Chapter 9.2, Problem 8WE

a.

To determine

$m\angle TSR$ and $m\angle TRS$ .

Answer to Problem 8WE

  $m\angle TSR=72°$ and $m\angle TRS=72°$ .

Explanation of Solution

Given:

  $TR$ and $TS$ are tangents to the circle at $O$ from $T$ .

  $m\angle RTS=36°$ .

  

  $TR=TS$ (Tangents from a same point towards the circle are equal).

In $\Delta RTS$ ,

  $m\angle RTS=36°$ .

Since $TR=TS$ ,

  $\Rightarrow \angle TSR=\angle TRS=x$ (angles opposite to equal sides are equal).

Again, since $m\angle RTS+m\angle TRS+m\angle TSR={180}^0$ (angle sum property of a triangle),

  $\Rightarrow 180°=36°+x+x$ ,

  $\Rightarrow 180°=36°+2x$ ,

  $\Rightarrow x=72°$

  $\therefore m\angle TSR=m\angle TRS=72°$ .

Conclusion:

The measures are: $m\angle TSR=72°$ and $m\angle TRS=72°$ .

b.

To determine

$m\angle ORS$ and $m\angle OSR$ .

Answer to Problem 8WE

  $m\angle ORS=m\angle OSR=18°$ .

Explanation of Solution

Given:

  $TR$ and $TS$ are tangents to then circle at $O$ from $T$ .

  $m\angle RTS=36°$ .

  

Calculation:

In quadrilateral $RTSO$ ,

  $\angle RTS+\angle TSO+\angle SOR+\angle ORT=360°$ (angle sum property of a quadrilateral)

  $\angle TRO=\angle TSO=90°$ (a tangent form an angle of $90°$ with the surface of the circle.)

By plugging the values, we have

  $\Rightarrow 36°+90°+90°+\angle ROS=360°$

  $\Rightarrow 216°+\angle ROS=360°$

  $\Rightarrow m\angle ROS=144°$

Now in $\Delta SOR$ ,

  $RO=SO$ (radii of the circle),

  $\Rightarrow \angle ORS=\angle OSR=x$ (angle opposite to equal sides are equal).

Now,

  $\angle ROS+\angle ORS+\angle OSR=180°$ (angle sum property of a triangle),

  $\Rightarrow 144°+x+x=180°$ ,

  $\Rightarrow 2x=36°$

  $\Rightarrow x=18°$

Conclusion:

Therefore, $m\angle ORS=m\angle OSR=18°$ .

c.

To determine

To find the value of $m\angle ROS$ .

Answer to Problem 8WE

  $m\angle ROS=144°$

Explanation of Solution

Given:

  $TR$ and $TS$ are tangents to circle $O$ from $T$ .

  $m\angle RTS=36°$ .

  

In quadrilateral $RTSO$ ,

  $\angle RTS+\angle TSO+\angle SOR+\angle ORT=360°$ (angle sum property of a quadrilateral)

  $\angle TRO=\angle TSO=90°$ (a tangent form an angle of $90°$ with the surface of the circle.)

By putting the values,

  $\Rightarrow 36°+90°+90°+\angle ROS=360°$ ,

  $\Rightarrow 216°+\angle ROS=360°$ ,

  $\Rightarrow m\angle ROS=144°$ .

Conclusion:

Therefore, the $m\angle ROS=144°.$