Congruence

Two figures are congruent when one can be transformed into the other using only rigid motions — translation, rotation, reflection — without scaling. They have the same shape and size.

Notation: $\triangle ABC \cong \triangle DEF$. Distinguish from similarity (same shape, possibly different size — congruence is similarity with scale factor $1$).

Triangle congruence shortcuts:

• SSS: three sides equal.
• SAS: two sides + included angle equal.
• ASA: two angles + included side equal.
• AAS: two angles + a non-included side equal.
• HL (right triangles only): hypotenuse + one leg equal.

SSA (side-side-angle) is not sufficient — the famous "ambiguous case" can produce 0, 1, or 2 valid triangles. Congruence generalises in algebra to modular arithmetic ($a \equiv b \pmod n$).