Triangle Proofs

Overview

A proof is a logical, step-by-step explanation showing that the truth of a hypothesis guarantees the truth of the conclusion. In a formal two-column proof, every statement is paired with a reason — a definition, property, postulate, or theorem — that justifies it. Triangle proofs combine the congruence shortcuts from the previous lessons with supporting reasons: a midpoint produces congruent segments, parallel lines produce congruent alternate interior angles, intersecting lines produce congruent vertical angles, and a shared side is congruent to itself. The proof ends with a congruence statement, and CPCTC can then deliver any final part.

Vocabulary

Proof: A logical, step-by-step explanation that shows the truth of a hypothesis (the given information) guarantees the truth of the conclusion (what is to be proved).
Two-Column Proof: A formal proof organized in two columns: statements on the left, in logical order, and the reason justifying each statement on the right.
Definition of Midpoint: A midpoint is the point that divides a segment into two congruent segments. Naming a point as a midpoint immediately gives one pair of congruent segments for a proof.
Vertical Angles: The opposite angles formed when two lines intersect. Vertical angles are always congruent, whether or not any lines are parallel.
Reflexive Property: A side or angle shared by two triangles is congruent to itself, so it can serve as one of the congruent parts a proof needs.
CPCTC: Corresponding Parts of Congruent Triangles are Congruent — the reason used after a congruence statement to conclude that one specific pair of corresponding parts is congruent.

Main Ideas

Start a proof with the given information, work through new information step by step, and end with the statement you were asked to prove — every step carries a reason.
Before writing, mark the diagram: the given parts, the pairs a midpoint creates, and the angles that must be congruent.
Three reliable sources of "free" congruent parts: the definition of midpoint (two congruent segments), vertical angles (congruent), and a shared side or angle (reflexive property).
Parallel lines cut by a transversal give congruent alternate interior angles — a common way to earn an angle pair in a proof.
Collect three qualifying parts, finish the congruence with SSS, SAS, ASA, AAS, or HL, and only then use CPCTC if the conclusion asks for one specific part.
Algebraic steps appear in proofs too, justified by properties of equality such as distribution, subtraction, and division.

Lesson Notes

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Interactive Activity

Build two-column triangle proofs step by step, choosing each statement and reason.

Triangle Proofs — Interactive ActivityOpen ›

Practice Problems

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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.