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.
