Base Angles of Isosceles Triangles are Congruent

Proof is probably one of the most difficult concepts to teach in high school mathematics. In this short post, we are going to discuss a simple activity that can be used in teaching mathematical proofs in Geometry. First, we can use a rectangular paper, fold it in the middle, and cut through the diagonal. Of course, before cutting the paper, we should clarify our basic assumptions.

Assumptions

1.) The paper is a rectangle.

2.) The fold is straight and connects the middle of the opposite edges.

3.) The cut is a straight line.

Discussion

First, we have learned that triangles with two congruent sides are called isosceles triangles. This is called a definition.  Continue reading

Triangles with Non Congruent Sides

We have shown that in an isosceles triangle, the angles opposite the congruent sides are congruent. This was the Isosceles Triangle Theorem which we proved two weeks ago.

In this post, we are gong to learn a slightly related theorem: a theorem that states that if in a triangle, two sides are not congruent, then the angles opposite these sides are not congruent and the angle opposite to the larger side is the larger angle.

 triangle

In the triangle above, $latex \overline{AB}$ is greater than $latex \overline{BC}$. We will show that they are not equal and $latex \angle C > \angle B$. Continue reading

The Proof of the Isosceles Triangle Theorem

An isosceles triangle is a triangle whose two sides are congruent. In the figure below, $latex ABC$ is an isosceles triangle and $latex \overline{AB} \cong \overline{BC}$.

The Isosceles Triangle Theorem states that the angles opposite to the two congruent sides of an isosceles triangle are congruent. In the figure above, the theorem states that since $latex AB \cong BC$, $latex \angle A \cong \angle C$.

The proof to this theorem uses the SSS triangle congruence. The SSS Triangle Congruence Theorem states that if the three corresponding sides of two triangles are congruent, then the two triangles are congruent. Continue reading