Congruent Figures and CPCTC
Define congruence in terms of corresponding parts and rigid motion. Identify all six pairs of corresponding parts between two congruent triangles. Apply CPCTC (Corresponding Parts of Congruent Triangles are Congruent) in proofs.
A lesson of Geometry, in Triangles and Congruence, about 10 minutes.
By the end of this lesson, you will be able to identify corresponding parts of congruent figures, write correct congruence statements preserving vertex correspondence, and apply CPCTC (Corresponding Parts of Congruent Triangles are Congruent) in proofs.
The Big Idea
Congruent figures have the same shape and size — one can be mapped exactly onto the other by a rigid motion (translation, rotation, reflection). For triangles, congruence means all three sides and all three angles are pairwise equal.
The order of vertices in a congruence statement encodes which parts correspond — this is not arbitrary.
Corresponding Parts
When we write $\triangle ABC \cong \triangle DEF$, the correspondence is:
- $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$
- $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\overline{AC} \cong \overline{DF}$
- $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$
Six pairs of congruent parts: 3 pairs of sides, 3 pairs of angles.
CPCTC
CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
This is used as a reason in the final steps of a proof, after triangle congruence has been established:
Step 1–N: Prove $\triangle ABC \cong \triangle DEF$ (using SSS, SAS, ASA, AAS, or HL) Step N+1: $\overline{AB} \cong \overline{DE}$ — CPCTC
CPCTC is never used before congruence is established — it's the payoff, not the setup.
Example 1 — Correspondence
$\triangle PQR \cong \triangle STU$. List all congruent corresponding parts.
Sides: $\overline{PQ} \cong \overline{ST}$, $\overline{QR} \cong \overline{TU}$, $\overline{PR} \cong \overline{SU}$
Angles: $\angle P \cong \angle S$, $\angle Q \cong \angle T$, $\angle R \cong \angle U$
Example 2 — Writing the Congruence Statement
In the figure, $\angle A = \angle X$, $\angle B = \angle Y$, $\angle C = \angle Z$, $AB = XY$, $BC = YZ$, $AC = XZ$. Write the congruence statement.
Match vertex by vertex: $A \leftrightarrow X$, $B \leftrightarrow Y$, $C \leftrightarrow Z$.
$$\triangle ABC \cong \triangle XYZ$$
Not $\triangle ABC \cong \triangle YXZ$ — vertex order matters.
Example 3 — CPCTC in a Proof
Given: $\triangle ABD \cong \triangle CBD$
Prove: $\overline{AB} \cong \overline{CB}$