Chapter 9.5, Problem 7WE

To determine

To find: the values of $x,y$ and $z$ .

Answer to Problem 7WE

The angles are as follows:

  $\begin{array}{l}x=104°\\ y=57°\\ z=104°\end{array}$

Explanation of Solution

Given information:

In the diagrams that follow, $O$ is the center of the circle.

Calculation:

Consider the theorem,

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

Also, if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

  $\begin{array}{c}x°=180°-76°\\ =104°\end{array}$

Consider the figure,

From figure, by angle sum property,

  $\begin{array}{c}y°+y°+76°=180°\\ 2y°=104°\\ y=57°\end{array}$

Also, by theorem,

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

  $\begin{array}{c}z°=2y°\\ =104°\end{array}$

The angles are as follows:

  $\begin{array}{l}x=104°\\ y=57°\\ z=104°\end{array}$