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Geometry: Congruence

More Ways to Prove Congruence

Problems

Problems

Aside from the traditional SSS, SAS, and ASA techniques for proving the congruence of triangles, there are two more ways to prove congruence of triangles. These methods are really only different applications of SSS, SAS, and ASA, but these new applications may be more suitable in certain situations.

AAS (Angle-Angle-Side)

AAS, or angle-angle-side, is another way to prove congruence by ASA, it is just disguised a bit. The measure of the third angle in a triangle can easily be calculated if two of the angle measures are known. With AAS, the measures of two angles are known along with the measure of a side not included by the angles. By calculating the measure of the third angle, you can create a situation in which ASA is applicable. As long as two angles and one side are known, ASA can be used to prove the congruence (or non-congruence, for that matter) of two triangles. AAS is just another way to think of ASA. It's basically a shortcut for a shortcut.

Hy-Leg (Hypotenuse-Leg)

When two triangles are right, their congruence can be proved using the hy-leg method of proof. Hy-leg states that if a leg and the hypotenuse of two right triangles are congruent, then the triangles are congruent. Can you see which of the original three techniques hy-leg comes from? If you know the lengths of the corresponding legs and hypotenuses, because the triangles are right, by definition you know that the triangles share corresponding angles of 90 degress. For this reason, Hy-leg is a simple take-off of the SAS method of proof. Hence, hy-leg is only a shortcut based on the SAS congruence of both legs of a right triangle and their included angle, the right angle.

Also, anytime two sides of a right triangle are known, the length of the third side can be easily calculated using the Pythagorean Theorem. Hy-leg takes less time, however, since it saves time calculating the length of the other leg.

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