## A Proof of The Trapezoid Angle Theorem

A trapezoid is a quadrilateral whose two sides are parallel. In the figure below, $ABCD$ is a trapezoid and $AB$ is parallel to $CD$. In this post, we are going to prove that in a trapezoid, the consecutive angles between a pair of parallel sides are complementary. That is, we are going to prove in the figure above that $m \angle A = m \angle D = 180$ degrees and $m \angle B + m \angle C = 180$.

The Trapezoid Angle Theorem.

In a trapezoid, the consecutive angles between a pair of parallel sides are complementary.  Continue reading…

## The Proof of the Kite Symmetry Theorem

A kite is a quadrilateral with two distinct pairs of congruent sides. The common vertices of its congruent sides are called its ends. In quadrilateral $ABCD$ below, the distinct pairs of congruent sides are $\overline{AB}$ & $\overline{BC}$ and $\overline{AD}$ & $\overline{CD}$. The ends are $B$ and $D$.

Exercise: Locate the ends and the pairs of the distinct pairs of the remaining quadrilaterals.

In this post, we are going to prove the Kite Symmetry Theorem. The theorem states that the line containing the ends of a kite is a symmetry line for the kite. In proving this theorem, we are going to use the Figure Reflection Theorem which is stated as follows. Continue reading…

## Proof that The Diagonals of a Rhombus are Perpendicular

In the previous post, we have proved the converse of the Pythagorean Theorem.  In this post, we will prove that the diagonals of a rhombus are perpendicular to each other. That is, if we have parallelogram ABCD with diagonal $\overline{AC}$ and $\overline{BD}$, then $\overline{AC}$ is perpendicular to $\overline{BD}$.

What We Know

A rhombus is a parallelogram with four congruent sides. So, all sides of rhombus ABCD are congruent. That is

$\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}$Continue reading…