Most proofs about triangles follow one pattern: find three pairs of congruent parts, prove the triangles congruent with one of the criteria from AAS and Hypotenuse-Leg and SSS, SAS and ASA, then use the fact that corresponding parts of congruent triangles are congruent (CPCTC) to get the part you really want.
- A shared side: $\overline{BD} \cong \overline{BD}$ by the Reflexive Property.
- Vertical angles where two segments cross: congruent by the Vertical Angles Theorem.
- Parallel lines: alternate interior angles are congruent.
- Midpoints and bisectors: they give two congruent segments or angles, by definition.
- Perpendicular lines: they give right angles, and all right angles are congruent. cap: Hidden congruent parts to look for.
Why it works
CPCTC is not a new idea: once a rigid motion carries one triangle onto the other (that is what congruent means), every part lands on its partner. The criteria let us know such a motion exists without finding it.
Worked example: A kite (SSS, then CPCTC)
Given: $\overline{AB} \cong \overline{CB}$ and $\overline{AD} \cong \overline{CD}$. Prove: $\angle A \cong \angle C$.
- 1. $\overline{AB} \cong \overline{CB}$, $\overline{AD} \cong \overline{CD}$ — Given.
- 2. $\overline{BD} \cong \overline{BD}$ — Reflexive Property.
- 3. $\triangle ABD \cong \triangle CBD$ — SSS.
- 4. $\angle A \cong \angle C$ — CPCTC.
Answer: Proved: the triangles are congruent first, then their matching angles.
Worked example: Crossing segments (SAS)
Given: $E$ is the midpoint of $\overline{AD}$ and of $\overline{BC}$. Prove: $\overline{AB} \cong \overline{DC}$.
- 1. $E$ is the midpoint of $\overline{AD}$ and $\overline{BC}$ — Given.
- 2. $\overline{AE} \cong \overline{DE}$ and $\overline{BE} \cong \overline{CE}$ — Definition of midpoint.
- 3. $\angle AEB \cong \angle DEC$ — Vertical Angles Theorem.
- 4. $\triangle ABE \cong \triangle DCE$ — SAS (the vertical angles are between the pairs of sides).
- 5. $\overline{AB} \cong \overline{DC}$ — CPCTC.
Answer: $\overline{AB} \cong \overline{DC}$
Key idea
Plan backward from the goal: which two triangles contain the parts you want? Find three pairs of congruent parts (look for shared sides, vertical angles, parallel lines, midpoints), name the criterion, and finish with CPCTC.
Common mistake
Using CPCTC before the triangles are proved congruent. CPCTC is only a valid reason on a line after the congruence statement. If a proof says "$\overline{AB} \cong \overline{DC}$ by CPCTC" in step 2, it is using what it has not yet shown. Check: every CPCTC line should point back to an earlier "$\triangle \ldots \cong \triangle \ldots$" line.