kongruente

Kongruente is a term used in mathematics to denote equality up to a specific relation. In geometry, two figures are congruent when they have the same size and shape; equivalently, one can be moved to coincide with the other by a rigid motion, such as a translation, rotation, or reflection. In modular arithmetic, two integers a and b are congruent modulo n, written a ≡ b (mod n), if n divides their difference a − b.

The congruence relation is an equivalence relation, meaning it is reflexive, symmetric, and transitive. In geometry, congruence preserves lengths and angles, so congruent figures have equal corresponding sides and angles. In the arithmetic setting, congruence classes modulo n partition the integers into residue classes and form a foundational tool in number theory.

In plane and solid geometry, triangle congruence is a central concept. Two triangles are congruent if they have the same side lengths and angle measures. There are several standard criteria for establishing triangle congruence: SSS (side–side–side), SAS (side–angle–side), ASA (angle–side–angle), AAS (angle–angle–side), and, for right triangles, HL (hypotenuse–leg).

Practical applications of congruence include geometric proofs, tiling and design in art and architecture, and algorithmic tasks in computer graphics. In number theory and cryptography, modular congruence underpins algorithms, error detection, and the study of integer patterns.