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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
∠1∠2∠3∠4

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

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
132456∥∥

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

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
cabd

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

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).

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
BDEACBD/BA = BE/BC = DE/AC = k

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.

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
b (opp)a (adj)c (hyp)θ

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

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
ParallelogramRectangleRhombusSquare+ 4 right angles+ 4 ≅ sides

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

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
centercentralinscribed

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

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
(x₁,y₁)(x₂,y₂)run (Δx)rise (Δy)

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

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
P (x, y)(x, −y) over x-axis(−x, y) over y-axis(−x, −y) 180°

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

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
cone: ⅓πr²hcylinder: πr²h

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

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Unit 1: Foundations: Points, Lines & Angles

Complementary angles
Two angles whose measures add up to 90°.
Supplementary angles
Two angles whose measures add up to 180°.
Vertical angles
Opposite angles formed by two intersecting lines. They are always congruent (equal).
Linear pair
Two adjacent angles that form a straight line. They are supplementary (sum 180°).
Midpoint
The point that divides a segment into two congruent halves.
Angle bisector
A ray that divides an angle into two congruent angles.
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
Acute / Right / Obtuse
Acute < 90°, Right = 90°, Obtuse between 90° and 180°.

Unit 2: Parallel Lines & Transversals

Corresponding angles
Same position at each intersection of a transversal. Congruent when lines are parallel.
Alternate interior angles
Between two parallel lines on opposite sides of the transversal. Congruent.
Same-side interior angles
Between two parallel lines on the same side of the transversal. Supplementary (add to 180°).
Transversal
A line that crosses two or more other lines, creating angle pairs.
Parallel lines (∥)
Lines in a plane that never meet and have equal slopes.

Unit 3: Triangles: Angles & Inequalities

Triangle Angle-Sum
The three interior angles of a triangle add to 180°.
Exterior Angle Theorem
An exterior angle equals the sum of the two remote (non-adjacent) interior angles.
Isosceles Base Angles Theorem
If two sides of a triangle are congruent, the angles opposite them are congruent.
Triangle Inequality
The sum of any two sides of a triangle must be greater than the third side.
Median of a triangle
A segment from a vertex to the midpoint of the opposite side. The three meet at the centroid.
Centroid
Where the three medians meet; the balance point. Divides each median in a 2:1 ratio.
Altitude of a triangle
A perpendicular segment from a vertex to the opposite side. The three meet at the orthocenter.

Unit 4: Congruent Triangles & Proofs

Congruence shortcuts
SSS, SAS, ASA, AAS, HL prove triangles congruent. SSA and AAA do NOT.
SAS
Two sides and the INCLUDED angle (the angle between them) are congruent.
ASA vs AAS
ASA: side is BETWEEN the two angles. AAS: side is NOT between the two angles.
HL
For right triangles only: hypotenuse and one leg congruent proves congruence.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent — used AFTER proving congruence.
Reflexive Property
A segment or angle is congruent to itself (used for a shared side in a proof).

Unit 5: Similarity & Proportions

Similar figures (~)
Same shape, sizes proportional: equal angles, sides in a common ratio (scale factor k).
AA Similarity
Two pairs of congruent angles prove two triangles are similar.
Scale factor effects
Length scales by k, area by k², volume by k³.
Side-Splitter Theorem
A line parallel to one side of a triangle divides the other two sides proportionally.
Geometric mean (altitude)
Altitude to the hypotenuse = √(p·q), where p and q are the two hypotenuse segments.
Leg geometric-mean rule
In a right triangle, each leg² = (whole hypotenuse)(segment of hypotenuse next to that leg).

Unit 6: Right Triangles & Trigonometry

Pythagorean Theorem
In a right triangle, a² + b² = c² (c is the hypotenuse).
SOH-CAH-TOA
sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj.
Inverse trig
Use sin⁻¹, cos⁻¹, or tan⁻¹ to find an ANGLE from two known sides.
Co-function relationship
sin(x) = cos(90° − x): the sine of an angle equals the cosine of its complement.
45-45-90 triangle
Legs equal; hypotenuse = leg · √2. Side ratio 1 : 1 : √2.
30-60-90 triangle
Side ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).
Pythagorean triples
3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.
Cofunction identity
sin(x) = cos(90° − x). cos(a) = sin(b) exactly when a + b = 90°.

Unit 7: Polygons & Quadrilaterals

Interior angle sum
Sum of interior angles of an n-gon = (n − 2)·180°.
Exterior angle sum
The exterior angles of ANY polygon add to 360°.
Parallelogram properties
Opposite sides ≅ and ∥, opposite angles ≅, diagonals bisect each other.
Rectangle vs Rhombus
Rectangle adds congruent diagonals; rhombus adds perpendicular diagonals + 4 ≅ sides.
Square
A parallelogram that is both a rectangle and a rhombus — has all their properties.
Trapezoid
Exactly one pair of parallel sides. Isosceles trapezoid has ≅ legs and ≅ diagonals.
Trapezoid midsegment (median)
The median of a trapezoid = ½(base₁ + base₂) — the average of the two parallel sides.

Unit 8: Circles

Inscribed angle
Vertex ON the circle; equals HALF its intercepted arc.
Central angle
Vertex at the center; equals the FULL intercepted arc.
Angle in a semicircle
An inscribed angle that intercepts a diameter is a right angle (90°).
Two chords (inside angle)
Angle = half the SUM of the two intercepted arcs.
Two secants (outside angle)
Angle = half the DIFFERENCE of the intercepted arcs.
Circle equation
(x − h)² + (y − k)² = r², center (h, k), radius r.
Arc length / sector
Arc = (n/360)·2πr; Sector area = (n/360)·πr².
Tangent line
Touches a circle at one point and is perpendicular to the radius there.
Tangent–chord angle
An angle formed by a tangent and a chord at the point of tangency = ½ the intercepted arc.
Cyclic quadrilateral
If a quadrilateral is inscribed in a circle, its opposite angles are supplementary (add to 180°).
Secant–secant power
From an external point: (whole secant)(external part) = (whole secant)(external part).
Tangent–secant power
From an external point: tangent² = (whole secant)(external part).
Perpendicular from center
A radius or diameter perpendicular to a chord bisects the chord and its arc.
Parallel chords
Two parallel chords in a circle intercept congruent arcs between them.

Unit 9: Coordinate Geometry

Distance formula
d = √[(x₂−x₁)² + (y₂−y₁)²] — the Pythagorean Theorem on a grid.
Midpoint formula
M = ((x₁+x₂)/2 , (y₁+y₂)/2) — average the x's and the y's.
Slope formula
m = (y₂−y₁)/(x₂−x₁) = rise/run.
Parallel vs perpendicular slope
Parallel: same slope. Perpendicular: negative reciprocals (product −1).
Slope-intercept form
y = mx + b, where m is slope and b is the y-intercept.
Partition a segment
To divide A→B in ratio a:b, move a/(a+b) of the way from A toward B.

Unit 10: Transformations & Constructions

Rigid motion
Translation, reflection, or rotation — preserves size and shape (image is congruent).
Translation rule
(x, y) → (x + a, y + b): every point slides the same amount.
Reflection rules
x-axis: (x,−y); y-axis: (−x,y); line y = x: (y,x).
Rotation rules (origin)
90° CCW: (−y,x); 180°: (−x,−y); 270° CCW: (y,−x).
Dilation
(x,y)→(kx,ky) about the origin. Changes size; image is SIMILAR. Lengths ×k, area ×k².
Perpendicular bisector construction
Equal arcs from both endpoints; the line through the crossings is perpendicular at the midpoint.
Carry a polygon onto itself
A regular n-gon maps onto itself after a rotation of 360/n° (and multiples). Hexagon = 60°, octagon = 45°.
Dilating a line
A dilation keeps a line's slope. If the line passes through the center, it maps onto itself; otherwise it maps to a PARALLEL line.

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

Prism / Cylinder volume
V = Bh (base area × height). Cylinder: V = πr²h.
Pyramid / Cone volume
V = ⅓Bh. Cone: V = ⅓πr²h. (One third of the matching prism/cylinder.)
Sphere
Volume = 4⁄3 πr³; Surface area = 4πr².
Cylinder surface area
2πr² (two circles) + 2πrh (the wrap-around side).
Cross-section
The 2-D shape formed when a plane slices a solid.
Density
Density = mass ÷ volume. (Population density = people ÷ area.)
Cavalieri's Principle
If two solids have the same height and the same cross-sectional area at every level, they have equal volume.
Press 1–4 to answer · Enter for next
Try each problem on your own first — then reveal the solution one step at a time. Mark “Got it” to track your progress.

Unit 1: Foundations: Points, Lines & Angles

Linear pair — solve for x
∠AOB and ∠BOC form a linear pair. ∠AOB = (3x + 20)° and ∠BOC = (5x)°. Find x and both angles.
Complement and supplement
An angle measures 37°. Find its complement and its supplement.

Unit 2: Parallel Lines & Transversals

Parallel lines — alternate interior
Two parallel lines are cut by a transversal. Alternate interior angles measure (2x + 30)° and (4x − 10)°. Find x and the angle.
Same-side interior angles
Co-interior (same-side interior) angles measure x° and (x + 40)°. Find x.

Unit 3: Triangles: Angles & Inequalities

Triangle angles from a ratio
The angles of a triangle are in the ratio 2 : 3 : 5. Find all three angles.
Exterior angle theorem
An exterior angle of a triangle is (4x)°. Its two remote interior angles are 50° and (2x)°. Find x.

Unit 4: Congruent Triangles & Proofs

Pick the congruence shortcut
Two triangles share side BD. You're given AB ≅ CB and AD ≅ CD. Which shortcut proves △ABD ≅ △CBD?
Two-column proof (SAS)
Given: AB ≅ CB and BD bisects ∠ABC. Prove △ABD ≅ △CBD.

Unit 5: Similarity & Proportions

Find a side of a similar triangle
△ABC ~ △DEF. AB = 4 corresponds to DE = 10, and BC = 6 corresponds to EF = x. Find x.
Scale factor: length, area, volume
A model is built at scale factor k = 3. If the original has perimeter 8, area 5, and volume 2, find the model's perimeter, area, and volume.
Geometric mean (altitude)
The altitude to the hypotenuse of a right triangle splits it into segments of length 4 and 16. Find the altitude.

Unit 6: Right Triangles & Trigonometry

Pythagorean — find a leg
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
Trig — find a side
In a right triangle, an acute angle is 35° and the hypotenuse is 20. Find the side OPPOSITE the 35° angle (nearest tenth).
Trig — find an angle
A right triangle has the side opposite θ = 7 and the adjacent side = 24. Find θ (nearest degree).
Angle of elevation
You stand 60 ft from a flagpole. The top is at a 37° angle of elevation. How tall is the pole (nearest foot)?

Unit 7: Polygons & Quadrilaterals

Polygon — interior angle sum
Find the sum of the interior angles of an octagon (8 sides), and each angle if it is regular.
Find the number of sides
Each interior angle of a regular polygon is 150°. How many sides does it have?
Parallelogram algebra
In a parallelogram, two opposite angles are (3x)° and (x + 50)°. Find x.

Unit 8: Circles

Inscribed angle
An inscribed angle intercepts an arc of 140°. Find the inscribed angle.
Two chords — segment products
Two chords cross inside a circle. One is split into 3 and x; the other into 4 and 6. Find x.
Circle equation — complete the square
Find the center and radius of x² + y² − 4x + 6y − 12 = 0.
Arc length
A circle has radius 10. Find the length of an arc cut off by a 72° central angle (in terms of π).
Power of a point — secants
From an external point, two secants are drawn. Secant 1 has whole length 8 and external part 3. Secant 2 has external part 4; find its whole length.

Unit 9: Coordinate Geometry

Distance formula
Find the distance between A(2, 3) and B(7, 15).
Perpendicular line through a point
Write the equation of the line through (6, 1) perpendicular to y = 3x + 2.
Partition a segment
Find the point that divides the segment from A(2, 3) to B(12, 8) in the ratio 2 : 3.
Directed line segment / partition (Jan 2026 #16)
R(6, 6) and Z(−12, −3). Point A divides RZ so that RA:AZ = 5:4. Find A.

Unit 10: Transformations & Constructions

Rotate 90° counterclockwise
Rotate the point (3, −2) 90° counterclockwise about the origin.
Composition of transformations
Reflect (4, 5) over the x-axis, then translate by (x + 2, y − 1). Find the final image.
Dilation with a fraction
Dilate (−3, 6) by scale factor k = 1/3 about the origin.
Dilating a line (Jan 2026 #21)
Line t is y = 2x − 1. Dilate it by scale factor 3 about the origin. What is the image?

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

Cylinder volume
Find the volume of a cylinder with radius 5 and height 12 (in terms of π and rounded).
Cone — find the height
A cone has radius 6 and volume 96π. Find its height.
Density / modeling
A solid gold bar has a volume of 20 cm³. Gold's density is 19.3 g/cm³. Find its mass.
Composite solid + density (Jan 2026 #34)
A prism 12×6×3 cm is topped by a pyramid (same 12×6 base) of height 10 cm. Glass density = 2.5 g/cm³. Find the mass.

Facts You Must Know Cold

Carry onto itself
regular n-gon: 360 ⁄ n°
Trapezoid median
½(base₁ + base₂)
Cofunctions
sin x = cos(90° − x)
Leg geom. mean
leg² = (hyp)(adjacent segment)
Altitude geom. mean
alt = √(seg₁ · seg₂)
Cavalieri
same height + cross-sections ⇒ same volume
Dilate a line
slope kept; off-center ⇒ parallel image
Centroid
divides each median 2 : 1
SAS Area
½ · a · b · sin(C)

⭕ Circle Segment & Angle Rules (must-know)

SituationRule
Central angle= intercepted arc
Inscribed angle= ½ intercepted arc
Inscribed in a semicircle= 90° (subtends a diameter)
Tangent–chord angle= ½ intercepted arc
Cyclic quadrilateralopposite angles are supplementary
Two chords meet insideangle = ½(sum of arcs); (part)(part) = (part)(part)
Two secants meet outsideangle = ½(difference of arcs); (whole)(ext) = (whole)(ext)
Tangent + secanttangent² = (whole secant)(external part)
Tangent & radiusmeet at 90° at the point of tangency
⊥ from center to a chordbisects the chord and its arc

✏️ Constructions You Must Know

  • Copy a segment / angle — reproduce a length or angle with arcs of equal radius.
  • Perpendicular bisector — equal arcs from both endpoints; the line through the crossings is ⊥ at the midpoint.
  • Angle bisector — arc from the vertex, then equal arcs from the two crossings.
  • Perpendicular through a point (on or off the line) — equal arcs to set up a perpendicular bisector.
  • Equilateral triangle — two same-radius arcs (radius = the side) from each endpoint.
  • Square / hexagon inscribed in a circle, and parallel line through a point.
  • The Geometry exam has one or two constructions — leave ALL construction marks.

Area & Perimeter Formulas

Triangle Area
A = ½ b h
Rectangle Area
A = l w
Parallelogram
A = b h
Trapezoid
A = ½ (b₁ + b₂) h
Circle Area
A = π r²
Circumference
C = 2π r = π d
Regular Polygon
A = ½ a P (apothem × perimeter)
Equilateral △
A = (√3 ⁄ 4) s²

Surface Area & Volume

Prism / Cylinder V
V = B h (cyl: πr²h)
Pyramid / Cone V
V = ⅓ B h (cone: ⅓πr²h)
Sphere Volume
V = 4⁄3 π r³
Cube Volume
V = s³
Cylinder SA
2πr² + 2πr h
Sphere SA
4π r²
Density
D = mass ⁄ volume
Pop. Density
people ⁄ area

Coordinate Geometry (memorize these 3)

Distance
√[(x₂−x₁)² + (y₂−y₁)²]
Midpoint
((x₁+x₂)⁄2 , (y₁+y₂)⁄2)
Slope
(y₂−y₁) ⁄ (x₂−x₁)
Slope-Intercept
y = m x + b
Point-Slope
y − y₁ = m(x − x₁)
Parallel slopes
equal (m₁ = m₂)
Perpendicular slopes
negative reciprocals (m₁·m₂ = −1)
Circle
(x−h)² + (y−k)² = r²

Right Triangle Trig

Pythagorean
a² + b² = c²
sin (SOH)
Opposite ⁄ Hypotenuse
cos (CAH)
Adjacent ⁄ Hypotenuse
tan (TOA)
Opposite ⁄ Adjacent
Find an angle
use sin⁻¹, cos⁻¹, tan⁻¹
Co-function
sin x = cos(90° − x)

Special Right Triangles

TriangleSide RatioRule
45-45-901 : 1 : √2hypotenuse = leg · √2
30-60-901 : √3 : 2short leg : long leg : hypotenuse

⭕ Circle Angle Rules (where is the vertex?)

Vertex locationAngle 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 semicircle90° (right angle)

Transformation Rules (about the origin)

TransformationRule (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 Angle Sums

PolygonSides (n)Interior Sum (n−2)·180°Each angle if regular
Triangle3180°60°
Quadrilateral4360°90°
Pentagon5540°108°
Hexagon6720°120°
Octagon81080°135°

Exterior angles of ANY polygon always add to 360°. Each exterior of a regular n-gon = 360 ⁄ n.

Triangle Congruence vs. Similarity

  • Congruent (same size): SSS, SAS, ASA, AAS, HL.
  • Similar (same shape): AA, SAS~, SSS~.
  • NOT valid: SSA and AAA (AAA only proves similar).
  • CPCTC: use only AFTER proving triangles congruent.
  • Scale factor k: length × k, area × k², volume × k³.

Quadrilateral Family Properties

ShapeKey diagonal / side facts
Parallelogramopp sides ∥ & ≅; diagonals bisect each other
Rectangleparallelogram + 4 right angles + ≅ diagonals
Rhombusparallelogram + 4 ≅ sides + ⊥ diagonals
Squarerectangle AND rhombus (all properties)
Isosceles Trapezoid1 pair ∥ sides; ≅ legs; ≅ diagonals; ≅ base angles
Quick ways to lock in the facts you keep forgetting. Read the big trick, then the small note tells you what it unlocks. Say them out loud — silly is memorable.

Trigonometry

SOH-CAH-TOA
Sin=Opp/Hyp · Cos=Adj/Hyp · Tan=Opp/Adj. Say: “Some Old Hippie Caught Another Hippie Tripping On Acid.”
“Co” = Complementary
Cosine & sine are cofunctions: sin(x)=cos(90−x). When you see cos = sin, the two angles ADD to 90.
Side? sin/cos/tan. Angle? hit the −1
Looking for a missing SIDE → use sin/cos/tan. Looking for a missing ANGLE → use sin⁻¹/cos⁻¹/tan⁻¹.
Hyp is always across from the 90°
The hypotenuse faces the right angle and is the longest side — never plug it in as a leg.

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

Center = Whole, On = Half
Vertex at the center → angle = the whole arc. Vertex on the circle (inscribed) → HALF the arc.
IN you ADD, OUT you SUBTRACT
Vertex inside (two chords) → ½(arc + arc). Vertex outside (secants/tangents) → ½(big arc − small arc).
Diameter = right angle
An angle inscribed in a semicircle (its arc is a diameter) is always 90°.
Tangent & radius hold hands at 90°
A tangent line is perpendicular to the radius at the point it touches.

Area, Circumference & Volume

“Cherry Pie Delicious, Apple Pies Are 2”
C=πd (Cherry Pie Delicious) and A=πr² (Apple Pies Are r-squared).
Pointy solids get a THIRD
Cones & pyramids come to a point → V = ⅓Bh. Flat-topped prisms & cylinders → V = Bh (no third).
“Four-thirds pi r cubed”
Sphere volume = 4⁄3 πr³. Sphere surface = 4πr². (Volume is the one with the cube.)
Volume is CUBIC, area is SQUARE
Answer in cm³ for volume, cm² for surface area — match the unit to the dimension.

Transformations

Reflect over an axis → that letter STAYS
Over the x-axis: x stays, y flips → (x, −y). Over the y-axis: y stays, x flips → (−x, y).
y = x → just SWAP
Reflecting over y = x swaps the coordinates: (x, y) → (y, x).
180° → negate BOTH
Rotate 180°: (x, y) → (−x, −y). For 90° CCW: swap then negate the new first number → (−y, x).
Slide–Flip–Turn keep size; Dilation changes it
Translations, reflections, rotations are rigid (congruent). Only a dilation resizes (similar).

Lines & Coordinate Geometry

Slope = “rise over run”
m = change in y ÷ change in x. The y’s go on top.
Perpendicular? “Flip it & switch the sign”
Negative reciprocal: 2 → −½, and ¾ → −4/3. Parallel lines just keep the SAME slope.
Midpoint = “average the points”
Average the x’s and average the y’s. (Distance = secretly the Pythagorean theorem.)
Circle: opposite signs inside
(x−h)²+(y−k)²=r². The center flips the signs: (x−2)²+(y+3)² → center (2, −3).

Triangle Congruence & Similarity

No “Donkey Theorem” (no ASS)
Valid: SSS, SAS, ASA, AAS, HL. SSA doesn’t work — and reversed it spells the donkey. Avoid it.
AAA = “All Angles → just Similar”
Three equal angles prove SIMILAR (same shape), not congruent (same size).
CPCTC = the “after” move
Only use CPCTC AFTER you’ve proven two triangles congruent, to grab one more equal part.
Add a dimension, add a power
Scale factor k: lengths ×k, area ×k², volume ×k³.

⬛ Quadrilaterals & Polygons

A square is BOTH
A square is a rectangle AND a rhombus, so it has every property of both.
Rhombus = X (perpendicular)
Rhombus diagonals cross at 90° (and bisect the angles). Rectangle diagonals are equal in length.
Exteriors always lap the track = 360°
The exterior angles of ANY polygon add to 360°. Interior sum = (n − 2)·180°.
Triangle inequality: shorts beat the long
The two shortest sides must add to MORE than the longest, or the triangle can’t close.

Fast Facts to Memorize Cold

30-60-90 = 1 : √3 : 2
Short : long : hyp. Hyp is double the short leg. 45-45-90 = 1 : 1 : √2 (legs equal).
Triples: 3-4-5, 5-12-13, 8-15-17
Memorize these right-triangle sets (and their multiples like 6-8-10) to skip the Pythagorean work.
Centroid is 2/3 from the vertex
It cuts each median 2:1 (the bigger piece touches the vertex).
D = M / V triangle
Density = Mass ÷ Volume. Cover the one you want: M = D·V, V = M ÷ D.
Carry onto itself = 360 ÷ n
A regular n-gon maps onto itself every 360/n°. Hexagon = 60°, octagon = 45°.
Trapezoid median = average of bases
Midsegment = ½(base₁ + base₂).