Английская Википедия:Corresponding sides and corresponding angles

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Шаблон:Short description

Файл: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 Шаблон:Mvar corresponds to (in this case of congruence, equals in length) Шаблон:Mvar, Шаблон:Mvar corresponds to Шаблон:Mvar, Шаблон:Mvar corresponds to Шаблон:Mvar, and Шаблон:Mvar corresponds to Шаблон:Mvar, while angle Шаблон:Mvar corresponds to (equals) angle Шаблон:Mvar, Шаблон:Mvar corresponds to Шаблон:Mvar, Шаблон:Mvar corresponds to Шаблон:Mvar, and Шаблон:Mvar corresponds to Шаблон: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.[1]

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

See also

References

Шаблон:Reflist