CPCTC

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." It is the payoff step.

The order is what matters. You cannot use CPCTC to prove triangles congruent — you use one of the five criteria for that, and *then* CPCTC gives you the remaining three pairs.

So the standard proof shape is:
1. Establish three pairs of parts.
2. Conclude the triangles are congruent by SSS, SAS, ASA, AAS or HL.
3. Use CPCTC to get whichever leftover side or angle you actually wanted.

Whenever a problem asks you to prove two segments or two angles congruent and they sit in different triangles, this is almost always the route.

CPCTC is the converse direction of the definition of congruence: once △ABC ≅ △DEF is established, all six correspondences hold. It is a valid reason in a proof only on a line *after* the congruence statement.

Its role is structural. The congruence criteria each require three specific parts; CPCTC releases the other three. Chained proofs exploit this repeatedly — prove one pair of triangles congruent, use CPCTC to obtain a segment or angle congruence, then use that new fact to prove a second pair congruent.

CPCTC is also the standard route to properties that are not obviously about triangles at all. That a parallelogram's diagonals bisect each other, that the base angles of an isosceles triangle are congruent, and that a point on a perpendicular bisector is equidistant from the endpoints are all proved by constructing triangles, proving them congruent, and applying CPCTC.

An auxiliary line is often required to create the triangles in the first place, and drawing the right one is usually the whole difficulty of the proof.