Question

a. Given:$\overline{WZ}$is a diameter of $\odot O;$

Prove: $m\angle Z=n.$

Short answer

The final answer is $m\angle Z=\text{n.}$

Step-by-step solution

Step 1. Given information:

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

Step 2. Concept used:

We use basic geometric and angle concept

Step 3. Applying the concept:

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, $m\angle Z =n$