Exercise 39 Page 568

We are given the following diagram.

We want to find $m\angle P$ when $m\overgroup{WZY}$ is equal to $200^{\circ }.$ First, by the Angles Outside the Circle Theorem we can write an equation. With this equation we need to find either $m\overgroup{WZ}$ and $m\overgroup{XY},$ or $m\overgroup{WZ}-m\overgroup{XY}.$ To do so we will use the given measure of $\overgroup{WZY}$ and the measure of the entire circle. Notice that by the Arc Addition Postulate the sum of $m\overgroup{WZ}$ and $m\overgroup{YZ}$ is equal to $m\overgroup{WZY}.$ Also, since the measure of the entire circle is $360^{\circ },$ we can write the following equation by the Arc Addition Postulate. From here, since we know the sum of $m\overgroup{WZ}$ and $m\overgroup{YZ}$, we can find $m\overgroup{WX}+m\overgroup{XY}.$ We are also given two equal chords. By the Congruent Corresponding Chords Theorem, their minor arcs are congruent, so the measure of the minor arcs is equal. We can substitute $m\overgroup{YZ}$ for $m\overgroup{WX}.$ Now we have a system of equations. Let's solve the system algebraically by the Elimination Method. Next, we will substitute $40$ for $m\overgroup{WZ}-m\overgroup{XY}$ in the equation of the Angles Outside the Circle Theorem. Consequently, the measure of $\angle P$ is $20^{\circ }.$