An isosceles triangle is a triangle that has two equal sides.
The isosceles triangle theorem states the following:
Isosceles Triangle Theorem
In an isosceles triangle, the angles opposite to the equal sides are equal. Conversely, if the base angles of a triangle are equal, then the triangle is isosceles.
Consider isosceles triangle with and suppose the internal bisector of intersects at Now consider the triangles and . We have , and by construction. Hence, by the SAS congruence axiom. So .
Note: The converse holds, too. If we were given that , in a similar way we would get by the AAS congruence theorem. Thus, follows immediately.
This theorem gives an equivalence relation. In order to show that two lengths of a triangle are equal, it suffices to show that their opposite angles are equal. In fact, given any two segments and in the plane with as a common endpoint, we have . So in a geometry problem, if we are to show equality of two sides of a triangle, we can start chasing angles!