Geometry · Units 1, 3–10, 13 · Quick reference
GeometryPART 2
Logic, proofs, parallel lines, congruent and similar triangles, centers, quadrilaterals, circles, transformations, coordinate proofs, trig and constructions.
1Logic & Conditionalsthe grammar every proof is written in
- Conditional
- If p, then q (p → q). p = hypothesis, q = conclusion
- Converse
- If q, then p (q → p)
- Inverse
- If not p, then not q (~p → ~q)
- Contrapositive
- If not q, then not p (~q → ~p)
- Biconditional
- p if and only if q (p ↔ q): both directions true. Every good definition is one.
| Form | Statement | True? |
|---|---|---|
| p → q | If an angle is 30°, then it is acute. | ✓ |
| Converse | If an angle is acute, then it is 30°. | ✗ (50°) |
| Inverse | If an angle is not 30°, then it is not acute. | ✗ (50°) |
| Contrapositive | If an angle is not acute, then it is not 30°. | ✓ |
Swap, negate, both
Converse swaps, Inverse negates, Contrapositive does both.
The contrapositive is the true twin: it always shares the original's truth value; the converse and inverse share each other's.
| p | q | p → q | ~q → ~p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
When is p → q false?
Only when p is true and q is false. A false hypothesis makes the conditional true (a promise not tested is not broken).
Deductive laws
- Detachment: p → q is true and p is true, so q is true.
- Syllogism: p → q and q → r give p → r.
- Counterexample: one case with p true and q false disproves p → q.
Chain it Worked
Reasoning traps
- Affirming the conclusion: p → q and q true does not give p.
- Inductive reasoning (patterns) suggests a conjecture; only deductive reasoning proves it.
2Two-Column Proofsevery statement needs a reason you could defend
Anatomy of a proof
- Given: what you may assume. Prove: the goal.
- Mark the figure first: ticks, arcs, right-angle boxes.
- Left column statements; right column reasons: a given, definition, postulate or proved theorem.
- Other formats: paragraph, flow (boxes and arrows), indirect (assume the opposite; reach a contradiction).
| Property | Equality form |
|---|---|
| Reflexive | a = a |
| Symmetric | a = b → b = a |
| Transitive | a = b, b = c → a = c |
| Substitution | a = b → b may replace a |
| +, −, ×, ÷ | do the same to both sides (÷ by c ≠ 0) |
| Distributive | a(b + c) = ab + ac |
Givens first, CPCTC last
Open with the givens, close with what was asked; a shared side or angle is free by the reflexive property.
| When you see… | Cite |
|---|---|
| X shares a side or angle | reflexive property |
| two lines cross | vertical ∠s are ≅ |
| a midpoint | def. of midpoint (two ≅ halves) |
| a bisector | def. of bisector |
| ⊥ lines | def. of ⊥ (right ∠s); all right ∠s ≅ |
| ∥ lines + transversal | alt. interior / corresponding ∠s ≅ |
| two ∠s forming a line | linear pair → supplementary |
| parts after ≅ triangles | CPCTC |
| Statement | Reason |
|---|---|
| 1. M is the midpoint of AB and CD | given |
| 2. AM ≅ MB, CM ≅ MD | def. of midpoint |
| 3. ∠AMC ≅ ∠BMD | vertical angles are ≅ |
| 4. △AMC ≅ △BMD | SAS (2, 3, 2) |
| 5. AC ≅ BD | CPCTC |
Algebraic proof Worked
Proof slips
- Using what you must prove as a reason.
- Reading facts off the picture that are not marked or given.
- Citing SSA or AAA: neither proves congruence.
3Parallel Lines & Transversalsone crossing angle gives all eight
- Transversal
- a line that crosses two or more lines
- Interior
- between ℓ and m: ∠3, ∠4, ∠5, ∠6
- Exterior
- outside ℓ and m: ∠1, ∠2, ∠7, ∠8
- Alternate
- opposite sides of t
- Same-side
- the same side of t
Only for parallel lines
The ≅ and 180° rules need ℓ ∥ m. If the lines are not parallel, only vertical angles and linear pairs still work.
Converses prove lines parallel
If corresponding ∠s are ≅, or alternate interior ∠s are ≅, or same-side interior ∠s are supplementary, then ℓ ∥ m.
| Pair (ℓ ∥ m) | Example | Rule |
|---|---|---|
| Corresponding | 1 & 5, 2 & 6, 3 & 7, 4 & 8 | ≅ |
| Alternate interior | 3 & 6, 4 & 5 | ≅ |
| Alternate exterior | 1 & 8, 2 & 7 | ≅ |
| Same-side interior | 3 & 5, 4 & 6 | sum 180° |
| Same-side exterior | 1 & 7, 2 & 8 | sum 180° |
| Vertical | 1 & 4, 2 & 3 | ≅ (any lines) |
| Linear pair | 1 & 2, 5 & 7 | sum 180° (any lines) |
Letters F, Z, C
Trace an F for corresponding angles, a Z for alternate interior, a C (or U) for same-side interior.
Only two sizes appear: every acute angle is equal, every obtuse angle is equal, and acute + obtuse = 180°.
Same-side interior Worked
Alternate interior Worked
Parallel & perpendicular facts
- Two lines ∥ to the same line are ∥ to each other.
- In a plane, two lines ⊥ to the same line are ∥.
- A transversal ⊥ to one of two ∥ lines is ⊥ to the other: all eight angles are 90°.
4Triangle Facts & Inequalitieswhat any three sides or angles can and cannot do
Angle facts
- Interior angles sum to 180°.
- Exterior angle = sum of the two remote interior angles.
- Isosceles: base angles ≅ (and the converse). Equilateral ⟺ equiangular (60° each).
Isosceles Worked
Exterior angle algebra Worked
Triangle inequality
Each side is shorter than the sum of the other two and longer than their difference.
EXsides 3 and 5: 2 < x < 8. Can 4, 5, 10 be a triangle? 4 + 5 = 9 < 10: no.
Two short sides must reach
The two shorter sides together must out-reach the longest one; if they only tie, the "triangle" lies flat.
Side–angle order
The largest angle is opposite the longest side; the smallest faces the shortest.
EXsides 7, 9, 12 → the angle opposite 12 is the largest.
Big faces big
The biggest angle always faces the biggest side.
Hinge theorem (SAS inequality)
Two sides of one triangle ≅ two sides of another: the larger included angle sits across from the longer third side. The converse also holds (SSS inequality).
Open the jaw
Open the jaw wider and the gap between the teeth gets longer.
Midsegment theorem
The segment joining two sides' midpoints is ∥ to the third side and half as long.
EXthird side 14 → midsegment 7.
Sides 7, 9, 12 at a glance Worked
5Congruent Trianglessame size and shape: every matching part equal
Congruence statements
△ABC ≅ △DEF matches in order: A↔D, B↔E, C↔F, so AB ≅ DE and ∠C ≅ ∠F. The order tells you which parts match.
| Shortcut | Needs | Note |
|---|---|---|
| SSS | three pairs of sides | |
| SAS | two sides and the included angle | angle must be between |
| ASA | two angles and the included side | |
| AAS | two angles and a non-included side | third angles match anyway |
| HL | hypotenuse and one leg | right triangles only |
Solve with CPCTC Worked
SSA swings, AAA scales
SSA can swing the side into two different triangles; AAA only fixes the shape, not the size. Neither proves ≅.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent: prove the triangles first, then the parts.
Which shortcut? Worked
Proof strategy
- Mark everything given; look for shared sides (reflexive) and vertical angles.
- Count S and A marks; match one of the five shortcuts.
- Overlapping triangles: redraw them apart.
Isosceles & bisector facts
- The bisector of an isosceles vertex angle is also the ⊥ bisector of the base.
- Points on a segment's ⊥ bisector are equidistant from its endpoints.
- Points on an angle bisector are equidistant from its sides.
6Similar Trianglessame shape, sizes scaled by k
| Test | Needs |
|---|---|
| AA~ | two pairs of ≅ angles |
| SAS~ | two side ratios equal + included ∠ ≅ |
| SSS~ | all three side ratios equal |
Scale factor
Corresponding sides are proportional: 64 = 96 = 10.57 = 1.5. Perimeters scale by k; areas by k² (Part 1).
EXperimeters 17 and 25.5 (ratio 1.5); areas in ratio 1.5² = 2.25.
Side splitter
A line ∥ to one side cuts the other two sides proportionally: ADDB = AEEC. Converse: proportional cuts mean the line is ∥.
Angle bisector theorem
An angle bisector splits the opposite side in the ratio of the adjacent sides.
EXsides 6 and 9, opposite side 10 → pieces 4 and 6.
AA is all you need
Two matching angles force the third, so two angles prove similarity.
Shadow problem Worked
Altitude to the hypotenuse
- Makes three similar right triangles.
- h = √(piece₁ · piece₂) (geometric mean).
- Each leg = √(its piece · hypotenuse): √(4 · 13) = 2√13, √(9 · 13) = 3√13.
7Triangle Centers & Segmentsfour special points where three lines meet
| Center | Where 3 … meet | Special property | Location |
|---|---|---|---|
| Centroid G | medians (vertex to opposite midpoint) | balance point; 2 : 1 split of each median | always inside |
| Incenter I | angle bisectors | equidistant from the sides; center of the inscribed circle | always inside |
| Circumcenter O | ⊥ bisectors of the sides | equidistant from the vertices; center of the circumscribed circle | in, on (right △: midpoint of hypotenuse), or out |
| Orthocenter H | altitudes | — | in, at the right-angle vertex, or out |
Cool Ice Cream Overflows; Mom Always Packs Aprons
Centroid–Medians · Incenter–Angle bisectors · Circumcenter–Perpendicular bisectors · Orthocenter–Altitudes.
Two-thirds to the vertex
The centroid sits ⅔ of the way from each vertex to the opposite midpoint.
Centroid split Worked
Centers of A(2,6), B(0,0), C(8,0) Worked
Euler line
O, G and H always lie on one line, with HG = 2 · GO. Above: all three have y = 2; HG = 43, GO = 23.
Altitude ≠ median
An altitude is ⊥ to the opposite side (it may fall outside); a median hits the midpoint. They agree only from the vertex of an isosceles triangle.
8Polygons & Quadrilateralseach step down the tree adds properties
Exterior angles
- One exterior angle per vertex always sums to 360°, for any convex polygon.
- Regular n-gon: each exterior = 360n, each interior = 180 − 360n.
EXeach interior 140° → exterior 40° → n = 36040 = 9
EXinterior sum 1440°: (n − 2)·180 = 1440 → n = 10
| Property | ▱ | Rect | Rhom | Sq |
|---|---|---|---|---|
| Opposite sides ∥ and ≅ | ✓ | ✓ | ✓ | ✓ |
| Opposite ∠s ≅; consecutive supp. | ✓ | ✓ | ✓ | ✓ |
| Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ |
| Four right angles | ✓ | ✓ | ||
| Diagonals ≅ | ✓ | ✓ | ||
| Four ≅ sides | ✓ | ✓ | ||
| Diagonals ⊥; bisect the angles | ✓ | ✓ |
The square wins
The square inherits every property in the table: it is the most special quadrilateral.
Prove a parallelogram (any one)
- both pairs of opposite sides ∥, or both ≅
- one pair both ∥ and ≅
- diagonals bisect each other
- both pairs of opposite angles ≅
Trapezoids & kites
- Trapezoid: exactly one pair of ∥ sides (bases). Midsegment = b₁ + b₂2.
- Isosceles trapezoid: legs ≅, base angle pairs ≅, diagonals ≅.
- Kite: two pairs of ≅ adjacent sides; diagonals ⊥; one pair of opposite ∠s ≅.
EXbases 8 and 14 → midsegment 8 + 142 = 11
EXrhombus diagonals 10, 24 → side √(5² + 12²) = 13 (halves meet at 90°)
Parallelogram & rectangle Worked
Kite angles Worked
9Circlesangles from arcs, lengths from products
- Chord
- both endpoints on the circle
- Secant
- line cutting the circle twice
- Tangent
- line touching it once
- Central ∠
- vertex at the center
- Inscribed ∠
- vertex on the circle
- Arcs
- minor < 180° < major; semicircle 180°
Inscribed is half
An inscribed angle is half the central angle (and arc) it shares.
Inscribed-angle facts
- Inscribed ∠s on the same arc are ≅.
- An angle inscribed in a semicircle is 90°.
- Inscribed quadrilateral: opposite ∠s supplementary (85° ↔ 95°).
Tangent facts
- A tangent is ⊥ to the radius at the point of tangency.
- Two tangents from one outside point are ≅.
- Tangent–chord angle = ½ its arc (arc 130° → 65°).
- A radius ⊥ to a chord bisects it; ≅ chords ⟺ ≅ arcs.
| Vertex | Angle | EX |
|---|---|---|
| center | = arc | arc 100° → 100° |
| on circle | ½ · arc | arc 100° → 50° |
| inside | ½ (sum of arcs) | ½(80 + 40) = 60° |
| outside | ½ (difference) | ½(150 − 50) = 50° |
Inside: average · outside: half the difference
Vertex inside, average the two arcs; vertex outside, take half their difference.
Equation of a circle
Center (h, k), radius r: (x − h)² + (y − k)² = r².
EXx² + y² − 6x + 4y − 12 = 0 → complete the squares: (x − 3)² + (y + 2)² = 25 → center (3, −2), r = 5.
| Case | Rule |
|---|---|
| two chords | part · part = part · part |
| two secants | outside · whole = outside · whole |
| tangent + secant | tangent² = outside · whole |
Part × whole
Outside the circle: outside part × whole = outside part × whole (a tangent counts as both).
10Transformations & Symmetrymove it, flip it, turn it, scale it
| Transformation | Rule | Image of (1, 3) |
|---|---|---|
| Translation ⟨a, b⟩ | (x + a, y + b) | ⟨−5, 1⟩: (−4, 4) |
| Reflect over x-axis | (x, −y) | (1, −3) |
| Reflect over y-axis | (−x, y) | (−1, 3) |
| Reflect over y = x | (y, x) | (3, 1) |
| Rotate 90° CCW | (−y, x) | (−3, 1) |
| Rotate 180° | (−x, −y) | (−1, −3) |
| Rotate 270° CCW | (y, −x) | (3, −1) |
| Dilate by k (center O) | (kx, ky) | k = 2: (2, 6) |
The axis keeps its coordinate
Reflect over the x-axis: x stays, y flips. Over the y-axis: y stays, x flips.
Swap, then negate
Rotate 90° CCW: swap x and y, then negate the new first coordinate: (1, 3) → (3, 1) → (−3, 1).
Rigid motions (isometries)
Translations, reflections, rotations and glide reflections keep lengths and angles: image ≅ preimage. A dilation keeps angles but scales lengths: image ~ preimage. Rigid motions keep size and shape; dilations keep shape only.
Compose Worked
Compositions
- Reflect over two ∥ lines → a translation of twice the gap.
- Reflect over two lines meeting at θ → a rotation of 2θ.
- Glide reflection: translate, then reflect over a line ∥ to the slide.
- Order matters: do the first one first.
Symmetry
- Line: a reflection maps the figure onto itself.
- Rotational: a turn under 360° does.
- Regular n-gon: n lines; turns of 360n°.
EXsquare 4 lines, 90° · rectangle 2 lines, 180° · regular pentagon 5 lines, 72°
11Coordinate Proofsalgebra does the measuring
Pick the right tool
Slopes for ∥ and ⊥ · distance for ≅ · midpoint for bisect.
Classify ABCD Worked
Midpoint of a hypotenuse Worked
Placing the figure
- Put a vertex at the origin and a side on the x-axis.
- Use variables (a, b, c), not numbers, so the proof covers every case.
- Use 2a, 2b when midpoints are coming: no fractions.
Right triangle? Worked
12Right-Triangle Trig & the Lawsratios of sides name the angles
SOH-CAH-TOA
Sine = Opp/Hyp · Cosine = Adj/Hyp · Tangent = Opp/Adj.
| θ | sin | cos | tan |
|---|---|---|---|
| 30° | ½ | √3⁄2 | √3⁄3 |
| 45° | √2⁄2 | √2⁄2 | 1 |
| 60° | √3⁄2 | ½ | √3 |
Count up the roots
sin 30°, 45°, 60° = √12, √22, √32; cos runs the same list backward.
3-4-5 Worked
Angle of elevation Worked
Area with sine
Area = ½ab sin C.
EX½ · 5 · 8 · sin 60° = 10√3 ≈ 17.32.
Law of sines
asin A = bsin B = csin C: use with AAS, ASA or SSA.
EXA = 40°, B = 75°, a = 10 → C = 65°, b ≈ 15.03, c ≈ 14.10.
Law of cosines
c² = a² + b² − 2ab cos C: use with SAS or SSS.
EXsides 5, 8, 7: cos θ = 25 + 64 − 4980 = 0.5 → the angle facing 7 is 60°.
Pythagoras with a correction
Law of cosines = Pythagoras minus 2ab cos C; at C = 90° the correction is 0.
SSA: the ambiguous case
A = 30°, a = 5, b = 8: sin B = 8 · 0.55 = 0.8 → B ≈ 53.13° or 126.87°: two triangles.
13Constructionscompass and straightedge only, no measuring
Compass for equal, straightedge for straight
Every arc copies a length; every line just joins two points.
Why they work
The crossing points are equidistant from A and B, so they lie on the ⊥ bisector (§5). The angle bisector uses SSS.
| Construct | Key steps |
|---|---|
| ⊥ bisector of AB | equal arcs from A and B (radius > ½AB) cross twice; join the crossings |
| Angle bisector | arc from the vertex cuts both sides; equal arcs from those points cross; join to the vertex |
| Copy a segment / angle | transfer the length with the compass; for an angle, copy the arc and its chord |
| ⊥ through a point | arcs mark two points equally far on the line; bisect between them |
| ∥ through a point | copy a corresponding angle at the point |
| Equilateral △ | radius AB arcs from A and from B; they meet at the third vertex |
| Regular hexagon | step the radius 6 times around the circle |