Geometry tricks that make proofs click

Indirect proof (proof by contradiction): assume opposite is true → reach a contradiction

Proof by Contradiction — Assume the negation of what you want to prove. Use valid reasoning steps. Reach a statement that is false or contradicts a known fact. Conclude the original assumption was wrong — so the original statement must be true.

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

CPCTC — Once you prove two triangles congruent (SSS, SAS, ASA, AAS, HL (Hypotenuse-Leg)), state any corresponding side or angle is congruent using CPCTC. It's the final step in most triangle proofs.

Triangle congruence shortcuts: SSS SAS ASA AAS HL — not AAA or SSA

SSSSide-Side-Side

SASSide-Angle-Side

ASAAngle-Side-Angle

AASAngle-Angle-Side

HLHypotenuse-Leg (right triangles)

Parallel lines + transversal: alternate interior angles are equal (Z-shape)

Parallel Line Angles — Corresponding angles: equal (same position). Alternate interior: equal (Z-shape). Alternate exterior: equal. Co-interior (same-side interior): supplementary (add to 180°).

Triangle interior angles sum to 180°. Exterior angle = sum of two non-adjacent interiors.

Triangle Angle Theorems — All three interior angles of any triangle sum to 180°. An exterior angle of a triangle equals the sum of the two non-adjacent interior angles. These two facts solve almost every triangle angle problem.

Postulate = accepted without proof. Theorem = proven from postulates. Definition = meaning of a term.

Postulates vs Theorems — Postulates (axioms): statements accepted as true without proof — the starting points. Euclid's 5 postulates include: two points determine a line, all right angles are equal. Theorems: statements proven from postulates and previously proven theorems. Definitions: precise meanings of geometric terms.

Segment addition: if B is between A and C, then AB + BC = AC. Angle addition: same concept for angles.

Segment and Angle Addition Postulates — Segment Addition: B is between A and C → AB + BC = AC. Angle Addition: ray BD is inside angle ABC → angle ABD + angle DBC = angle ABC. These postulates let you break segments and angles into parts or combine parts into wholes — used constantly in proofs.

Vertical angles are congruent. Linear pair is supplementary (adds to 180°).

Vertical Angles and Linear Pairs — Vertical angles: opposite angles formed by two intersecting lines — always congruent. Linear pair: two adjacent angles forming a straight line — always supplementary (sum = 180°). Supplementary: add to 180°. Complementary: add to 90°. These appear in almost every proof involving intersecting lines.

Transitive property: if a=b and b=c, then a=c. Substitution: replace one equal expression with another.

Properties Used in Proofs — Reflexive: a=a (any figure is congruent to itself). Symmetric: if a=b then b=a. Transitive: if a=b and b=c then a=c. Addition property: if a=b then a+c=b+c. Subtraction property: if a=b then a-c=b-c. Substitution: if a=b, replace a with b anywhere. Division/Multiplication: same for both sides.

Isosceles triangle theorem: if two sides are equal, the base angles are equal. Converse is also true.

Isosceles Triangle Theorem — Isosceles triangle: two congruent sides (legs). Theorem: angles opposite the congruent sides (base angles) are congruent. Converse: if two angles of a triangle are congruent, the sides opposite them are congruent. Equilateral triangle: all three sides equal → all three angles equal (60° each).

Two-column proof: statements in left column, reasons in right. Each reason justifies the statement.

Two-Column Proof Format — Left column: numbered statements (geometric facts). Right column: reasons (given, definition, postulate, theorem, property). Start with 'Given.' End with what you're proving. Each statement must follow logically from previous statements plus the reason cited. Plan proof backwards: what do you need to prove the conclusion?

AIA (Alternate Interior Angles) theorem: alternate interior angles are congruent if and only if lines are parallel.

Parallel Line Theorems — If lines are parallel: corresponding angles congruent, alternate interior angles congruent, alternate exterior angles congruent, co-interior angles supplementary. Converses are also true — use angle relationships to PROVE lines are parallel. These theorems are bidirectional (if and only if).