Chapter 12.1, Problem 8E

To determine

To Calculate: The measure of an interior angle of a triangle when the measure of an exterior angle and the measure of vertical angle of another angle are given

Answer to Problem 8E

The measure of $\angle 5$ is ${75}^o$ .

Explanation of Solution

Given Information: The measures $m\angle 1={35}^o$ and $m\angle 3={110}^o$ in the figure

  

Concept Used:

• The measures of a pair of vertical angles are equal.
• The sum of measures of a pair of adjacent angles forming a straight angle (supplementary angles) is ${180}^o$ .
• The sum of the measures of the three angles of a triangle is ${180}^o$ .

  

In the above figure, $\angle 1$ and $\angle 2$ are vertical angles. So

  $m\angle 2=m\angle 1$

Put the given value of $m\angle 1={35}^o$ in the above equation to write

  $m\angle 2=35^o$

Now, in the figure, $\angle 3$ and $\angle 4$ are supplementary angles as they together form a straight angle. So

  $m\angle 3+m\angle 4={180}^o$

Put the given value of $m\angle 3={110}^o$ in the above equation to write

  $m\angle 4+{110}^o={180}^o$

  $\Rightarrow m\angle 4=70^o$

Finally, since in the figure $\angle 2,\angle 4,\angle 5$ form a triangle, the sum of their measures should be equal to ${180}^o$ . So

  $m\angle 2+m\angle 4+m\angle 5={180}^o$

Put the earlier computed values of $m\angle 2$ and $m\angle 4$ in the above equation to write

  $m\angle 5+{70}^o+{35}^o={180}^o$

  $m\angle 5={180}^o-\left({35}^o+{70}^o\right)={180}^o-{105}^o=75^o$

  $$