Triangles [Congruency in Triangles]

207. CPCTC: The Mirror Rule · Congruent triangles keep matching parts equal

Corresponding sides and angles of congruent triangles are always equal.

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. Once two triangles are proved congruent (by SSS, SAS, ASA, AAS or RHS), every pair of corresponding sides and corresponding angles is automatically equal. CPCTC is the step that lets you *use* a congruence: first prove the triangles congruent, then read off the equal parts.

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For subscribers · Selina ICSE: Triangles [Congruency in Triangles]

What this lesson covers

Try to break it

Drag C to reshape △ABC. △DEF morphs in perfect sync — every matching side stays equal in length and every matching angle stays equal in measure. Try to find a position where AB ≠ DE or ∠A ≠ ∠D — impossible. CPCTC: Corresponding Parts of Congruent Triangles are Congruent.

How you build it

Construct two triangles that are guaranteed to be congruent using SSS, SAS, or ASA criteria.

Point tool: mark point A near the lower-left.
Point tool: mark point B to the right of A — AB is the base of your triangle.
Point tool: mark point C above AB — this is the apex. Any spot makes a valid triangle.
Segment tool: join A to B.
Segment tool: join B to C.
Segment tool: join C to A. Triangle ABC — your given triangle — is done.
Point tool: mark point D well to the right — the start of your copy.
Ray tool: click D, then click further right to draw the base ray that DE will lie on.
Arc tool: click centre A, drag until the arc snaps to B. This locks the compass to radius AB.
Arc tool: click centre D. With radius AB locked, this arc crosses the base ray — that crossing is E.
Point tool: mark E where the arc crosses the base ray. Now DE = AB.
Arc tool: click centre A, drag until the arc snaps to C. This locks the compass to radius AC.
Arc tool: click centre D. This arc (radius AC) sweeps above the base — F will lie on it.
Arc tool: click centre B, drag until the arc snaps to C. This locks the compass to radius BC.
Arc tool: click centre E. This arc (radius BC) crosses the D-arc from step 13 at F.
Point tool: mark F where the two arcs meet, above the base.
Segment tool: join D to F.
Segment tool: join E to F. Triangle DEF ≅ triangle ABC by SSS — and every matching angle is equal too. That is CPCTC!

The proof, step by step

Prove that corresponding parts of congruent triangles are equal.

Given: ΔABC ≅ ΔDEF
By definition of congruence, all corresponding sides and angles are equal.
Therefore, AB = DE, BC = EF, AC = DF, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.
This result is abbreviated as C.P.C.T.C.

Worked example

In ΔABC and ΔDEF, if ΔABC ≅ ΔDEF, AB = 6 cm, BC = 8 cm, and ∠B = 70°. What is the measure of ∠E?

Since ΔABC ≅ ΔDEF, corresponding angles are equal. ∠B corresponds to ∠E. Therefore, ∠E = ∠B = 70°.

60°
70° — correct
80°
90°