kongruent

Kongruent is an adjective used in German and some other languages to denote agreement or compatibility, and in mathematics it describes equality of shape and size under a transformation.

In geometry, two figures are congruent if one can be transformed into the other by rigid motions—translations, rotations, and reflections. For polygons, congruence means corresponding sides and angles are equal. In triangles, there are standard criteria that establish congruence: side–side–side (SSS), side–angle–side (SAS), angle–angle–side (AAS/ASA), and, for right triangles, the hypotenuse–leg (HL) rule. If two triangles are congruent, there is a one-to-one correspondence between their vertices that preserves distance and angle.

In modular arithmetic, a and b are congruent modulo n if their difference is a multiple of n, written a ≡ b (mod n). This partitions the integers into congruence classes and underpins many number-theoretic results.

In algebra, a congruence relation is an equivalence relation that is compatible with the algebraic operations of a structure, enabling the construction of quotient structures such as quotient groups and quotient rings. These relations respect addition and multiplication, ensuring that operations on equivalence classes are well defined.

Etymology: kongruent comes from Latin congruentem, from con- “together” and gruent- related to gruere “to grow together.” The term is closely related to the English word congruent.

Usage notes: In German, kongruent can also describe consistency or alignment beyond geometry, as in “eine kongruente Lösung” (a congruent/consistent solution). In English, congruent is predominantly a geometric and abstract-algebra term.