Congruence (geometry)

In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. More generally, two subsets A and B of Euclidean space Rⁿ are called congruent if there exists an isometry f : Rⁿ → Rⁿ with f(A) = B. Congruence is an equivalence relation.

Two sets that are not congruent are called non-congruent.

For instance:

   *                *                   *
   *                *         *        * *
   *****        *****         ***      ***
   *                *         *
   *                *

The first two figures are congruent to each other. The third is a different size, and so is similar but not congruent to the first two; the fourth is different altogether. Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distances and angles. The latter sort of properties are called invariants and studying them is the essence of geometry.

Congruence of triangles

Two triangles are congruent if their corresponding sides and angles are equal in measure. Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles:

SAS Axiom (Side-Angle-Side): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.

SSS Theorem (Side-Side-Side): Two triangles are congruent if their corresponding sides are equal.

ASA Theorem (Angle-Side-Angle): Two triangles are congruent if a pair of corresponding angles and the included side are equal.

While the AAS (Angle-Angle-Side) condition also guarantees congruence, SSA (Side-Side-Angle) does not, as there are often two dissimilar triangles with a pair of corresponding sides and a non-included angle equal. This is known as the ambiguous case. Of course, AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence shows only similarity and not congruence.

However, a derivative of the SSA condition is the HL (Hypotenuse-Leg) condition. This is true because all right triangles (which this condition is used with) have a congruent angle (the right angle). If the hypotenuse and a certain leg of a triangle are congruent to the corresponding hypotenuse and leg of a different triangle, the two triangles are congruent.