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 (Hypotenuse-Leg)), 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 (Alternate Interior Angles) 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).
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🎓 Common Exam Questions
Q: What are the five triangle congruence shortcuts and which ones do NOT work?
A: Valid shortcuts: SSS (Side-Side-Side) — all three pairs of sides equal. SAS (Side-Angle-Side) — two sides and the included angle equal (angle must be BETWEEN the two sides). ASA (Angle-Side-Angle) — two angles and the included side equal. AAS (Angle-Angle-Side) — two angles and a non-included side equal. HL (Hypotenuse-Leg) — right triangles only, hypotenuse and one leg equal. Does NOT work: AAA — proves similarity but not congruence (triangles could be different sizes). SSA — the ambiguous case, one side-side-angle configuration can produce two different triangles.
Q: What does CPCTC mean and how is it used in a two-column proof?
A: CPCTC = Corresponding Parts of Congruent Triangles are Congruent. It is a CONCLUSION tool, not a proof tool — you cannot use CPCTC to prove triangles congruent, only to draw conclusions AFTER proving congruence. Typical proof structure: Given information (setup the triangles), prove the triangles congruent using SSS/SAS/ASA/AAS/HL, then use CPCTC to state that specific corresponding sides or angles are congruent. Example: prove that the diagonals of a parallelogram bisect each other — prove two triangles congruent, then CPCTC that the diagonal segments are equal.
Q: Explain parallel line angle relationships — which pairs are congruent and which are supplementary?
A: When parallel lines are cut by a transversal: Congruent pairs: alternate interior angles (Z-shape, on opposite sides of transversal between the parallels), alternate exterior angles (outside the parallels on opposite sides), corresponding angles (same position at each intersection — F-shape). Supplementary pairs (add to 180): co-interior angles also called same-side interior or consecutive interior angles (between the parallels on the same side — C-shape). Linear pairs are also supplementary. The AIA (Alternate Interior Angles) theorem and its converse are if and only if — alternate interior angles are congruent exactly when lines are parallel.
Q: How do you write a two-column proof? Explain the structure with an example.
A: Two-column proof format: left column = numbered statements, right column = reasons justifying each statement. The proof begins with the given information and ends with the statement to be proven. Valid reasons include: Given, definitions, postulates, theorems already proven, and properties (reflexive, symmetric, transitive, substitution). Example — prove vertical angles are congruent: (1) Lines AB and CD intersect at E — Given. (2) Angle AEC and angle AED form a linear pair — definition of linear pair. (3) Angle AEC + angle AED = 180 — linear pairs are supplementary. (4) Similarly angle AED + angle DEB = 180. (5) So angle AEC = angle DEB — subtraction property. This is the structure for all two-column proofs.
Q: What is proof by contradiction (indirect proof) and when do you use it?
A: Indirect proof (reductio ad absurdum): assume the negation of what you want to prove is true. Then derive a logical contradiction with the given information, a theorem, or a known fact. Since the assumption leads to a contradiction, the assumption must be false — so the original statement must be true. When to use it: when direct proof seems difficult, especially for proving something CANNOT happen or a UNIQUENESS result. Classic example: prove that the square root of 2 is irrational. Assume it equals p/q in lowest terms. Then 2 = p squared/q squared, so p squared = 2q squared, so p is even. But then q must also be even, contradicting that p/q was in lowest terms. Contradiction — so the assumption was wrong.