Chapter 9.5, Problem 5CE

To determine

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

Answer to Problem 5CE

The measure of the angles is;

  $\begin{array}{l}x=25°\\ y=65°\end{array}$

Explanation of Solution

Given information:

Given figure,

  $O$ 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.

Hence,

The measure of the arc $AB$ is $50°$ .

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

That is,

  $\begin{array}{c}x=\frac{1}{2}\times 50°\\ x=25°\end{array}$

Also, in a circle, any angle in a semi0circle is a right angle.

And by angle sum property,

  $\begin{array}{c}x+y+90°=180°\\ 25°+y+90°=180°\\ 115°+y=180°\\ y=65°\end{array}$

Therefore the measure of the angles is,

  $\begin{array}{l}x=25°\\ y=65°\end{array}$