Chapter 12.1, Problem 35E

To determine

To find: The value of $x$ when $\angle 1$ and $\angle 2$ are complementary angles, and $\angle 2$ and $\angle 3$ are supplementary angles.

Answer to Problem 35E

The value of $x$ is ${30}^{\circ }$

Explanation of Solution

Given:

  $m\angle 2=2x^{\circ }\text{ and }m\angle 1=x^{\circ }$ .

Calculation:

Since, complementary angles are those whose sum of the angles is ${90}^{\circ }$ . That is $m\angle 1+m\angle 2={90}^{\circ }$ ……………….. (1)

Find $x$ using equation (1)

  $m\angle 1+m\angle 2={90}^{\circ }$

  $\begin{array}{l}\Rightarrow x^{\circ }+2x^{\circ }={90}^{\circ }\\ \Rightarrow 3x^{\circ }={90}^{\circ }\\ \Rightarrow x=\frac{{90}^{\circ }}{3^{\circ }}\\ \Rightarrow x={30}^{\circ }\end{array}$

Thus, the value of $x$ is ${30}^{\circ }$ .

Therefore, the angle measures are ${30}^{\circ },{60}^{\circ }$

Also, supplementary angles are those whose sum of the angles is ${180}^{\circ }$ . That is $m\angle 2+m\angle 3={180}^{\circ }$ …………………… (2)

Find $x$ using equation (2)

  $m\angle 2+m\angle 3={180}^{\circ }$

  $\text{since }m\angle 2={60}^{\circ }$

Therefore, the equation (2) becomes

  ${60}^{\circ }+m\angle 3={180}^{\circ }$

  $\begin{array}{l}\Rightarrow m\angle 3={180}^{\circ }-{60}^{\circ }\\ \Rightarrow m\angle 3={120}^{\circ }\end{array}$

Therefore,

  $m\angle 3={120}^{\circ }$

Hence, the angles are $m\angle 1={30}^{\circ },m\angle 2={60}^{\circ },m\angle 3={120}^{\circ }$