
- 6
Corresponding sides are equal (CPCTC): 4x+2 = 2x+14 → x = 6 → AB = 26.
△ABC ≅ △DEF with AB = 5x + 23 and DE = 6x + 19. AB =
- 43
- 23
- 4
- 38
Corresponding sides are equal (CPCTC): 5x+23 = 6x+19 → x = 4 → AB = 43.
△ABD ≅ △CBD share side BD. If AD = 3x + 7 and CD = 2x + 15, then AD =
- 15
- 8
- 31
- 34
CPCTC: AD = CD → 3x+7 = 2x+15 → x = 8 → AD = 31.
△ABC ≅ △DEF with AB = 4x + 25 and DE = 5x + 6. AB =
- 19
- 96
- 101
- 25
Corresponding sides are equal (CPCTC): 4x+25 = 5x+6 → x = 19 → AB = 101.

Unit 5: Similarity & Proportions (40)

Two similar triangles have corresponding sides in the ratio 4:9. If the smaller triangle's perimeter is 16, the larger triangle's perimeter is
- 21
- 40
- 36
- 16
Perimeters scale with the side ratio: 16 × 9/4 = 36.
Two similar triangles have corresponding sides in the ratio 4:3. If the smaller triangle's perimeter is 36, the larger triangle's perimeter is
- 31
- 36
- 48
- 27
Perimeters scale with the side ratio: 36 × 3/4 = 27.
Two similar triangles have corresponding sides in the ratio 9:6. If the smaller triangle's perimeter is 63, the larger triangle's perimeter is
- 68
- 42
- 51
- 63
Perimeters scale with the side ratio: 63 × 6/9 = 42.
Two similar triangles have corresponding sides in the ratio 9:2. If the smaller triangle's perimeter is 81, the larger triangle's perimeter is
- 86
- 27
- 81
- 18
Perimeters scale with the side ratio: 81 × 2/9 = 18.
Two similar triangles have corresponding sides in the ratio 5:7. If the smaller triangle's perimeter is 20, the larger triangle's perimeter is
- 20
- 33
- 25
- 28
Perimeters scale with the side ratio: 20 × 7/5 = 28.
Two similar polygons have corresponding sides in the ratio 6:4. Their areas are in the ratio
- 24:24
- 12:8
- 36 : 16
- 6:4
Area ratio = (side ratio)² = 6²:4² = 36 : 16.
Two similar polygons have corresponding sides in the ratio 2:5. Their areas are in the ratio
- 4:10
- 10:10
- 4 : 25
- 2:5
Area ratio = (side ratio)² = 2²:5² = 4 : 25.
Two similar polygons have corresponding sides in the ratio 5:3. Their areas are in the ratio
- 15:15
- 10:6
- 25 : 9
- 5:3
Area ratio = (side ratio)² = 5²:3² = 25 : 9.
Two similar polygons have corresponding sides in the ratio 5:2. Their areas are in the ratio
- 5:2
- 25 : 4
- 10:10
- 10:4
Area ratio = (side ratio)² = 5²:2² = 25 : 4.
Two similar polygons have corresponding sides in the ratio 6:2. Their areas are in the ratio
- 6:2
- 36 : 4
- 12:12
- 12:4
Area ratio = (side ratio)² = 6²:2² = 36 : 4.
The altitude to the hypotenuse of a right triangle splits it into segments of 12 and 3. The altitude measures
- 15
- 6
- 9
- 7
Altitude = √(segment₁ × segment₂) = √(12×3) = √36 = 6.
The altitude to the hypotenuse of a right triangle splits it into segments of 2 and 18. The altitude measures
- 16
- 10
- 6
- 20
Altitude = √(segment₁ × segment₂) = √(2×18) = √36 = 6.
The altitude to the hypotenuse of a right triangle splits it into segments of 18 and 2. The altitude measures
- 6
- 10
- 20
- 16
Altitude = √(segment₁ × segment₂) = √(18×2) = √36 = 6.
The altitude to the hypotenuse of a right triangle splits it into segments of 4 and 16. The altitude measures
- 8
- 10
- 12
- 20
Altitude = √(segment₁ × segment₂) = √(4×16) = √64 = 8.
The altitude to the hypotenuse of a right triangle splits it into segments of 16 and 4. The altitude measures
- 10
- 8
- 12
- 20
Altitude = √(segment₁ × segment₂) = √(16×4) = √64 = 8.
In a right triangle, the whole hypotenuse measures 12 and one hypotenuse segment (adjacent to a leg) measures 3. That leg measures
- 12
- 9
- 3
- 6
leg² = (adjacent segment)(whole hypotenuse) = 3 × 12 = 36 → leg = 6.
In a right triangle, the whole hypotenuse measures 25 and one hypotenuse segment (adjacent to a leg) measures 16. That leg measures
- 9
- 20
- 25
- 16
leg² = (adjacent segment)(whole hypotenuse) = 16 × 25 = 400 → leg = 20.
In a right triangle, the whole hypotenuse measures 25 and one hypotenuse segment (adjacent to a leg) measures 9. That leg measures
- 25
- 9
- 15
- 16
leg² = (adjacent segment)(whole hypotenuse) = 9 × 25 = 225 → leg = 15.
In a right triangle, the whole hypotenuse measures 18 and one hypotenuse segment (adjacent to a leg) measures 2. That leg measures
- 2
- 6
- 18
- 16
leg² = (adjacent segment)(whole hypotenuse) = 2 × 18 = 36 → leg = 6.
In a right triangle, the whole hypotenuse measures 36 and one hypotenuse segment (adjacent to a leg) measures 9. That leg measures
- 36
- 9
- 18
- 27
leg² = (adjacent segment)(whole hypotenuse) = 9 × 36 = 324 → leg = 18.
DE ∥ AC in △ABC, with BD = 15, DA = 6, BE = 10. Then EC =
- 15
- 10
- 4
- 6
Side-Splitter: BD/DA = BE/EC → 15/6 = 10/EC → EC = 4.
DE ∥ AC in △ABC, with BD = 10, DA = 15, BE = 8. Then EC =
- 10
- 12
- 8
- 14
Side-Splitter: BD/DA = BE/EC → 10/15 = 8/EC → EC = 12.
DE ∥ AC in △ABC, with BD = 12, DA = 8, BE = 9. Then EC =
- 12
- 8
- 9
- 6
Side-Splitter: BD/DA = BE/EC → 12/8 = 9/EC → EC = 6.
DE ∥ AC in △ABC, with BD = 5, DA = 8, BE = 15. Then EC =
- 5
- 24
- 15
- 26
Side-Splitter: BD/DA = BE/EC → 5/8 = 15/EC → EC = 24.
Two similar solids have volumes in the ratio 1:8. Their surface areas are in the ratio
- 2:4
- 1:4
- 1:8
- 1:2
Side ratio = ∛(1:8) = 1:2 → area ratio = 1²:2² = 1:4.
Two similar solids have volumes in the ratio 8:1. Their surface areas are in the ratio
- 4:1
- 2:1
- 4:2
- 8:1
Side ratio = ∛(8:1) = 2:1 → area ratio = 2²:1² = 4:1.
Two similar solids have volumes in the ratio 8:64. Their surface areas are in the ratio
- 4:8
- 2:4
- 4:16
- 8:64
Side ratio = ∛(8:64) = 2:4 → area ratio = 2²:4² = 4:16.
Two similar solids have volumes in the ratio 125:1. Their surface areas are in the ratio
- 5:1
- 10:2
- 125:1
- 25:1
Side ratio = ∛(125:1) = 5:1 → area ratio = 5²:1² = 25:1.
Two similar solids have surface areas in the ratio 16:9. Their volumes are in the ratio
- 64:27
- 16:9
- 128:54
- 4:3
Side ratio = √(16:9) = 4:3 → volume ratio = 4³:3³ = 64:27.
Two similar solids have surface areas in the ratio 4:25. Their volumes are in the ratio
- 2:5
- 4:25
- 16:250
- 8:125
Side ratio = √(4:25) = 2:5 → volume ratio = 2³:5³ = 8:125.
Two similar solids have surface areas in the ratio 25:4. Their volumes are in the ratio
- 125:8
- 25:4
- 250:16
- 5:2
Side ratio = √(25:4) = 5:2 → volume ratio = 5³:2³ = 125:8.
Two similar solids have surface areas in the ratio 16:4. Their volumes are in the ratio
- 128:16
- 64:8
- 4:2
- 16:4
Side ratio = √(16:4) = 4:2 → volume ratio = 4³:2³ = 64:8.
Solve the proportion: 4/8 = 2/x
- 8
- 7
- -2
- 4
Cross-multiply: 4·x = 8·2 → x = 16/4 = 4.
Solve the proportion: 4/4 = 7/x
- 10
- 28
- 7
- 3
Cross-multiply: 4·x = 4·7 → x = 28/4 = 7.
Solve the proportion: 2/7 = 22/x
- 77
- 20
- 80
- 44
Cross-multiply: 2·x = 7·22 → x = 154/2 = 77.
Solve the proportion: 2/8 = 29/x
- 58
- 119
- 27
- 116
Cross-multiply: 2·x = 8·29 → x = 232/2 = 116.
A figure is dilated by scale factor 1.5. Its area is multiplied by
- 3.2
- 1.5
- 2.2
- 3
Area scales by k² = 1.5² = 2.2.
A figure is dilated by scale factor 5. Its area is multiplied by
- 5
- 10
- 25
- 26
Area scales by k² = 5² = 25.
A figure is dilated by scale factor 3. Its area is multiplied by
- 9
- 10
- 6
- 3
Area scales by k² = 3² = 9.
A figure is dilated by scale factor 4. Its area is multiplied by
- 16
- 4
- 17
- 8
Area scales by k² = 4² = 16.

Unit 6: Right Triangles & Trigonometry (40)

A right triangle has legs 12 and 16. The hypotenuse is
- 28
- 20
- 22
- 4
√(12²+16²) = √400 = 20.
A right triangle has legs 20 and 48. The hypotenuse is
- 54
- 28
- 68
- 52
√(20²+48²) = √2704 = 52.
A right triangle has legs 6 and 8. The hypotenuse is
- 12
- 2
- 10
- 14
√(6²+8²) = √100 = 10.
A right triangle has legs 10 and 24. The hypotenuse is
- 14
- 26
- 28
- 34
√(10²+24²) = √676 = 26.
A right triangle has hypotenuse 5 and one leg 3. The other leg is
- 3
- 8
- 2
- 4
√(5²−3²) = √16 = 4.
A right triangle has hypotenuse 25 and one leg 7. The other leg is
- 32
- 18
- 7
- 24
√(25²−7²) = √576 = 24.
A right triangle has hypotenuse 10 and one leg 6. The other leg is
- 8
- 16
- 6
- 4
√(10²−6²) = √64 = 8.
A right triangle has hypotenuse 20 and one leg 12. The other leg is
- 16
- 12
- 8
- 32
√(20²−12²) = √256 = 16.
In a 45-45-90 triangle, each leg measures 6. The hypotenuse measures
- 6√3
- 6√2
- 12
- 12√2
45-45-90: hypotenuse = leg × √2 = 6√2.
In a 45-45-90 triangle, each leg measures 9. The hypotenuse measures
- 9√2
- 18√2
- 9√3
- 18
45-45-90: hypotenuse = leg × √2 = 9√2.
In a 45-45-90 triangle, each leg measures 3. The hypotenuse measures
- 6√2
- 3√2
- 3√3
- 6
45-45-90: hypotenuse = leg × √2 = 3√2.
In a 45-45-90 triangle, each leg measures 10. The hypotenuse measures
- 10√3
- 20
- 20√2
- 10√2
45-45-90: hypotenuse = leg × √2 = 10√2.
In a 30-60-90 triangle, the short leg measures 5. The hypotenuse measures
- 10
- 5√2
- 5√3
- 15
30-60-90: hypotenuse = 2 × short leg = 10.
In a 30-60-90 triangle, the short leg measures 4. The hypotenuse measures
- 8
- 4√2
- 4√3
- 12
30-60-90: hypotenuse = 2 × short leg = 8.
In a 30-60-90 triangle, the short leg measures 7. The hypotenuse measures
- 7√3
- 7√2
- 14
- 21
30-60-90: hypotenuse = 2 × short leg = 14.
In a 30-60-90 triangle, the short leg measures 2. The hypotenuse measures
- 2√2
- 4
- 6
- 2√3
30-60-90: hypotenuse = 2 × short leg = 4.
In a 30-60-90 triangle, the hypotenuse measures 18. The LONGER leg measures
- 9√3
- 18
- 9√2
- 18√3
Short leg = hyp/2 = 9; longer leg = short × √3 = 9√3.
In a 30-60-90 triangle, the hypotenuse measures 14. The LONGER leg measures
- 14
- 7√2
- 14√3
- 7√3
Short leg = hyp/2 = 7; longer leg = short × √3 = 7√3.
In a 30-60-90 triangle, the hypotenuse measures 10. The LONGER leg measures
- 5√3
- 10√3
- 10
- 5√2
Short leg = hyp/2 = 5; longer leg = short × √3 = 5√3.
In a 30-60-90 triangle, the hypotenuse measures 6. The LONGER leg measures
- 3√2
- 3√3
- 6√3
- 6
Short leg = hyp/2 = 3; longer leg = short × √3 = 3√3.
In right triangle ABC, ∠C = 90°, the side opposite ∠A is 10 and adjacent to ∠A is 8. tan(A) =
- 8/10
- 8/√164
- 10/8
- 10/9
tan(A) uses opposite=10, adjacent=8, hypotenuse=√164.
In right triangle ABC, ∠C = 90°, the side opposite ∠A is 13 and adjacent to ∠A is 17. cos(A) =
- 17/√458
- 13/18
- 17/13
- 13/√458
cos(A) uses opposite=13, adjacent=17, hypotenuse=√458.
In right triangle ABC, ∠C = 90°, the side opposite ∠A is 11 and adjacent to ∠A is 6. sin(A) =
- 6/√157
- 11/√157
- 11/7
- 6/11
sin(A) uses opposite=11, adjacent=6, hypotenuse=√157.
In right triangle ABC, ∠C = 90°, the side opposite ∠A is 9 and adjacent to ∠A is 15. cos(A) =
- 15/√306
- 15/9
- 9/16
- 9/√306
cos(A) uses opposite=9, adjacent=15, hypotenuse=√306.
From a point 50 ft from the base of a tower, the angle of elevation to the top is 35°. The tower's height, to the nearest foot, is
- 35 ft
- 50 ft
- 71 ft
- 29 ft
h = 50·tan(35°) ≈ 35 ft.
From a point 200 ft from the base of a tower, the angle of elevation to the top is 15°. The tower's height, to the nearest foot, is
- 54 ft
- 52 ft
- 746 ft
- 69 ft
h = 200·tan(15°) ≈ 54 ft.
From a point 80 ft from the base of a tower, the angle of elevation to the top is 35°. The tower's height, to the nearest foot, is
- 46 ft
- 56 ft
- 71 ft
- 114 ft
h = 80·tan(35°) ≈ 56 ft.
From a point 200 ft from the base of a tower, the angle of elevation to the top is 20°. The tower's height, to the nearest foot, is
- 68 ft
- 88 ft
- 73 ft
- 549 ft
h = 200·tan(20°) ≈ 73 ft.
From the top of a 100-ft cliff, the angle of depression to a boat is 15°. The horizontal distance to the boat, to the nearest foot, is
- 27 ft
- 393 ft
- 386 ft
- 373 ft
d = 100/tan(15°) ≈ 373 ft.
From the top of a 180-ft cliff, the angle of depression to a boat is 35°. The horizontal distance to the boat, to the nearest foot, is
- 126 ft
- 314 ft
- 277 ft
- 257 ft
d = 180/tan(35°) ≈ 257 ft.
From the top of a 150-ft cliff, the angle of depression to a boat is 25°. The horizontal distance to the boat, to the nearest foot, is
- 70 ft
- 342 ft
- 355 ft
- 322 ft
d = 150/tan(25°) ≈ 322 ft.
From the top of a 200-ft cliff, the angle of depression to a boat is 15°. The horizontal distance to the boat, to the nearest foot, is
- 773 ft
- 54 ft
- 746 ft
- 766 ft
d = 200/tan(15°) ≈ 746 ft.
If sin(3x + 17)° = cos(4x + 3)°, then x =
- 17
- 3
- 10
- 12
Cofunctions: (3x+17) + (4x+3) = 90 → x = 10.
If sin(2x + 11)° = cos(2x + 19)°, then x =
- 15
- 11
- 17
- 2
Cofunctions: (2x+11) + (2x+19) = 90 → x = 15.
If sin(4x + 16)° = cos(3x + 39)°, then x =
- 5
- 4
- 7
- 16
Cofunctions: (4x+16) + (3x+39) = 90 → x = 5.
If sin(3x + 2)° = cos(3x + 22)°, then x =
- 13
- 2
- 11
- 3
Cofunctions: (3x+2) + (3x+22) = 90 → x = 11.
A ramp rises 2 ft over a horizontal run of 13 ft. The angle it makes with the ground, to the nearest tenth of a degree, is
- 8.7°
- 4.4°
- 12°
- 81.3°
θ = tan⁻¹(2/13) ≈ 8.7°.
A ramp rises 9 ft over a horizontal run of 12 ft. The angle it makes with the ground, to the nearest tenth of a degree, is
- 53.1°
- 40°
- 18.4°
- 36.9°
θ = tan⁻¹(9/12) ≈ 36.9°.
A ramp rises 10 ft over a horizontal run of 19 ft. The angle it makes with the ground, to the nearest tenth of a degree, is
- 62.2°
- 27.8°
- 13.9°
- 31°
θ = tan⁻¹(10/19) ≈ 27.8°.
A ramp rises 4 ft over a horizontal run of 27 ft. The angle it makes with the ground, to the nearest tenth of a degree, is
- 81.6°
- 8.4°
- 4.2°
- 11°
θ = tan⁻¹(4/27) ≈ 8.4°.

Unit 7: Polygons & Quadrilaterals (40)

Each interior angle of a regular 8-gon measures
- 135°
- 45°
- 125°
- 145°
(n−2)·180/n = (8−2)·180/8 = 135°.
Each interior angle of a regular 9-gon measures
- 140°
- 130°
- 150°
- 40°
(n−2)·180/n = (9−2)·180/9 = 140°.
Each interior angle of a regular 20-gon measures
- 162°
- 18°
- 172°
- 152°
(n−2)·180/n = (20−2)·180/20 = 162°.
Each interior angle of a regular 12-gon measures
- 30°
- 140°
- 150°
- 160°
(n−2)·180/n = (12−2)·180/12 = 150°.
Each interior angle of a regular 10-gon measures
- 144°
- 154°
- 36°
- 134°
(n−2)·180/n = (10−2)·180/10 = 144°.
Each exterior angle of a regular polygon measures 24°. The polygon has
- 17 sides
- 18 sides
- 15 sides
- 13 sides
n = 360/24 = 15.
Each exterior angle of a regular polygon measures 18°. The polygon has
- 22 sides
- 18 sides
- 20 sides
- 23 sides
n = 360/18 = 20.
Each exterior angle of a regular polygon measures 60°. The polygon has
- 9 sides
- 6 sides
- 8 sides
- 4 sides
n = 360/60 = 6.
Each exterior angle of a regular polygon measures 20°. The polygon has
- 16 sides
- 20 sides
- 18 sides
- 21 sides
n = 360/20 = 18.
Each exterior angle of a regular polygon measures 15°. The polygon has
- 27 sides
- 24 sides
- 22 sides
- 26 sides
n = 360/15 = 24.
Parallelogram diagonals meet at E. If AE = 4x + 6 and EC = 3x + 20, diagonal AC =
- 14
- 124
- 62
- 128
Diagonals bisect each other: 4x+6 = 3x+20 → x = 14 → AE = 62 → AC = 124.
Parallelogram diagonals meet at E. If AE = 6x + 5 and EC = 4x + 25, diagonal AC =
- 134
- 130
- 65
- 10
Diagonals bisect each other: 6x+5 = 4x+25 → x = 10 → AE = 65 → AC = 130.
Parallelogram diagonals meet at E. If AE = 6x + 17 and EC = 4x + 27, diagonal AC =
- 5
- 94
- 98
- 47
Diagonals bisect each other: 6x+17 = 4x+27 → x = 5 → AE = 47 → AC = 94.
Parallelogram diagonals meet at E. If AE = 5x + 1 and EC = 4x + 14, diagonal AC =
- 66
- 136
- 13
- 132
Diagonals bisect each other: 5x+1 = 4x+14 → x = 13 → AE = 66 → AC = 132.
Parallelogram diagonals meet at E. If AE = 3x + 19 and EC = 2x + 38, diagonal AC =
- 19
- 76
- 156
- 152
Diagonals bisect each other: 3x+19 = 2x+38 → x = 19 → AE = 76 → AC = 152.
A rhombus has perimeter 104 and one diagonal of length 20. The other diagonal measures
- 52
- 20
- 48
- 26
Side = 104/4 = 26. Half-diagonals 10.0 and √(26²−10.0²) = 24 → other diagonal = 48.
A rhombus has perimeter 52 and one diagonal of length 24. The other diagonal measures
- 10
- 13
- 24
- 14
Side = 52/4 = 13. Half-diagonals 12.0 and √(13²−12.0²) = 5 → other diagonal = 10.
A rhombus has perimeter 52 and one diagonal of length 10. The other diagonal measures
- 28
- 13
- 24
- 10
Side = 52/4 = 13. Half-diagonals 5.0 and √(13²−5.0²) = 12 → other diagonal = 24.
A rhombus has perimeter 20 and one diagonal of length 6. The other diagonal measures
- 8
- 5
- 12
- 6
Side = 20/4 = 5. Half-diagonals 3.0 and √(5²−3.0²) = 4 → other diagonal = 8.
A rhombus has perimeter 60 and one diagonal of length 18. The other diagonal measures
- 24
- 18
- 15
- 28
Side = 60/4 = 15. Half-diagonals 9.0 and √(15²−9.0²) = 12 → other diagonal = 24.
A kite has one pair of opposite angles measuring 127° and 75°. The other two (equal) angles each measure
- 127°
- 75°
- 158°
- 202°
Quadrilateral angles sum to 360: 127+75+2x = 360 → x = 158°.
A kite has one pair of opposite angles measuring 97° and 147°. The other two (equal) angles each measure
- 116°
- 147°
- 244°
- 97°
Quadrilateral angles sum to 360: 97+147+2x = 360 → x = 116°.
A kite has one pair of opposite angles measuring 147° and 122°. The other two (equal) angles each measure
- 269°
- 147°
- 122°
- 91°
Quadrilateral angles sum to 360: 147+122+2x = 360 → x = 91°.
A kite has one pair of opposite angles measuring 135° and 109°. The other two (equal) angles each measure
- 109°
- 135°
- 116°
- 244°
Quadrilateral angles sum to 360: 135+109+2x = 360 → x = 116°.
A kite has one pair of opposite angles measuring 65° and 136°. The other two (equal) angles each measure
- 201°
- 65°
- 136°
- 159°
Quadrilateral angles sum to 360: 65+136+2x = 360 → x = 159°.
A trapezoid has parallel bases 11 and 7. Its midsegment measures
- 11
- 18
- 9
- 4
Midsegment = ½(base₁+base₂) = ½(11+7) = 9.
A trapezoid has parallel bases 8 and 12. Its midsegment measures
- 4
- 10
- 12
- 20
Midsegment = ½(base₁+base₂) = ½(8+12) = 10.
A trapezoid has parallel bases 17 and 13. Its midsegment measures
- 15
- 17
- 30
- 4
Midsegment = ½(base₁+base₂) = ½(17+13) = 15.
A trapezoid has parallel bases 9 and 5. Its midsegment measures
- 9
- 4
- 14
- 7
Midsegment = ½(base₁+base₂) = ½(9+5) = 7.
A trapezoid has parallel bases 11 and 15. Its midsegment measures
- 26
- 13
- 4
- 15
Midsegment = ½(base₁+base₂) = ½(11+15) = 13.
A rectangle has length 6 and width 8. Each diagonal measures
- 7
- 10
- 14
- 9
Diagonal = √(6²+8²) = √100 = 10.
A rectangle has length 12 and width 16. Each diagonal measures
- 19
- 14
- 28
- 20
Diagonal = √(12²+16²) = √400 = 20.
A rectangle has length 9 and width 12. Each diagonal measures
- 15
- 10
- 14
- 21
Diagonal = √(9²+12²) = √225 = 15.
A rectangle has length 18 and width 24. Each diagonal measures
- 29
- 21
- 42
- 30
Diagonal = √(18²+24²) = √900 = 30.
A rectangle has length 3 and width 4. Each diagonal measures
- 3
- 5
- 7
- 4
Diagonal = √(3²+4²) = √25 = 5.
A convex polygon has 6 sides. The total number of diagonals is
- 6
- 11
- 9
- 15
Diagonals = n(n−3)/2 = 6(6−3)/2 = 9.
A convex polygon has 10 sides. The total number of diagonals is
- 35
- 45
- 37
- 10
Diagonals = n(n−3)/2 = 10(10−3)/2 = 35.
A convex polygon has 8 sides. The total number of diagonals is
- 20
- 28
- 8
- 22
Diagonals = n(n−3)/2 = 8(8−3)/2 = 20.
A convex polygon has 12 sides. The total number of diagonals is
- 56
- 12
- 66
- 54
Diagonals = n(n−3)/2 = 12(12−3)/2 = 54.
A convex polygon has 7 sides. The total number of diagonals is
- 16
- 21
- 7
- 14
Diagonals = n(n−3)/2 = 7(7−3)/2 = 14.

Unit 8: Circles (40)

An inscribed angle intercepts an arc of 200°. The angle measures
- 110°
- 200°
- 190°
- 100°
Inscribed angle = ½ its intercepted arc = ½(200) = 100°.
An inscribed angle intercepts an arc of 152°. The angle measures
- 152°
- 76°
- 304°
- 86°
Inscribed angle = ½ its intercepted arc = ½(152) = 76°.
An inscribed angle intercepts an arc of 320°. The angle measures
- 160°
- 320°
- 170°
- 310°
Inscribed angle = ½ its intercepted arc = ½(320) = 160°.
An inscribed angle intercepts an arc of 60°. The angle measures
- 30°
- 40°
- 120°
- 60°
Inscribed angle = ½ its intercepted arc = ½(60) = 30°.
An inscribed angle intercepts an arc of 324°. The angle measures
- 324°
- 172°
- 314°
- 162°
Inscribed angle = ½ its intercepted arc = ½(324) = 162°.
Two chords intersect inside a circle, forming an angle that intercepts arcs of 154° and 69°. The angle measures
- 111.5°
- 223°
- 154°
- 69°
Chord-chord angle = ½(sum of intercepted arcs) = ½(154+69) = 111.5°.
Two chords intersect inside a circle, forming an angle that intercepts arcs of 111° and 63°. The angle measures
- 111°
- 63°
- 87°
- 174°
Chord-chord angle = ½(sum of intercepted arcs) = ½(111+63) = 87°.
Two chords intersect inside a circle, forming an angle that intercepts arcs of 49° and 160°. The angle measures
- 104.5°
- 209°
- 49°
- 160°
Chord-chord angle = ½(sum of intercepted arcs) = ½(49+160) = 104.5°.
Two chords intersect inside a circle, forming an angle that intercepts arcs of 66° and 87°. The angle measures
- 76.5°
- 66°
- 153°
- 87°
Chord-chord angle = ½(sum of intercepted arcs) = ½(66+87) = 76.5°.
Two chords intersect inside a circle, forming an angle that intercepts arcs of 94° and 144°. The angle measures
- 119°
- 144°
- 238°
- 94°
Chord-chord angle = ½(sum of intercepted arcs) = ½(94+144) = 119°.
Two secants meet outside a circle, intercepting arcs of 114° and 62°. The angle formed measures
- 34°
- 26°
- 52°
- 62°
Exterior angle = ½(far arc − near arc) = ½(114−62) = 26°.
Two secants meet outside a circle, intercepting arcs of 156° and 24°. The angle formed measures
- 74°
- 24°
- 132°
- 66°
Exterior angle = ½(far arc − near arc) = ½(156−24) = 66°.
Two secants meet outside a circle, intercepting arcs of 152° and 27°. The angle formed measures
- 62.5°
- 27°
- 70.5°
- 125°
Exterior angle = ½(far arc − near arc) = ½(152−27) = 62.5°.
Two secants meet outside a circle, intercepting arcs of 298° and 16°. The angle formed measures
- 16°
- 141°
- 282°
- 149°
Exterior angle = ½(far arc − near arc) = ½(298−16) = 141°.
Two secants meet outside a circle, intercepting arcs of 102° and 30°. The angle formed measures
- 44°
- 30°
- 36°
- 72°
Exterior angle = ½(far arc − near arc) = ½(102−30) = 36°.
A tangent and a chord meet at the point of tangency, forming an angle that intercepts an arc of 216°. The angle measures
- 123°
- 108°
- 144°
- 216°
Tangent-chord angle = ½ intercepted arc = ½(216) = 108°.
A tangent and a chord meet at the point of tangency, forming an angle that intercepts an arc of 20°. The angle measures
- 20°
- 10°
- 25°
- 340°
Tangent-chord angle = ½ intercepted arc = ½(20) = 10°.
A tangent and a chord meet at the point of tangency, forming an angle that intercepts an arc of 144°. The angle measures
- 216°
- 72°
- 87°
- 144°
Tangent-chord angle = ½ intercepted arc = ½(144) = 72°.
A tangent and a chord meet at the point of tangency, forming an angle that intercepts an arc of 308°. The angle measures
- 52°
- 154°
- 308°
- 169°
Tangent-chord angle = ½ intercepted arc = ½(308) = 154°.
A tangent and a chord meet at the point of tangency, forming an angle that intercepts an arc of 148°. The angle measures
- 74°
- 148°
- 212°
- 89°
Tangent-chord angle = ½ intercepted arc = ½(148) = 74°.
Two chords intersect inside a circle. One chord is split into segments 8 and 7; the other into 14 and x. Find x.
- 6
- 1
- 14
- 4
Chord segments: 8·7 = 14·x → x = 56/14 = 4.
Two chords intersect inside a circle. One chord is split into segments 6 and 12; the other into 8 and x. Find x.
- 11
- 8
- 9
- 10
Chord segments: 6·12 = 8·x → x = 72/8 = 9.
Two chords intersect inside a circle. One chord is split into segments 10 and 14; the other into 7 and x. Find x.
- 22
- 7
- 17
- 20
Chord segments: 10·14 = 7·x → x = 140/7 = 20.
Two chords intersect inside a circle. One chord is split into segments 15 and 5; the other into 3 and x. Find x.
- 17
- 27
- 25
- 3
Chord segments: 15·5 = 3·x → x = 75/3 = 25.
The circle (x − 3)² + (y − 5)² = 25 has radius
- 25
- 6
- 4
- 5
r = √25 = 5.
The circle (x − 6)² + (y − -1)² = 64 has radius
- 7
- 8
- 9
- 64
r = √64 = 8.
The circle (x − 3)² + (y − -1)² = 64 has radius
- 9
- 7
- 8
- 64
r = √64 = 8.
The circle (x − 0)² + (y − 1)² = 49 has radius
- 49
- 7
- 8
- 6
r = √49 = 7.
A circle has radius 12 and a central angle of 120°. The length of the intercepted arc, in terms of π, is
- 8π
- 10π
- 12π
- 24π
Arc length = (central/360)·2πr = (120/360)·2π(12) = 8π.
A circle has radius 10 and a central angle of 45°. The length of the intercepted arc, in terms of π, is
- 10π
- 7π/2
- 5π/2
- 20π
Arc length = (central/360)·2πr = (45/360)·2π(10) = 5π/2.
A circle has radius 4 and a central angle of 45°. The length of the intercepted arc, in terms of π, is
- 3π
- 4π
- 1π
- 8π
Arc length = (central/360)·2πr = (45/360)·2π(4) = 1π.
A circle has radius 8 and a central angle of 30°. The length of the intercepted arc, in terms of π, is
- 6π/3
- 4π/3
- 16π
- 8π
Arc length = (central/360)·2πr = (30/360)·2π(8) = 4π/3.
A circle has radius 12. The area of a sector with central angle 120° in terms of π is
- 24π
- 48π
- 144π
- 51π
Sector area = (central/360)·πr² = (120/360)·π(144) = 48π.
A circle has radius 9. The area of a sector with central angle 135° in terms of π is
- 243π/8
- 246π/8
- 81π
- 18π
Sector area = (central/360)·πr² = (135/360)·π(81) = 243π/8.
A circle has radius 6. The area of a sector with central angle 150° in terms of π is
- 36π
- 12π
- 18π
- 15π
Sector area = (central/360)·πr² = (150/360)·π(36) = 15π.
A circle has radius 12. The area of a sector with central angle 135° in terms of π is
- 54π
- 57π
- 144π
- 24π
Sector area = (central/360)·πr² = (135/360)·π(144) = 54π.
A tangent and a chord meet at the point of tangency, forming an angle that intercepts an arc of 68°. The angle measures
- 49°
- 68°
- 292°
- 34°
Tangent-chord angle = ½ intercepted arc = ½(68) = 34°.
The circle (x − -5)² + (y − -6)² = 64 has radius
- 8
- 7
- 64
- 9
r = √64 = 8.
Two chords intersect inside a circle. One chord is split into segments 15 and 8; the other into 2 and x. Find x.
- 62
- 2
- 21
- 60
Chord segments: 15·8 = 2·x → x = 120/2 = 60.
Two secants meet outside a circle, intercepting arcs of 280° and 90°. The angle formed measures
- 95°
- 90°
- 190°
- 103°
Exterior angle = ½(far arc − near arc) = ½(280−90) = 95°.

Unit 9: Coordinate Geometry (40)

The distance between (-7, 1) and (0, 25) is
- 25
- 27
- 17
- 31
√((7)² + (24)²) = √625 = 25.
The distance between (-8, -4) and (12, 17) is
- 29
- 1
- 31
- 41
√((20)² + (21)²) = √841 = 29.
The distance between (-5, 3) and (3, 9) is
- 2
- 14
- 10
- 12
√((8)² + (6)²) = √100 = 10.
The distance between (-9, 2) and (-1, 8) is
- 14
- 10
- 12
- 2
√((8)² + (6)²) = √100 = 10.
The distance between (5, -5) and (14, 7) is
- 3
- 21
- 15
- 17
√((9)² + (12)²) = √225 = 15.
The midpoint of (3, -12) and (9, 2) is
- (-5, 6)
- (6, -5)
- (7, -5)
- (12, -10)
Average the x's and the y's.
The midpoint of (-10, -10) and (-6, 12) is
- (1, -8)
- (-16, 2)
- (-8, 1)
- (-7, 1)
Average the x's and the y's.
The midpoint of (-3, -5) and (-1, -9) is
- (-2, -7)
- (-4, -14)
- (-7, -2)
- (-1, -7)
Average the x's and the y's.
The midpoint of (-10, 9) and (-4, 7) is
- (-7, 8)
- (8, -7)
- (-6, 8)
- (-14, 16)
Average the x's and the y's.
The midpoint of (-7, -6) and (-3, 10) is
- (-4, 2)
- (2, -5)
- (-5, 2)
- (-10, 4)
Average the x's and the y's.
The slope of the line through (-1, 5) and (-4, 6) is
- 1/-2
- -1/-3
- 1/-3
- -3/1
slope = Δy/Δx = (1)/(-3) = 1/-3.
The slope of the line through (8, 8) and (-7, -5) is
- --13/-15
- -15/-13
- -13/-14
- -13/-15
slope = Δy/Δx = (-13)/(-15) = -13/-15.
The slope of the line through (7, -5) and (1, 2) is
- -7/-6
- 7/-5
- 7/-6
- -6/7
slope = Δy/Δx = (7)/(-6) = 7/-6.
The slope of the line through (-7, -2) and (-5, -1) is
- -1/2
- 2/1
- 1/3
- 1/2
slope = Δy/Δx = (1)/(2) = 1/2.
The slope of the line through (0, -2) and (-4, 0) is
- -1/-2
- 1/-2
- 1/-1
- -2/1
slope = Δy/Δx = (2)/(-4) = 1/-2.
Point P divides the segment from A(3, -3) to B(8, 7) in the ratio 2:3. P is
- (6, 1)
- (5, 2)
- (1, 5)
- (5, 1)
P = A + 2/5(B−A) = (3,-3) + 2/5(5,10) = (5, 1).
Point P divides the segment from A(-2, -3) to B(13, 9) in the ratio 1:2. P is
- (3, 1)
- (5, 3)
- (1, 3)
- (4, 1)
P = A + 1/3(B−A) = (-2,-3) + 1/3(15,12) = (3, 1).
Point P divides the segment from A(-7, -1) to B(3, 14) in the ratio 1:4. P is
- (2, -5)
- (-5, 2)
- (-2, 6)
- (-4, 2)
P = A + 1/5(B−A) = (-7,-1) + 1/5(10,15) = (-5, 2).
Point P divides the segment from A(-5, 3) to B(5, 18) in the ratio 1:4. P is
- (6, -3)
- (-2, 6)
- (0, 10)
- (-3, 6)
P = A + 1/5(B−A) = (-5,3) + 1/5(10,15) = (-3, 6).
Point P divides the segment from A(-9, -5) to B(0, 7) in the ratio 1:2. P is
- (-5, -1)
- (-5, 1)
- (-1, -6)
- (-6, -1)
P = A + 1/3(B−A) = (-9,-5) + 1/3(9,12) = (-6, -1).
A line contains the points (-2, -5) and (1, 0). An equation of a line PERPENDICULAR to it through (-5, -4) is
- y + -4 = -3/5(x + -5)
- y − -5 = -3/5(x − -4)
- y − -4 = -3/5(x − -5)
- y − -4 = 5/3(x − -5)
Original slope = 5/3. Perpendicular slope = −3/5. Through (-5,-4): y − -4 = -3/5(x − -5).
A line contains the points (0, 2) and (3, 7). An equation of a line PERPENDICULAR to it through (0, -3) is
- y − -3 = -3/5(x − 0)
- y + -3 = -3/5(x + 0)
- y − -3 = 5/3(x − 0)
- y − 0 = -3/5(x − -3)
Original slope = 5/3. Perpendicular slope = −3/5. Through (0,-3): y − -3 = -3/5(x − 0).
A line contains the points (3, 3) and (6, 7). An equation of a line PERPENDICULAR to it through (2, 1) is
- y + 1 = -3/4(x + 2)
- y − 2 = -3/4(x − 1)
- y − 1 = -3/4(x − 2)
- y − 1 = 4/3(x − 2)
Original slope = 4/3. Perpendicular slope = −3/4. Through (2,1): y − 1 = -3/4(x − 2).
A line contains the points (6, 4) and (8, 9). An equation of a line PERPENDICULAR to it through (-4, 4) is
- y − 4 = 5/2(x − -4)
- y − -4 = -2/5(x − 4)
- y + 4 = -2/5(x + -4)
- y − 4 = -2/5(x − -4)
Original slope = 5/2. Perpendicular slope = −2/5. Through (-4,4): y − 4 = -2/5(x − -4).
A line contains the points (0, -5) and (2, 0). An equation of a line PERPENDICULAR to it through (-6, -2) is
- y + -2 = -2/5(x + -6)
- y − -2 = 5/2(x − -6)
- y − -2 = -2/5(x − -6)
- y − -6 = -2/5(x − -2)
Original slope = 5/2. Perpendicular slope = −2/5. Through (-6,-2): y − -2 = -2/5(x − -6).
Quadrilateral ABCD has AB ∥ CD, AB ≅ CD, and diagonal AC ⊥ BD. It must be a
- trapezoid
- rhombus
- rectangle
- kite
One pair of opposite sides parallel AND congruent makes it a parallelogram; perpendicular diagonals in a parallelogram make it a rhombus.
A circle has a diameter with endpoints (-2, 3) and (6, 9). The radius is
- 4
- 7
- 5
- 10
Diameter = √(8²+6²) = 10 → radius = 5.
A circle has a diameter with endpoints (-7, -7) and (-1, 1). The radius is
- 4
- 10
- 7
- 5
Diameter = √(6²+8²) = 10 → radius = 5.
A circle has a diameter with endpoints (-3, -4) and (3, 4). The radius is
- 10
- 5
- 7
- 4
Diameter = √(6²+8²) = 10 → radius = 5.
A circle has a diameter with endpoints (0, 0) and (6, 8). The radius is
- 5
- 7
- 10
- 4
Diameter = √(6²+8²) = 10 → radius = 5.
A circle has a diameter with endpoints (-1, -3) and (5, 5). The radius is
- 4
- 5
- 7
- 10
Diameter = √(6²+8²) = 10 → radius = 5.
The slope of the line through (0, -4) and (-7, 7) is
- 11/-6
- -7/11
- 11/-7
- -11/-7
slope = Δy/Δx = (11)/(-7) = 11/-7.
The slope of the line through (-8, -7) and (0, -5) is
- 4/1
- -1/4
- 1/4
- 1/5
slope = Δy/Δx = (2)/(8) = 1/4.
The distance between (-1, 3) and (11, 12) is
- 21
- 17
- 3
- 15
√((12)² + (9)²) = √225 = 15.
The distance between (3, -8) and (12, 4) is
- 3
- 17
- 21
- 15
√((9)² + (12)²) = √225 = 15.
A line contains the points (6, 5) and (7, 7). An equation of a line PERPENDICULAR to it through (1, -1) is
- y − -1 = -1/2(x − 1)
- y − 1 = -1/2(x − -1)
- y + -1 = -1/2(x + 1)
- y − -1 = 2/1(x − 1)
Original slope = 2/1. Perpendicular slope = −1/2. Through (1,-1): y − -1 = -1/2(x − 1).
A line contains the points (6, 1) and (9, 5). An equation of a line PERPENDICULAR to it through (-3, 1) is
- y − -3 = -3/4(x − 1)
- y + 1 = -3/4(x + -3)
- y − 1 = -3/4(x − -3)
- y − 1 = 4/3(x − -3)
Original slope = 4/3. Perpendicular slope = −3/4. Through (-3,1): y − 1 = -3/4(x − -3).
A line contains the points (3, 6) and (7, 9). An equation of a line PERPENDICULAR to it through (-5, -6) is
- y − -5 = -4/3(x − -6)
- y + -6 = -4/3(x + -5)
- y − -6 = -4/3(x − -5)
- y − -6 = 3/4(x − -5)
Original slope = 3/4. Perpendicular slope = −4/3. Through (-5,-6): y − -6 = -4/3(x − -5).
The slope of the line through (7, -5) and (3, -8) is
- --3/-4
- -3/-3
- -4/-3
- -3/-4
slope = Δy/Δx = (-3)/(-4) = -3/-4.
A line contains the points (5, 2) and (6, 4). An equation of a line PERPENDICULAR to it through (4, 3) is
- y − 3 = 2/1(x − 4)
- y + 3 = -1/2(x + 4)
- y − 3 = -1/2(x − 4)
- y − 4 = -1/2(x − 3)
Original slope = 2/1. Perpendicular slope = −1/2. Through (4,3): y − 3 = -1/2(x − 4).

Unit 10: Transformations & Constructions (40)

Reflecting (-3, -9) over the line y = x gives
- (-9, -3)
- (9, 3)
- (3, -9)
- (-3, 9)
Over y = x: swap the coordinates.
Reflecting (9, -9) over the line y = x gives
- (9, -9)
- (-9, 9)
- (-9, -9)
- (9, 9)
Over y = x: swap the coordinates.
Reflecting (-7, 4) over the line y = x gives
- (4, -7)
- (7, 4)
- (-7, -4)
- (-4, 7)
Over y = x: swap the coordinates.
Reflecting (-3, -1) over the line y = x gives
- (1, 3)
- (-3, 1)
- (3, -1)
- (-1, -3)
Over y = x: swap the coordinates.
Reflecting (10, 1) over the line y = x gives
- (-10, 1)
- (-1, -10)
- (10, -1)
- (1, 10)
Over y = x: swap the coordinates.
Rotating (5, -6) 90° counterclockwise about the origin gives
- (-5, 6)
- (5, -6)
- (6, 5)
- (-6, -5)
90° CCW: (x, y) → (−y, x).
Rotating (-5, -5) 90° counterclockwise about the origin gives
- (-5, -5)
- (5, -5)
- (5, 5)
- (-5, 5)
90° CCW: (x, y) → (−y, x).
Rotating (1, 6) 90° counterclockwise about the origin gives
- (6, -1)
- (-6, 1)
- (1, 6)
- (-1, -6)
90° CCW: (x, y) → (−y, x).
Rotating (10, -8) 90° counterclockwise about the origin gives
- (-10, 8)
- (10, -8)
- (8, 10)
- (-8, -10)
90° CCW: (x, y) → (−y, x).
Rotating (-3, 4) 90° counterclockwise about the origin gives
- (3, -4)
- (-4, -3)
- (-3, 4)
- (4, 3)
90° CCW: (x, y) → (−y, x).
Rotating (9, 1) 180° about the origin gives
- (1, 9)
- (9, -1)
- (-9, -1)
- (-9, 1)
180°: (x, y) → (−x, −y).
Rotating (-6, 3) 180° about the origin gives
- (6, 3)
- (6, -3)
- (-6, -3)
- (3, -6)
180°: (x, y) → (−x, −y).
Rotating (-5, -8) 180° about the origin gives
- (5, 8)
- (-5, 8)
- (-8, -5)
- (5, -8)
180°: (x, y) → (−x, −y).
Rotating (10, 9) 180° about the origin gives
- (-10, 9)
- (9, 10)
- (-10, -9)
- (10, -9)
180°: (x, y) → (−x, −y).
Rotating (9, -4) 180° about the origin gives
- (-9, -4)
- (-9, 4)
- (9, 4)
- (-4, 9)
180°: (x, y) → (−x, −y).
Point (6, 3) is translated by (x+-3, y+-5), then by (x+-1, y+6). The final image is
- (2, 4)
- (3, 4)
- (3, -2)
- (5, 9)
Apply both shifts: (6+-3+-1, 3+-5+6) = (2, 4).
Point (-6, -2) is translated by (x+-4, y+5), then by (x+-3, y+-5). The final image is
- (-12, -2)
- (-10, 3)
- (-13, -2)
- (-9, -7)
Apply both shifts: (-6+-4+-3, -2+5+-5) = (-13, -2).
Point (2, -5) is translated by (x+-6, y+-1), then by (x+-2, y+-6). The final image is
- (-6, -12)
- (-5, -12)
- (-4, -6)
- (0, -11)
Apply both shifts: (2+-6+-2, -5+-1+-6) = (-6, -12).
Point (7, 6) is translated by (x+0, y+6), then by (x+5, y+0). The final image is
- (7, 12)
- (12, 12)
- (13, 12)
- (12, 6)
Apply both shifts: (7+0+5, 6+6+0) = (12, 12).
Point (-7, 1) is translated by (x+-1, y+6), then by (x+4, y+6). The final image is
- (-3, 7)
- (-4, 13)
- (-8, 7)
- (-3, 13)
Apply both shifts: (-7+-1+4, 1+6+6) = (-4, 13).
Dilating (-7, -9) by scale factor 4 centered at the origin gives
- (-28, -36)
- (-3, -5)
- (-26, -36)
- (-36, -28)
Multiply both coordinates by 4: (-7×4, -9×4) = (-28, -36).
Dilating (-5, -2) by scale factor 3 centered at the origin gives
- (-13, -6)
- (-6, -15)
- (-2, 1)
- (-15, -6)
Multiply both coordinates by 3: (-5×3, -2×3) = (-15, -6).
Dilating (0, 8) by scale factor 2 centered at the origin gives
- (0, 16)
- (2, 16)
- (2, 10)
- (16, 0)
Multiply both coordinates by 2: (0×2, 8×2) = (0, 16).
Dilating (2, -6) by scale factor 3 centered at the origin gives
- (5, -3)
- (-18, 6)
- (8, -18)
- (6, -18)
Multiply both coordinates by 3: (2×3, -6×3) = (6, -18).
Dilating (-3, -2) by scale factor 3 centered at the origin gives
- (-6, -9)
- (-9, -6)
- (0, 1)
- (-7, -6)
Multiply both coordinates by 3: (-3×3, -2×3) = (-9, -6).
Reflecting (-4, -5) over the x-axis gives
- (-4, -5)
- (-5, -4)
- (-4, 5)
- (4, 5)
Over the x-axis: y negates.
Reflecting (-1, -6) over the x-axis gives
- (-6, -1)
- (1, 6)
- (-1, 6)
- (-1, -6)
Over the x-axis: y negates.
Reflecting (10, 4) over the x-axis gives
- (4, 10)
- (10, -4)
- (10, 4)
- (-10, -4)
Over the x-axis: y negates.
Reflecting (1, -9) over the y-axis gives
- (1, -9)
- (-1, 9)
- (-1, -9)
- (-9, 1)
Over the y-axis: x negates.
Reflecting (7, 6) over the x-axis gives
- (7, -6)
- (-7, -6)
- (6, 7)
- (7, 6)
Over the x-axis: y negates.
How many lines of reflectional symmetry does a isosceles triangle (non-equilateral) have?
- 6
- 2
- 5
- 1
A isosceles triangle (non-equilateral) has 1 line(s) of reflectional symmetry.
How many lines of reflectional symmetry does a square have?
- 1
- 3
- 4
- 6
A square has 4 line(s) of reflectional symmetry.
How many lines of reflectional symmetry does a regular pentagon have?
- 0
- 5
- 4
- 1
A regular pentagon has 5 line(s) of reflectional symmetry.
How many lines of reflectional symmetry does a regular octagon have?
- 8
- 5
- 0
- 2
A regular octagon has 8 line(s) of reflectional symmetry.
How many lines of reflectional symmetry does a regular hexagon have?
- 3
- 6
- 4
- 2
A regular hexagon has 6 line(s) of reflectional symmetry.
A regular 12-gon carries onto itself after a minimum rotation of
- 40°
- 180°
- 25°
- 30°
A regular n-gon maps onto itself every 360/n°: 360/12 = 30°.
A regular 8-gon carries onto itself after a minimum rotation of
- 45°
- 40°
- 55°
- 180°
A regular n-gon maps onto itself every 360/n°: 360/8 = 45°.
A regular 9-gon carries onto itself after a minimum rotation of
- 180°
- 40°
- 35°
- 50°
A regular n-gon maps onto itself every 360/n°: 360/9 = 40°.
A regular 10-gon carries onto itself after a minimum rotation of
- 180°
- 46°
- 36°
- 31°
A regular n-gon maps onto itself every 360/n°: 360/10 = 36°.
A regular 3-gon carries onto itself after a minimum rotation of
- 130°
- 115°
- 180°
- 120°
A regular n-gon maps onto itself every 360/n°: 360/3 = 120°.

Unit 11: 3D Solids: Surface Area, Volume & Density (40)

A cylinder has radius 6 and height 4. Its volume is
- 144π
- 48π
- 150π
- 24π
V = πr²h = π(6²)(4) = 144π.
A cylinder has radius 12 and height 5. Its volume is
- 720π
- 120π
- 60π
- 732π
V = πr²h = π(12²)(5) = 720π.
A cylinder has radius 9 and height 8. Its volume is
- 648π
- 72π
- 144π
- 657π
V = πr²h = π(9²)(8) = 648π.
A cylinder has radius 5 and height 13. Its volume is
- 325π
- 65π
- 330π
- 130π
V = πr²h = π(5²)(13) = 325π.
A cylinder has radius 4 and height 15. Its volume is
- 120π
- 240π
- 60π
- 244π
V = πr²h = π(4²)(15) = 240π.
A sphere has radius 9. Its volume is
- 972π
- 729π
- 981π
- 324π
V = 4/3πr³ = 4/3π(9³) = 972π.
A sphere has radius 6. Its volume is
- 294π
- 216π
- 144π
- 288π
V = 4/3πr³ = 4/3π(6³) = 288π.
A sphere has radius 15. Its volume is
- 4515π
- 900π
- 4500π
- 3375π
V = 4/3πr³ = 4/3π(15³) = 4500π.
A sphere has radius 12. Its volume is
- 2304π
- 1728π
- 576π
- 2316π
V = 4/3πr³ = 4/3π(12³) = 2304π.
A rectangular prism measures 7 by 6 by 4 cm. If the material has density 0.5 g/cm³, its mass is
- 168 g
- 42 g
- 94 g
- 84 g
V = 7×6×4 = 168 cm³. Mass = volume × density = 168 × 0.5 = 84 g.
A rectangular prism measures 5 by 4 by 3 cm. If the material has density 1.2 g/cm³, its mass is
- 36 g
- 72 g
- 60 g
- 82 g
V = 5×4×3 = 60 cm³. Mass = volume × density = 60 × 1.2 = 72 g.
A rectangular prism measures 5 by 8 by 7 cm. If the material has density 1.2 g/cm³, its mass is
- 168 g
- 346 g
- 336 g
- 280 g
V = 5×8×7 = 280 cm³. Mass = volume × density = 280 × 1.2 = 336 g.
A rectangular prism measures 9 by 4 by 7 cm. If the material has density 2.5 g/cm³, its mass is
- 315 g
- 630 g
- 252 g
- 640 g
V = 9×4×7 = 252 cm³. Mass = volume × density = 252 × 2.5 = 630 g.
A rectangular prism measures 6 by 6 by 2 cm. If the material has density 2.5 g/cm³, its mass is
- 190 g
- 90 g
- 72 g
- 180 g
V = 6×6×2 = 72 cm³. Mass = volume × density = 72 × 2.5 = 180 g.
Two similar solids have a linear scale factor of 1:4. If the smaller solid's volume is 27, the larger's volume is
- 1738
- 1728
- 27
- 108
Volume ratio = (b/a)³ = (4/1)³. 27 × (4³/1³) = 1728.
Two similar solids have a linear scale factor of 1:5. If the smaller solid's volume is 125, the larger's volume is
- 15625
- 15635
- 625
- 125
Volume ratio = (b/a)³ = (5/1)³. 125 × (5³/1³) = 15625.
Two similar solids have a linear scale factor of 2:4. If the smaller solid's volume is 125, the larger's volume is
- 1000
- 125
- 1010
- 250
Volume ratio = (b/a)³ = (4/2)³. 125 × (4³/2³) = 1000.
Two similar solids have a linear scale factor of 1:5. If the smaller solid's volume is 27, the larger's volume is
- 3375
- 27
- 3385
- 135
Volume ratio = (b/a)³ = (5/1)³. 27 × (5³/1³) = 3375.
Two similar solids have a linear scale factor of 4:1. If the smaller solid's volume is 64, the larger's volume is
- 64
- 1
- 11
- 16
Volume ratio = (b/a)³ = (1/4)³. 64 × (1³/4³) = 1.
The cross-section of a cylinder, cut perpendicular to its base through the center is
- a rectangle
- a circle
- a triangle
- an ellipse
Slicing a cylinder, cut perpendicular to its base through the center produces a rectangle.
The cross-section of a rectangular pyramid, cut parallel to its base is
- a rectangle
- an ellipse
- a circle
- a triangle
Slicing a rectangular pyramid, cut parallel to its base produces a rectangle.
The cross-section of a cone, cut parallel to its base is
- an ellipse
- a rectangle
- a triangle
- a circle
Slicing a cone, cut parallel to its base produces a circle.
The cross-section of a sphere, cut through its center is
- a circle
- a triangle
- a rectangle
- an ellipse
Slicing a sphere, cut through its center produces a circle.
The cross-section of a cylinder, cut parallel to its base is
- a circle
- a triangle
- a rectangle
- an ellipse
Slicing a cylinder, cut parallel to its base produces a circle.
A cylinder has radius 8 and height 12. Its total surface area is
- 320π
- 768π
- 192π
- 328π
SA = 2πr² + 2πrh = 2π(8²) + 2π(8)(12) = 320π.
A cylinder has radius 5 and height 12. Its total surface area is
- 175π
- 120π
- 300π
- 170π
SA = 2πr² + 2πrh = 2π(5²) + 2π(5)(12) = 170π.
A cylinder has radius 5 and height 3. Its total surface area is
- 30π
- 85π
- 75π
- 80π
SA = 2πr² + 2πrh = 2π(5²) + 2π(5)(3) = 80π.
A cylinder has radius 8 and height 9. Its total surface area is
- 144π
- 280π
- 272π
- 576π
SA = 2πr² + 2πrh = 2π(8²) + 2π(8)(9) = 272π.
A cylinder has radius 7 and height 15. Its total surface area is
- 210π
- 315π
- 308π
- 735π
SA = 2πr² + 2πrh = 2π(7²) + 2π(7)(15) = 308π.
A cylinder has radius 3 and height 4. Its total surface area is
- 45π
- 36π
- 24π
- 42π
SA = 2πr² + 2πrh = 2π(3²) + 2π(3)(4) = 42π.
A cylinder has radius 9 and height 6. Its volume is
- 495π
- 54π
- 108π
- 486π
V = πr²h = π(9²)(6) = 486π.
A cylinder has radius 6 and height 9. Its volume is
- 54π
- 108π
- 330π
- 324π
V = πr²h = π(6²)(9) = 324π.
A cylinder has radius 8 and height 11. Its total surface area is
- 304π
- 312π
- 176π
- 704π
SA = 2πr² + 2πrh = 2π(8²) + 2π(8)(11) = 304π.
A cylinder has radius 5 and height 11. Its volume is
- 55π
- 280π
- 110π
- 275π
V = πr²h = π(5²)(11) = 275π.
A rectangular prism measures 6 by 2 by 5 cm. If the material has density 1.2 g/cm³, its mass is
- 36 g
- 60 g
- 72 g
- 82 g
V = 6×2×5 = 60 cm³. Mass = volume × density = 60 × 1.2 = 72 g.
Two similar solids have a linear scale factor of 1:4. If the smaller solid's volume is 10, the larger's volume is
- 640
- 40
- 10
- 650
Volume ratio = (b/a)³ = (4/1)³. 10 × (4³/1³) = 640.
A rectangular prism measures 2 by 3 by 4 cm. If the material has density 0.5 g/cm³, its mass is
- 12 g
- 24 g
- 6 g
- 22 g
V = 2×3×4 = 24 cm³. Mass = volume × density = 24 × 0.5 = 12 g.
A cylinder has radius 4 and height 6. Its volume is
- 100π
- 48π
- 96π
- 24π
V = πr²h = π(4²)(6) = 96π.
A cylinder has radius 4 and height 5. Its total surface area is
- 40π
- 80π
- 76π
- 72π
SA = 2πr² + 2πrh = 2π(4²) + 2π(4)(5) = 72π.
A cylinder has radius 9 and height 9. Its volume is
- 162π
- 81π
- 729π
- 738π
V = πr²h = π(9²)(9) = 729π.

Situation	Rule
Central angle	= intercepted arc
Inscribed angle	= ½ intercepted arc
Inscribed in a semicircle	= 90° (subtends a diameter)
Tangent–chord angle	= ½ intercepted arc
Cyclic quadrilateral	opposite angles are supplementary
Two chords meet inside	angle = ½(sum of arcs); (part)(part) = (part)(part)
Two secants meet outside	angle = ½(difference of arcs); (whole)(ext) = (whole)(ext)
Tangent + secant	tangent² = (whole secant)(external part)
Tangent & radius	meet at 90° at the point of tangency
⊥ from center to a chord	bisects the chord and its arc

Vertex location	Angle equals…
At the CENTER (central angle)	the whole intercepted arc
ON the circle (inscribed angle)	½ the intercepted arc
INSIDE (two chords cross)	½ the SUM of the two arcs
OUTSIDE (secants/tangents)	½ the DIFFERENCE of the arcs
Inscribed in a semicircle	90° (right angle)

Shape	Key diagonal / side facts
Parallelogram	opp sides ∥ & ≅; diagonals bisect each other
Rectangle	parallelogram + 4 right angles + ≅ diagonals
Rhombus	parallelogram + 4 ≅ sides + ⊥ diagonals
Square	rectangle AND rhombus (all properties)
Isosceles Trapezoid	1 pair ∥ sides; ≅ legs; ≅ diagonals; ≅ base angles

Geometry Study Guide — EXPANDED EDITION

See what your study plan could look like

Unit 1: Foundations: Points, Lines & Angles

Unit 2: Parallel Lines & Transversals

Unit 3: Triangles: Angles & Inequalities

Unit 4: Congruent Triangles & Proofs

Unit 5: Similarity & Proportions

Unit 6: Right Triangles & Trigonometry

Unit 7: Polygons & Quadrilaterals

Unit 8: Circles

Unit 9: Coordinate Geometry

Unit 10: Transformations & Constructions

Unit 11: 3D Solids: Surface Area, Volume & Density

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Session Progress by Unit

Trigonometry

⭕ Circle Angles — “Where’s the vertex?”

Area, Circumference & Volume

Transformations

Lines & Coordinate Geometry

Triangle Congruence & Similarity

⬛ Quadrilaterals & Polygons

Fast Facts to Memorize Cold

Facts You Must Know Cold

⭕ Circle Segment & Angle Rules (must-know)

✏️ Constructions You Must Know

Area & Perimeter Formulas

Surface Area & Volume

Coordinate Geometry (memorize these 3)

Right Triangle Trig

Special Right Triangles

⭕ Circle Angle Rules (where is the vertex?)

Transformation Rules (about the origin)

⬡ Polygon Angle Sums

Triangle Congruence vs. Similarity

Quadrilateral Family Properties

Triangle	Side Ratio	Rule
45-45-90	1 : 1 : √2	hypotenuse = leg · √2
30-60-90	1 : √3 : 2	short leg : long leg : hypotenuse

Transformation	Rule (x, y) →	Size?
Translation	(x + a, y + b)	same
Reflect over x-axis	(x, −y)	same
Reflect over y-axis	(−x, y)	same
Reflect over y = x	(y, x)	same
Rotate 90° CCW	(−y, x)	same
Rotate 180°	(−x, −y)	same
Rotate 270° CCW	(y, −x)	same
Dilation (factor k)	(kx, ky)	CHANGES

Polygon	Sides (n)	Interior Sum (n−2)·180°	Each angle if regular
Triangle	3	180°	60°
Quadrilateral	4	360°	90°
Pentagon	5	540°	108°
Hexagon	6	720°	120°
Octagon	8	1080°	135°