What Is Congruence?

How congruent figures agree

In triangle geometry, congruence is usually established through tests such as SSS, SAS, ASA, AAS, and HL rather than by checking every single part separately from scratch.

This page keeps the compared figures aligned with their matching labels so congruence can be read as a statement about correspondence and rigid motion, not just about a rough visual fit.

A triangle with side lengths 3, 4, and 5 is congruent to another triangle with the same three side lengths when the corresponding sides are matched correctly. It may be drawn in a different place or reflected, but no stretch is needed to make the two figures coincide.

The order of correspondence matters. If vertex A matches vertex D, then the angle at A must equal the angle at D and the sides touching those vertices must match in the same order. A correct congruence statement tells the reader exactly which parts belong together.

• Congruent figures have the same size and shape.
• Congruent figures can coincide exactly under a rigid motion such as a translation, rotation, or reflection.
• Corresponding parts of congruent figures are equal in measure.
• In triangle proofs, the congruence criteria justify the entire figure match, which then supports CPCTC-style conclusions about remaining parts.

Where congruence helps

• Use congruence in triangle proofs where a full figure match unlocks new equal sides or angles.
• Use it in construction and transformation work to explain why a moved figure is still the same size and shape.
• Use congruence when checking whether two coordinate figures match exactly rather than only proportionally.

Congruence checks

• Do not call figures congruent just because they look alike; corresponding parts must match exactly.
• Do not confuse congruence with similarity, where size can change.
• Do not mix up the order of corresponding vertices when writing a congruence statement.