Chapter 9.3, Problem 14WE

To determine

To prove: Arc length $WX\cong$arc length XY.

Explanation of Solution

Given information :

$\overline{WZ}$ is a diameter of circle and $\overline{OX}\left|\right|\overline{ZY}$

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

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

Let $\angle WOX=x^{\circ }$ , since $\overline{OX}\left|\right|\overline{ZY}$ then $\angle OZY=x^{\circ }$ and also $\Delta OZY$ is isosceles triangles as $\overline{OY}=\overline{OZ}$ then $\angle OYZ=x^{\circ }$

From $\Delta OZY$ , we have $\angle OZY+\angle ZYO+\angle YOZ={180}^0$

$\begin{array}{l}\Rightarrow x^{\circ }+x^{\circ }+\angle ZOY={180}^{\circ }\\ \Rightarrow \angle ZOY={180}^{\circ }-2x^{\circ }\end{array}$

Also from straight line $\overline{WZ}$ , we have $\angle WOX+\angle XOY+\angle ZOY={180}^0$

$\begin{array}{l}\Rightarrow x^{\circ }+\angle XOY+{180}^{\circ }-2x^{\circ }={180}^{\circ }\\ \Rightarrow \angle XOY-x^{\circ }=180-180\\ \therefore \angle XOY=x^{\circ }\end{array}$

Thus, $\angle WOX=x^{\circ }=\angle XOY$

$\Rightarrow$ Arcs made by same angle in a circle are congruent.

Hence, Arc length $WX\cong$arc length XY.