Yhdenmukukaisiin

Yhdenmukukaisiin is the illative plural form of the Finnish adjective yhdenmukainen, used in mathematical Finnish to describe objects that are congruent. In geometry, congruent figures have the same shape and size, and one can be mapped onto the other by a rigid motion.

Two figures are yhdenmukaiset (congruent) if there exists an isometry—such as a translation, rotation, or reflection (and combinations thereof)—that takes one figure onto the other. Under this mapping, corresponding distances and angles are preserved. The concept is central to proofs and constructions, since congruent figures behave identically in geometric relations.

In the plane, congruence is commonly tested for triangles using criteria such as SSS (three sides), SAS (two sides and the included angle), ASA (two angles and a side), AAS (two angles and a non-included side), and RHS (right triangle hypothesis, side and angle). For polygons with more than three sides, congruence is established by corresponding side lengths and interior angles, or by showing a suitable sequence of rigid motions.

Congruence is distinct from similarity. Similar figures have the same shape but not necessarily the same size; congruent figures must also share size. The symbol ≅ is used to denote congruence, e.g., ΔABC ≅ ΔA'B'C'. In practice, yhdenmukaisuus justifies replacing one figure with a congruent counterpart in reasoning, constructions, and measurements.

See also: isometry, congruence transformation, rigidity, similarity, Euclidean geometry.