Chapter 11, Problem 31CUR

To determine

To Find: The angle $m\angle 2$ .

Answer to Problem 31CUR

  $61°$

Explanation of Solution

Given:

  

$\overline{PQ},\overline{PR}$ are tangents. $m\angle 1=58$ .

By construction, referring to the below figure:

  

  $O$ is the center of the circle.

  $OQ,OR$ are the two radii of the circle.

  $m\angle 1\cong m\angle QPR$ (Vertically opposite angles)

  $m\angle QPR$ is the circumscribed angle.

  $m\angle 2$ is the inscribed angle.

Angle at center is twice the inscribed angle.

Therefore, $m\angle QOR=2m\angle 2$ .

Angle between tangents and radius is always $90°$ .

And sum of all interior angles of a quadrilateral is $360°$

In quadrilateral $PQOR$ :

  $\begin{array}{l}58+90+90+2m\angle 2=360\\ \Rightarrow 2m\angle 2=122\\ \Rightarrow m\angle 2=61\end{array}$