congruentem

Congruentem is a form encountered in Latin-language mathematical writing, serving as the accusative singular of the adjective meaning congruent. In Latin texts, congruentem would be used to describe two geometric figures that are congruent, that is, two figures that have the same shape and size. The underlying concept, congruence, is central to Euclidean geometry and to the study of rigid motions.

In geometry, two figures are congruent if there exists an isometry that maps one onto the other. Such a transformation preserves distances and angles, so corresponding lengths and angles are equal. Consequently, congruent figures have identical shapes and sizes, even if their positions or orientations differ.

Congruence is distinct from similarity. Similar figures have the same shape but may differ in size, while congruent figures share both shape and size. This distinction is often emphasized in geometric proofs and constructions.

For triangles, several standard criteria can establish congruence without direct measurement of all parts: SSS (three corresponding sides equal), SAS (two sides and the included angle equal), ASA (two angles and a side), AAS (two angles and a non-included side), and, in right triangles, the RHS or HL criterion. These criteria enable proofs that two triangles are congruent and thus have corresponding parts equal.

Notationally, modern English-language mathematics typically uses the symbol ≅ to denote congruence. In Latin texts, congruence would be described in prose using congruentem or related forms rather than a dedicated symbol.