# 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.

If a figure determined by certain points, then its reflection image is the corresponding figure determined by the images of those points.

Now, we proceed with the proof.

The Kite Symmetry Theorem

The line containing the ends of a kite is a symmetry line for the kite.

Proof

Let $ABCD$ be a kite with with ends $B$ and $D$. $AB = BC$ and $AD = DC$ since the ends of the kite are $B$ and $D$ (definition of the ends of a kite).

Triangle $ABC$ and triangle $ADC$ are isosceles since each triangle has two pairs of congruent sides (definition of isosceles triangle).

We draw $AC$, and let line $m$ be the angle bisector of $AC$. The reflection of A through $m$ is $C$ and the reflection of $C$ through $m$ is $A$ (definition of reflection).

Now, line $m$ contains $B$ and $D$ since the bisector of an isosceles triangle is the bisector of the vertex angle and therefore contains the vertex.

There reflection of $B$ through $m$ is itself and the reflection of $D$ is itself (definition of reflection).

The reflection of $ABCD$ through $m$ is $CBAD$ by the Figure Reflection Theorem.

Therefore, line m which is also line $BD$ is a symmetry line for $ABCD$ (definition of symmetry line).