Question

Let $\angle 1$and $\angle 2$be complementary angles, and let $\angle 2$and $\angle 3$be supplementary angles. If $m\angle 1=x°$ and $m\angle 2=2x°$, find x. Then find $m\angle 1$, $m\angle 2$, and $m\angle 3$.

Short answer

The value of x is$30°$. $m\angle 1=30{}^{\circ },m\angle 2=60{}^{\circ },$and $m\angle 3=120°$

Step-by-step solution

Step 1. Given Information.

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

Step 2. Calculation.

Since, complementary angles are those whose sum of the angles is $90°$. That is,

$m\angle 1+m\angle 2=90°$

…… (1)

Find x using equation (1).

$\begin{array}{c}x°+2x°=90°\\ 3x°=90°\\ x=\frac{90°}{3°}\\ x=30°\end{array}$

Thus, the value of x is $30°$.

Therefore, the angle measures are $30°$, $60°$.

Also, supplementaryangles are those whose sum of the angles is $180°$. That is,

$m\angle 2+m\angle 3=180°$ ……. (2)

Find x using equation (2).

$m\angle 2+m\angle 3=180°$

Since, $m\angle 2=60°$

Therefore, the equation (2) becomes,

$\begin{array}{c}60°+m\angle 3=180°\\ m\angle 3=180°-60°\\ m\angle 3=120°\end{array}$

Therefore, $m\angle 3=120°$.

Step 3. Conclusion.

The angles are $m\angle 1=30{}^{\circ },m\angle 2=60{}^{\circ },$ and $m\angle 3=120°$.