congruent triangles

Two triangles are congruent when they are the same triangle in two places — identical in shape and in size, so that one could be picked up and laid exactly on top of the other (allowing a flip). Same angles, same side lengths, just possibly moved or mirrored. Congruence is the geometric word for 'truly identical', as opposed to merely the same shape at a different scale, which is similarity.

Precisely: triangle ABC is congruent to triangle DEF, written ABC = DEF (with the congruence symbol), if there is a correspondence A to D, B to E, C to F under which all three pairs of sides are equal (|AB| = |DE|, |BC| = |EF|, |CA| = |FD|) and all three pairs of angles are equal. The order of the letters carries the matching, so writing the correspondence correctly is part of stating the congruence. In transformation language, two figures are congruent exactly when one maps to the other by a rigid motion (a translation, rotation, reflection, or glide reflection).

You almost never check all six equalities by hand. Instead, a handful of shortcuts — the congruence criteria SSS, SAS, ASA, AAS, and HL — let three well-chosen equalities force the other three. Once two triangles are known congruent, every remaining pair of corresponding parts is automatically equal, the principle abbreviated CPCTC, which is the real engine of triangle proofs.