Chapter 9.3, Problem 15WE

To determine

To prove: $m\angle Z=n$.

Explanation of Solution

Given information :

$\overline{WZ}$ is a diameter of circle and $m\left(WX\right)=m\left(XY\right)=n$

Proof: First let us draw $\overline{OY}$ in the given figure.

Then $\overline{OY}$ and $\overline{OZ}$ becomes radii of the circles $\Rightarrow \overline{OY}=\overline{OZ}$

Since $m\left(XY\right)=m\angle XOY=n$

Since $\overline{OX}\left|\right|\overline{ZY}$ and $\overline{OY}$ transverse line

$\Rightarrow m\angle XOY=m\angle OYZ=n$

and also $\Delta OZY$ is isosceles triangles as $\overline{OY}=\overline{OZ}$ then $m\angle OYZ=m\angle OZY=n$

Hence, $m\angle Z=n$