DWG NO. 04.5 — Lesson 5 of 6

CPCTC and Proof Applications

Unit 4: Congruent Triangles · ~15–30 min

Objective

Use CPCTC to prove that individual parts of two triangles are congruent, once the triangles themselves have been shown congruent.

What CPCTC unlocks

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. It's not a new way to prove triangles congruent — it's what you get to use after you've already proven two triangles congruent by SSS, SAS, ASA, AAS, or HL.

CPCTC: Once triangle ABC ≅ triangle DEF is established, every corresponding pair of sides and angles is automatically congruent — AB ≅ DE, BC ≅ EF, angle A ≅ angle D, and so on.

This is the payoff move in a huge number of geometry proofs: prove the triangles congruent first using one of the five shortcuts, then use CPCTC to reach a conclusion about one specific side or angle that the original shortcut didn't directly address.

The typical proof shape

Most CPCTC proofs follow the same two-stage pattern: (1) prove two triangles congruent using SSS, SAS, ASA, AAS, or HL, then (2) cite CPCTC to conclude that a specific pair of corresponding parts — often the thing you were actually asked to prove — must be congruent.

Shared segment BD splits triangle ABC into triangle ABD and triangle CBD — prove those congruent, then use CPCTC for angle A ≅ angle C.

Worked Example 1 · Prove an angle congruence with CPCTC

ProblemGiven: AB ≅ CB and BD bisects angle ABC (so angle ABD ≅ angle CBD). Prove: angle A ≅ angle C.

1.AB ≅ CB is given. Angle ABD ≅ angle CBD is given (bisected angle).

2.BD ≅ BD by the Reflexive Property, since it's the same segment in both triangles.

3.Triangle ABD ≅ triangle CBD by SAS (two sides and the included angle).

4.Angle A and angle C are corresponding parts of those now-congruent triangles.

Angle A ≅ angle C, by CPCTC

Worked Example 2 · Prove a segment congruence with CPCTC

ProblemGiven: angle W ≅ angle Y and angle WXZ ≅ angle YXZ. Prove: WZ ≅ YZ.

1.Angle W ≅ angle Y is given. Angle WXZ ≅ angle YXZ is given. XZ ≅ XZ by the Reflexive Property, since it's shared by both triangles.

2.Triangle WXZ ≅ triangle YXZ by AAS (two angles and a non-included side, XZ).

3.WZ and YZ are corresponding sides of those congruent triangles.

WZ ≅ YZ, by CPCTC

Guided practice

1.What must be true before you're allowed to use CPCTC in a proof?
Show answer
The two triangles must already be proven congruent (by SSS, SAS, ASA, AAS, or HL) before CPCTC applies to any of their individual parts.
2.Triangle PQR ≅ triangle STU. Name the side in triangle STU that must be congruent to side PQ, and the theorem that guarantees it.
Show answer
Side ST (matching the order P↔S, Q↔T, R↔U), congruent to PQ by CPCTC.
3.A proof shows triangle ABC ≅ triangle DEF using SSS. Can you now conclude angle B ≅ angle E? Why?
Show answer
Yes — once the triangles are proven congruent by any valid shortcut, CPCTC guarantees every corresponding pair of angles (and sides) is congruent, including angle B and angle E.
4.Why can't you use CPCTC as the very first step of a proof, before showing any triangle congruence?
Show answer
CPCTC only describes a consequence of triangles already being congruent — it isn't itself a way to establish congruence. Using it first would be circular, assuming the conclusion you still need to prove.
5.Given isosceles triangle ABC with AB ≅ CB, and BD the bisector of angle B meeting AC at D. After proving triangle ABD ≅ triangle CBD, what can CPCTC tell you about segments AD and CD?
Show answer
AD ≅ CD by CPCTC — they're corresponding sides of the two congruent triangles, which also shows the angle bisector from the apex of an isosceles triangle bisects the base.

← Lesson 4.4 Unit 4 Overview Lesson 4.6 →