What Congruence Means

Congruent figures have the same shape and size. Modern geometry defines that precisely: two figures are congruent if you can slide, turn or flip one onto the other exactly.

Those three moves — translation, rotation, reflection — are the rigid motions. They preserve every length and every angle, which is why the image is identical to the original.

Order in a congruence statement is not decoration. Writing △ABC ≅ △DEF claims A matches D, B matches E and C matches F. From that single line you can read off six facts: three pairs of congruent sides and three pairs of congruent angles.

Getting the letters in the wrong order makes the statement false even when the triangles really are congruent.

A rigid motion (isometry) is a transformation preserving distance; translations, rotations and reflections generate all of them in the plane. Two figures are congruent precisely when some composition of rigid motions maps one onto the other. This definition replaces the older "same size and shape" and has the advantage of being checkable.

Because isometries preserve distance, they preserve angle measure, betweenness, collinearity and area as well.

A congruence statement encodes a specific vertex correspondence. △ABC ≅ △DEF asserts AB ≅ DE, BC ≅ EF, AC ≅ DF, ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F — the six facts abbreviated as CPCTC.

Verifying all six is unnecessary. The congruence criteria (SSS, SAS, ASA, AAS, HL) each establish congruence from three well-chosen pieces, and the reason they work is that three such constraints determine a triangle up to rigid motion.

Composition of an even number of reflections is a direct isometry (a translation or rotation, preserving orientation); an odd number reverses orientation.