congruent

Congruent is an adjective used in mathematics and other areas to indicate that two objects are identical in shape and size, or, more broadly, that they are in agreement. In geometry, congruence means that one figure can be transformed into another by a rigid motion—rotation, translation, or reflection—without changing distances or angles. As a result, corresponding sides and angles of congruent figures are equal.

In Euclidean geometry, two polygons are congruent if there exists a vertex correspondence that makes every corresponding side and angle equal. For triangles, several criteria establish congruence: side–side–side (SSS), side–angle–side (SAS), angle–side–angle (ASA), angle–angle–side (AAS), and the hypotenuse–leg (HL) rule for right triangles. Circles are congruent when their radii are equal.

In number theory and algebra, the concept of congruence appears in modular arithmetic. The relation a ≡ b (mod n) means that n divides the difference a − b; this is an equivalence relation and partitions the integers into congruence classes modulo n. Arithmetic operations respect congruence: if a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d and ac ≡ bd (mod n).

Congruence is distinct from similarity. Congruent figures have identical size and shape, while similar figures have the same shape but may differ in size.

In everyday language, congruent can describe harmony or agreement between ideas, plans, or styles, indicating compatibility or consistency.