Definition: CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent; once triangles are proven congruent, all their matching sides and angles are equal.
When can you use CPCTC?
Only after proving two triangles congruent by SSS, SAS, ASA, AAS, or HL. It is the reason that comes last in many proofs, used to show that two segments or two angles are equal.
How does a proof flow?
Find triangles that contain the segments or angles you care about. Prove them congruent with a criterion. Then cite CPCTC for the pair you need.
Example. Point M is the midpoint of segment AD, and angles ABM and DCM are right angles, with AB = DC given. In right triangles ABM and DCM, the hypotenuses AM and DM are equal because M is the midpoint, and the legs AB and DC are equal. The triangles are congruent by HL. By CPCTC, angle AMB equals angle DMC.
What goes wrong?
Using CPCTC before the triangles are proven congruent, which is circular reasoning. Another error is to match the wrong parts: the order of the vertices in the congruence statement tells which sides and angles correspond. Write that statement carefully, such as triangle ABM is congruent to triangle DCM.