Question

An angle is $5$ times its supplement. Find the angle.

Answer 1

Supplementary Angles.

When the sum of the measures of two angles is $180°$, then the angles are called supplementary angles. When two angles are supplementary, each angle is said to be the supplement of the other.

Thus, two angles $\alpha$ and $\beta$ are said to be supplementary angles if $\alpha +\beta =180°$.

Given, an angle is equal to $5$ times its supplement. Let the given angle be $\alpha$ and let $\beta$ be its supplementary angle, then

$\alpha =5\beta$

Since $\alpha$ and $\beta$ are supplementary angles, therefore

$\begin{array}{rcl}\alpha +\beta & =& 180°\\ & \Rightarrow & 5\beta +\beta =180°\left[\because \alpha =5\beta \right]\\ & \Rightarrow & 6\beta =180°\\ & \Rightarrow & \beta =30°\end{array}$

Therefore,

$\begin{array}{rcl}\alpha & =& 5\beta \\ & \Rightarrow & \alpha =5\left(30°\right)\\ & \Rightarrow & \alpha =150°\end{array}$

Hence, the measure of the given angle is $150°$.