Kongruentes

Kongruentes is a mathematical term used to describe objects that are equal in shape and size, or more broadly to describe a relation of equivalence under a specific rule, depending on the context. In geometry, two figures are congruent if one can be transformed into the other by a rigid motion, such as a translation, rotation, or reflection. Congruent figures have corresponding parts that are equal in length, angle, and area, and the symbol ≅ is commonly used to denote congruence.

In triangle geometry, several criteria guarantee that two triangles are congruent. The SSS (side–side–side) criterion states that if all three corresponding sides are equal, the triangles are congruent. SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse–leg, for right triangles) are other established criteria. When congruent, corresponding parts—sides and angles—match exactly.

In number theory, congruence modulo n is a relation on integers: a ≡ b (mod n) if the difference a − b is divisible by n. This partitions integers into residue classes modulo n and is compatible with addition and multiplication. For example, 5 ≡ 2 (mod 3) because 5 − 2 = 3 is divisible by 3.

Congruence is distinct from similarity: congruent figures have identical size and shape, while similar figures have the same shape but may differ in size. Kongruentes thus serves as a foundational concept in geometry, algebra, and number theory, with wide-ranging applications in proofs, cryptography, computer graphics, and calendar calculations. In German-language mathematics, Kongruentes is the adjective form, with Kongruenz as the noun for the concept of congruence.