Prove that the diagonals of kite UVWX are perpendicular.

Steps to Solve

1. Identify the Properties of a Kite

A kite has two pairs of adjacent sides that are equal in length. Label the vertices as follows: $U$, $V$, $W$, and $X$. The diagonals will be $UV$ and $WX$. We need to show that these diagonals intersect at right angles.

2. Label Key Points and Angles

Let the intersection of the diagonals be point $O$. Since diagonal $UV$ bisects diagonal $WX$, we can denote $WO = OX$. Let's call the lengths of these segments $a$ and $b$, respectively.

3. Use Triangle Congruence

Since $UO = VO$ (because sides $UV$ are equal) and $WO = OX$, we have two triangles: $\triangle UWO$ and $\triangle VXO$. By the Side-Side-Side (SSS) postulate, these triangles are congruent:

$$ \triangle UWO \cong \triangle VXO $$

4. Show that Angles are Equal

The congruence of triangles $\triangle UWO$ and $\triangle VXO$ implies that the corresponding angles are equal, specifically $\angle UWO = \angle VXO$.

5. Use the Fact that Angles Add Up to 180 Degrees

From the property of angles formed when two lines intersect, we know that $\angle UWO + \angle VXO = 180^\circ$. Since $\angle UWO = \angle VXO$, we can write:

$$ 2\angle UWO = 180^\circ $$

6. Conclude the Angles are Right Angles

Dividing both sides by 2 results in:

$$ \angle UWO = 90^\circ $$

This means that the diagonals $UV$ and $WX$ intersect at right angles.