Question

An angle is one-fifth of its supplement. Find the measure of 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 one-fifth of its supplement. Let the given angle be $\alpha$ and let $\beta$ be its supplementary angle, then

$\begin{array}{rcl}\alpha & =& \frac{1}{5}\times \beta \\ & \Rightarrow & \alpha =\frac{\beta }{5}\\ & \Rightarrow & \beta =5\alpha \end{array}$

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

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

Hence, the measure of the required angle is $30°$.