Triangle Congruence

Intuition and core definition

Congruent triangles have all corresponding sides and angles equal; one can be mapped onto the other by rigid motions. Sufficient tests are SSS, SAS, ASA, AAS, and HL for right triangles. AAA establishes only similarity, and SSA is generally ambiguous.

Notation, language, and conditions

A statement $\triangle ABC\cong\triangle DEF$ fixes correspondence $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. CPCTC means corresponding parts of congruent triangles are congruent and may be used only after congruence is established.

Why this idea matters

Congruence criteria prove that all corresponding parts match from limited but sufficient measurements, avoiding the need to measure every feature.

A dependable method

1. Mark all givens and shared sides or angles.
2. Match vertices consistently and write a tentative correspondence.
3. Identify a valid congruence criterion, checking included side/angle conditions.
4. State congruence in matching vertex order.
5. Use CPCTC for any requested remaining parts.

Worked example

Common mistakes