Chapter 2.4, Problem 18WE

To determine

The measure of $m\angle ZOT$

Answer to Problem 18WE

The measure of $m\angle ZOT$ is ${85}^{\circ }$

Explanation of Solution

Given information:

  

  $\overrightarrow{OT}$ bisects $\angle SOU,m\angle UOV=35,$ and $m\angle YOW=120.$

As the angles $\angle SOY$ and $\angle YOW$ forms a linear pair.

From the figure $\angle YOW={120}^{\circ }$

Thus,

  $\begin{array}{cc}SOY+\angle YOW={180}^{\circ }& \\ \angle SOY+{120}^{\circ }={180}^{\circ }& \\ \angle SOY={180}^{\circ }-{120}^{\circ }& \\ \angle SOY={60}^{\circ }& \end{array}$

Also $\angle SOY$ and $\angle UOW$ are vertically opposite angles and therefore are equal.

  $\angle SOY={60}^{\circ }$

Thus, $\angle UOW={60}^{\circ }$

Also, from the figure $m\angle UOV={35}^{\circ }$

From Angle addition postulate,

  $\begin{array}{l}m\angle UOW=m\angle UOV+m\angle VOW\\ {60}^{\circ }={35}^{\circ }+m\angle VOW\\ m\angle VOW={25}^{\circ }\end{array}$

Thus, the measure of the angle $\angle VOW$ is ${25}^{\circ }$

As $\angle ZOS$ and $\angle VOW$ are vertically opposite angles and therefore are equal.

From the figure $\angle VOW={25}^{\circ }$

This implies

  $\angle ZOS={25}^{\circ }$

Since $\angle SOU$ and $\angle YOW$ are vertically opposite angles and therefore are equal.

Therefore, $\angle YOW={120}^{\circ }$

This implies that

  $\angle SOU={120}^{\circ }$

Thus, the measure of $\angle \text{SOU}$ is ${120}^{\circ }$

As $\overrightarrow{OT}$ bisects $\angle SOU$

Thus, by the angle bisector theorem

  $\begin{array}{l}\angle TOU=\frac{1}{2}\angle SOU\\ =\frac{1}{2}\times {120}^{\circ }={60}^{\circ }\end{array}$

Thus, the measure of the angle $\angle TOU$ is ${60}^{\circ }$

From Angle addition postulate,

  $\begin{array}{l}m\angle ZOT=m\angle ZOS+m\angle SOT\\ ={25}^{\circ }+{60}^{\circ }\\ ={85}^{\circ }\end{array}$

Thus, the measure of $m\angle ZOT$ is ${85}^{\circ }$