CPCTC: Corresponding Parts of Congruent Triangles are Congruent
CPCTC
After proving triangles congruent, all their parts match
Once you prove two triangles congruent (SSS, SAS, ASA, AAS, HL), state any corresponding side or angle is congruent using CPCTC. It's the final step in most triangle proofs.
Congruence Shortcuts
Triangle congruence shortcuts: SSS SAS ASA AAS HL — not AAA or SSA
Congruence Shortcuts
Five valid ways to prove triangles congruent
SSS: three sides. SAS: two sides + included angle. ASA: two angles + included side. AAS: two angles + non-included side. HL: hypotenuse + leg (right triangles only). AAA only proves similarity, SSA proves nothing.
SSS
Side-Side-Side
SAS
Side-Angle-Side
ASA
Angle-Side-Angle
AAS
Angle-Angle-Side
HL
Hypotenuse-Leg (right triangles)
Parallel Line Angles
Parallel lines + transversal: alternate interior angles are equal (Z-shape)
Parallel Line Angles
Four angle relationships formed by a transversal and parallel lines
Triangle interior angles sum to 180°. Exterior angle = sum of two non-adjacent interiors.
Triangle Angle Theorems
Two fundamental triangle angle facts
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.
Proof by Contradiction
Indirect proof (proof by contradiction): assume opposite is true → reach a contradiction
Proof by Contradiction
Prove something by assuming its opposite and finding an impossibility
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.
Postulates vs Theorems
Postulate = accepted without proof. Theorem = proven from postulates. Definition = meaning of a term.
Postulates vs Theorems
The building blocks of geometric proof
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 and Angle Addition Postulates
Segment addition: if B is between A and C, then AB + BC = AC. Angle addition: same concept for angles.
Segment and Angle Addition Postulates
Fundamental postulates used in almost every proof
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 and Linear Pairs
Vertical angles are congruent. Linear pair is supplementary (adds to 180°).
Vertical Angles and Linear Pairs
Two angle relationships formed when lines intersect
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.
Properties Used in Proofs
Transitive property: if a=b and b=c, then a=c. Substitution: replace one equal expression with another.
Properties Used in Proofs
The algebraic properties that justify steps in geometric 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
Isosceles triangle theorem: if two sides are equal, the base angles are equal. Converse is also true.
Isosceles Triangle Theorem
Equal sides guarantee equal base angles — and vice versa
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 Format
Two-column proof: statements in left column, reasons in right. Each reason justifies the statement.
Two-Column Proof Format
The standard format for writing geometric proofs
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?
Parallel Line Theorems
AIA theorem: alternate interior angles are congruent IF AND ONLY IF lines are parallel.
Parallel Line Theorems
The theorems connecting parallel lines to angle relationships
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).