Isosceles Triangle Theorem

The field of geometry is comprised of many different types of questions. Some are construction based, such as “what is the area of this circle?” On the other hand, many are proof based, like “why are these two triangles congruent?” This is slightly different from the proofs I normally discuss; proofs I normally post are very generalized theorems while these questions are more similar to a specific algebra or arithmetic problem.

When writing a geometric proof, many different theorems come into play. One must use the generalized theorems, properties, and postulates to arrive at a conclusion. Let’s try a simple proof just to demonstrate the nature of these theorems. Let’s prove that triangle ABE is congruent to triangle CDE.

First, we would say that it is given that segment AC is parallel to segment BD and that segment AB is parallel to segment CD. We would then say that ABCD is a parallelogram by the definition of a parallelogram (which requires two sets of parallel sides). The definition of a parallelogram also requires AB and CD to be congruent. The vertical angle theorem can be used to say that angle AEB is congruent to angle CED. The alternate interior angle theorem says that angle ABE is congruent to angle DCE. The angle-angle-side triangle congruence postulate then concludes that triangle ABE is congruent to triangle CDE.

As you can see, there are many steps in this proof, and each one uses a different definition, theorem, or postulate. In high school geometry classes, students are told these theorems and postulates, and expected to memorize them for future examples. This makes geometry boring and pointless, when it can be quite fascinating. A way to easily spice up geometric proofs is to actually prove the theorems before they are used in class. If Euclid could do it, then we can do it.

A fun one to prove is the Isosceles Triangle Theorem. This theorem states that when a triangle has two congruent sides, it also has two congruent angles. This can be proven in a similar way as the congruent triangle question I posed earlier. Take an isosceles triangle:

If you were to bisect that top angle, it would create two new triangles. Since the original triangle is isosceles, it is given that the top left segment is congruent to the top right segment. The definition of a bisection (cutting an angle in half) states that the left part of the top angle is congruent to the right part of the top angle. The reflexive property of congruence states that the middle segment is congruent to itself. By the side-angle-side triangle congruence postulate, the left triangle is congruent to the right triangle. And finally, by CPCTC (common parts of congruent triangles are congruent), the bottom left angle is congruent to the bottom right angle.

This sort of geometric proof language sounds extremely long and boring. However, finding uses for it such as proving the Isosceles Triangle Theorem can make it a little more fun. Among many things, I think that schools should teach the reasons behind these theorems to make it more logical and fun to apply them to class.