Chapter 3.4, Problem 26WE

To determine

To find: The measures of $\angle 1$ and $\angle 2$

Answer to Problem 26WE

The value of measures are $\angle 1={35}^{\circ }$ and $\angle 2={25}^{\circ }$

Explanation of Solution

Given information:

Consider the following

  

  $\begin{array}{l}\angle TMN={85}^{\circ }\\ \angle MTX={30}^{\circ }\end{array}$

Calculation:

Consider that $\overline{MN}$ parallel to $\overline{TR}$ and $\overline{MT}$ is a traversal line.

So sum of the interior angles on the same side of the transversal line is supplementary.

  $\begin{array}{cc}\angle TMN+\angle MTR& =180\\ 85+\angle MTR& =180\\ \angle MTR& =180-85\\ & =95\end{array}$

In a triangle exterior angle is the sum of interior remote angles.

  $\begin{array}{l}30+\angle MTR=\angle S+\angle 1\\ \begin{array}{cc}30+95& =90+\angle 1\\ 125-90& =\angle 1\\ \angle 1& =35\end{array}\end{array}$

Since $\overline{TS}$ parallel to $\overline{NR}$ and $\overline{RS}$

Also as $\angle S={90}^{\circ }$ so the value is $\angle \text{R}={90}^{\circ }$

Consider that if two lines are parallel and the line is perpendicular to one line,

It is perpendicular to other line.

  $\begin{array}{cc}NRT+1& ={90}^{\circ }\\ NRT& ={90}^{\circ }-{35}^{\circ }\\ & ={55}^{\circ }\end{array}$

Since the sum of the same side interior angles is supplementary.

  $\begin{array}{cc}\angle 2+{55}^{\circ }& ={180}^{\circ }\\ \angle 2& ={180}^{\circ }-{55}^{\circ }\\ & ={25}^{\circ }\end{array}$

Therefore the value of angle is $\angle 2={25}^{\circ }$