Corresponding sides and corresponding angles

File:Quadrilateral congruence.png

The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established in that side Template:Mvar corresponds to (in this case of congruence, equals in length) Template:Mvar, Template:Mvar corresponds to Template:Mvar, Template:Mvar corresponds to Template:Mvar, and Template:Mvar corresponds to Template:Mvar, while angle Template:Mvar corresponds to (equals) angle Template:Mvar, Template:Mvar corresponds to Template:Mvar, Template:Mvar corresponds to Template:Mvar, and Template:Mvar corresponds to Template:Mvar.

In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency.

For example, if one polygon has sequential sides Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar and the other has sequential sides Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar, and if Template:Mvar and Template:Mvar are corresponding sides, then side Template:Mvar (adjacent to Template:Mvar) must correspond to either Template:Mvar or Template:Mvar (both adjacent to Template:Mvar). If Template:Mvar and Template:Mvar correspond to each other, then Template:Mvar corresponds to Template:Mvar, Template:Mvar corresponds to Template:Mvar, and Template:Mvar corresponds to Template:Mvar; hence the Template:Mvarth element of the sequence Template:Mvar corresponds to the Template:Mvarth element of the sequence Template:Mvar for Template:Math On the other hand, if in addition to Template:Mvar corresponding to Template:Mvar we have Template:Mvar corresponding to Template:Mvar, then the Template:Mvarth element of Template:Mvar corresponds to the Template:Mvarth element of the reverse sequence Template:Mvar.

Congruence tests look for all pairs of corresponding sides to be equal in length, though except in the case of the triangle this is not sufficient to establish congruence (as exemplified by a square and a rhombus that have the same side length). Similarity tests look at whether the ratios of the lengths of each pair of corresponding sides are equal, though again this is not sufficient. In either case equality of corresponding angles is also necessary; equality (or proportionality) of corresponding sides combined with equality of corresponding angles is necessary and sufficient for congruence (or similarity). The corresponding angles as well as the corresponding sides are defined as appearing in the same sequence, so for example if in a polygon with the side sequence Template:Mvar and another with the corresponding side sequence Template:Mvar we have vertex angle Template:Mvar appearing between sides Template:Mvar and Template:Mvar then its corresponding vertex angle Template:Mvar must appear between sides Template:Mvar and Template:Mvar.

See also

• Template:Section link