Exercise 23 Page 394

Let's find the measures of the numbered angles one at a time.

Measure of $\angle 1,$ $\angle 2,$ $\angle 3$ and $\angle 4$

Before we begin, let's name the vertices of our kite. We will call it quadrilateral $ABCD.$

If a quadrilateral is a kite, then its diagonals are perpendicular. Therefore $m\angle 1=90,$ $m\angle 2=90,$ $m\angle 3=90$ and $m\angle 4=90.$

Measure of $\angle 5$ and $\angle 6$

Notice that $△BAD$ and $△BCD$ are congruent by the Side-Side-Side (SSS) Congruence Theorem. Since corresponding parts of congruent triangles are congruent, $\angle ABD\cong \angle CBD$ and $\angle ADB\cong \angle CDB.$ Because these angles are congruent, we know that their measures are equal.