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Geometry: Congruence

Proving Congruence of Triangles



When proving that triangles are congruent, it is not necessary to prove that all three pairs of corresponding angles and all three pairs of corresponding sides are congruent. There are shortcuts. For example, if two pairs of corresponding angles are congruent, then the third angle pair is also congruent, since all triangles have 180 degrees of interior angles. The following three methods are shortcuts for determining congruence between triangles without having to prove the congruence of all six corresponding parts. They are called SSS, SAS, and ASA.

SSS (Side-Side-Side)

The simplest way to prove that triangles are congruent is to prove that all three sides of the triangle are congruent. When all the sides of two triangles are congruent, the angles of those triangles must also be congruent. This method is called side-side-side, or SSS for short. To use it, you must know the lengths of all three sides of both triangles, or at least know that they are equal.

SAS (Side-Angle-Side)

A second way to prove the congruence of triangles is to show that two sides and their included angle are congruent. This method is called side-angle-side. It is important to remember that the angle must be the included angle--otherwise you can't be sure of congruence. When two sides of a triangle and the angle between them are the same as the corresponding parts of another triangle there is no way that the triangles aren't congruent. When two sides and their included angle are fixed, all three vertices of the triangle are fixed. Therefore, two sides and their included angle is all it takes to define a triangle; by showing the congruence of these corresponding parts, the congruence of each whole triangle follows.

Figure %: Two sides and their included angle determine a triangle

ASA (Angle-Side-Angle)

The third major way to prove congruence between triangles is called ASA, for angle-side-angle. If two angles of a triangle and their included side are congruent, then the pair of triangles is congruent. When the side of a triangle is determined, and the two angles from which the other two sides point, the whole triangle is already determined, there is only one point, the third vertex, where those other sides could possibly meet. For this reason, ASA is also a valid shortcut/technique for proving the congruence of triangles.

Figure %: A triangle is determined by two angles and their included side