Chapter 9.5, Problem 9WE

To determine

To Find : The values of x, y and z.

Answer to Problem 9WE

The values of $x=50,y=170$ and $z=35$ .

Explanation of Solution

Given information:

In the figure O is center of the circle.

Consider the following labeled figure:

  

Two inscribed angles of the same intercept arc are congruent within a circle.

Consider the following theorem

The calculation of the inscribed angle is equal to half of its intercepted arc measurement.

And, the measurement of the angle created by the tangent and the chord is equal to half the measurement of the arc intercepted by it.

  $\begin{array}{l}x°=\frac{100°}{2}\hfill \\ =50°\hfill \\ \angle CBD=\frac{90°}{2}\hfill \\ \angle CBD=45°\hfill \end{array}$

By angle sum property,

  $\begin{array}{l}x°+\angle CBD+\angle CDB=180°\hfill \\ \angle CDB=85°\hfill \\ y°=2\times 85°\hfill \\ =170°\hfill \end{array}$

Also, if a quadrilateral is inscribed in a circle, then it's opposite angles are supplementary.

  $\begin{array}{l}\angle CAB+\angle CDB=180°\hfill \\ \angle CAB=95°\hfill \end{array}$

By angle sum property,

  $\begin{array}{l}\begin{array}{l}\angle CAB+\angle ACB+\angle ABC=180°\hfill \\ 95°+z°+x°=180°\hfill \end{array}\\ z°=35°\end{array}$