Question

what is congruence, how does that differ or compare to similarity.

Answer 1

Short answer
- Congruence (geometry): two figures are congruent if they have exactly the same shape and size. One can be moved to coincide with the other by rigid motions (translations, rotations, reflections). Notation: ≅.
- Similarity: two figures are similar if they have the same shape but not necessarily the same size. One can be obtained from the other by a rigid motion combined with a dilation (scaling). Corresponding angles are equal and corresponding side lengths are proportional. Notation: ~.

Key differences and comparisons
- Size: Congruent figures have equal size; similar figures may differ by a scale factor.
- Shape: Both congruent and similar figures have the same shape (corresponding angles equal).
- Transformations: Congruence = isometry (no resizing). Similarity = isometry + dilation (may resize).
- Implication: Congruence implies similarity (scale factor 1), but similarity does not imply congruence unless the scale factor is 1.
- Notation: congruent: ≅ ; similar: ~.

Triangle criteria (common quick tests)
- Congruence: SSS, SAS, ASA, AAS, HL (right triangles).
- Similarity: AA (two equal angles), SAS (two pairs of sides proportional and the included angles equal), SSS (all three pairs of sides proportional).

Consequences of similarity
- If scale factor is k, corresponding lengths multiply by k, perimeters multiply by k, areas multiply by k^2.
- Corresponding angles remain equal.

Examples
- Triangles with side lengths (3,4,5) and (6,8,10) are similar (scale factor 2) but not congruent.
- Two identical copies of a polygon placed differently (rotated/transposed) are congruent.

Both congruence and similarity are equivalence relations (reflexive, symmetric, transitive).