## 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…

## Quadrilaterals With Congruent Opposite Angles Are Parallelogram

In the previous post, we have proved that if the opposite sides of a quadrilateral are congruent, then they are parallel. In this post, we are going to show that if the opposite angles of a quadrilateral are congruent, then it is a parallelogram.

In the figure below, we have quadrilateral ABCD with $\angle A \cong \angle C$ and $\angle B \cong \angle D$. To show that it is a parallelogram, we have to show that $AB \parallel CD$ and $AD \parallel BC$.

Theorem: If two pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram.

Proof

It is given that $\angle A \cong \angle C$ and $\angle B \cong \angle D$Continue reading…

## Congruent Opposite Sides of Quadrilaterals are Parallel

In this post, we are going to discuss one basic concepts of quadrilaterals. That is, if we have a quadrilateral, and the opposite sides are congruent, then these opposite sides are parallel.  This is the same if the two opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Consider the quadrilateral below. The lengths of AB and CD are equal. Also, The lengths of AD and BC are also equal. So, we are going to show that these pairs of sides are parallel.

Theorem: If the two pairs of the opposite sides of a quadrilateral are congruent, then they are parallel.

Given: $AB = CD$  and $AD = BC$Continue reading…