Congruency: Congruent Triangles
124. The Mirror Match · When triangles are congruent, every part lines up perfectly
Corresponding sides of the two triangles remain equal as you drag the vertices.
CPCT stands for Corresponding Parts of Congruent Triangles (are equal). Here is the idea: if two triangles are congruent — written ΔABC ≅ ΔDEF — then *every* matching pair of parts must be equal. That means all 3 pairs of sides (AB = DE, BC = EF, CA = FD) and all 3 pairs of angles (∠A = ∠D, ∠B = ∠E, ∠C = ∠F) — six equalities in one go. Think of CPCT as a shortcut: once you prove two triangles congruent (using SSS, SAS, ASA, or RHS), you can immediately claim any corresponding side or angle is equal, without proving each one separately. The order of letters in ΔABC ≅ ΔDEF tells you which parts correspond: A↔D, B↔E, C↔F.
What this lesson covers
Try to break it
Drag A, B, or C to reshape triangle ABC. Triangle DEF follows the same change. Try to make a side of DEF different from its matching side in ABC — you can't. Corresponding parts of congruent triangles are always equal (CPCT).
How you build it
Make a triangle.
- Place point A.
- Place point B.
- Place point C.
- Connect A to B with a line.
- Connect B to C.
- Connect C to A.
The proof, step by step
Prove that the corresponding sides of the congruent triangles are equal.
- If ΔABC ≅ ΔDEF, then by definition they can be superimposed exactly.
- Therefore, side AB coincides with DE, BC with EF, and CA with FD.
- Thus, all corresponding sides and angles are equal. This rule is called **CPCT** — *Corresponding Parts of Congruent Triangles* (are equal).
Worked example
In ΔABC ≅ ΔPQR, if AB = 5 cm, ∠B = 40°, and ∠C = 60°, find the length of PQ and the measure of ∠R.
By CPCT, corresponding parts are equal. PQ corresponds to AB, so PQ = 5 cm. ∠R corresponds to ∠C, so ∠R = 60°.
- PQ = 5 cm, ∠R = 60° — correct
- PQ = 5 cm, ∠R = 80°
- PQ = 6 cm, ∠R = 60°
- PQ = 5 cm, ∠R = 40°