Chapter 9.6, Problem 10WE

To determine

Tocalculate: The measure of each numbered angle.

Answer to Problem 10WE

The value of each measure numbered angle = $60°$ .

Explanation of Solution

Giveninformation:

Line $\overline{AC}$ is a diameter.

Line $\overleftrightarrow{BZ}$ is tangent to $\odot \text{O}$ .

  $\begin{array}{l}m\stackrel{⌢}{AB}=90°\\ m\stackrel{⌢}{CD}=30°\\ m\stackrel{⌢}{DE}=20°\end{array}$

  

In triangle $△BOE$ , $\angle BOE=90°$ .

So the value of angle can be calculated as,

  $\begin{array}{l}m\angle BEO=\frac{1}{2}\left(m\stackrel{⌢}{AB}+m\stackrel{⌢}{CD}\right)\hfill \\ =\frac{1}{2}\left(90+30\right)\hfill \\ \begin{array}{l}=\frac{1}{2}\times 120\\ =60\end{array}\hfill \end{array}$

Therefore, the value of angle is $60°$ .

The sum of three angles of a triangle is $180°$ .

  $\begin{array}{l}\begin{array}{l}\angle OBE=180–\left(90+60\right)\hfill \\ =180–150\hfill \end{array} \\ =30°\end{array}$

The line which is perpendicular to the radius at the outer endpoint is known as tangent to the circle.

The value of other angle is,

  $\begin{array}{l}\angle EBZ=90°–30°\hfill \\ =60°\hfill \end{array}$

Hence, the value of each measure numbered angle = $60°$ .