Chapter 10.6, Problem 20PSB

To determine

Give the explanation for the given proof.

Explanation of Solution

Given:

The given diagram is

  

  $\Delta WXZ$ is isosceles.

  $\overline{WX}\cong \overline{WZ}$ .

The given proof is the $Y$ is the midpoint of $\overline{XZ}$ .

Calculation:

Draw a segment $\overline{WY}$ .

  

The angle inscribed in a semicircle is a right angle.

  $\begin{array}{l}m\angle WYZ={90}^{°}\\ m\angle WYX={180}^{°}-{90}^{°}={90}^{°}\end{array}$

  $\Delta WXZ$ is isosceles.

  $\overline{WX}\cong \overline{WZ}$ .

So, $m\angle WXY=m\angle WZY$

If the two angles are equal, then the third angle also be equal.

From the above result, $\overline{WX}=\overline{WZ}$ and $\overline{WY}=\overline{WY}$ .

From $RHS$ , $\Delta XYW\cong \Delta ZYW$

If $\Delta XYW\cong \Delta ZYW$ , then $\overline{XY}=\overline{YZ}$

Hence $Y$ is the midpoint of $\overline{XZ}$ .