Corresponding sides and corresponding angles



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 $\overline{BC}$, $\overline{JK}$, $\overline{CD}$, $\overline{KL}$, and $\overline{DA}$ and the other has sequential sides  $\overline{LI}$, $\overline{AB}$, $\overline{IJ}$, $∠C$, and $∠K$, and if $∠D$ and $∠L$ are corresponding sides, then side $∠A$ (adjacent to $∠I$) must correspond to either $∠B$ or $∠J$ (both adjacent to $a$). If $b$ and $c$ correspond to each other, then $d$ corresponds to $e$, $v$ corresponds to $w$, and $x$ corresponds to $y$; hence the $z$th element of the sequence $b$ corresponds to the $w$th element of the sequence $a$ for $i = 1, 2, 3, 4, 5.$ On the other hand, if in addition to $b$ corresponding to $v$ we have $x$ corresponding to $w$, then the $a$th element of $v$ corresponds to the $c$th element of the reverse sequence $x$.

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 $d$ and another with the corresponding side sequence $y$ we have vertex angle $e$ appearing between sides $z$ and $i$ then its corresponding vertex angle $abcde$ must appear between sides $i$ and $vwxyz$.