Question

An angle is equal to $8$ times of its complement. Find its measure.

Answer 1

Complementary Angles :

When the sum of the measures of two angles is $90°$, then the angles are called complementary angles. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles.

Thus, two angles $\alpha$ and $\beta$ are said to be complementary angles if $\alpha +\beta =90°$.

Given, an angle is equal to $8$ times of its complement. Let the given angle be $\alpha$ and let $\beta$ be its complementary angle, then

$\alpha =8\beta$

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

$\begin{array}{rcl}\alpha +\beta & =& 90°\\ & \Rightarrow & 8\beta +\beta =90°\left[\because \alpha =8\beta \right]\\ & \Rightarrow & 9\beta =90°\\ & \Rightarrow & \beta =10°\end{array}$

Therefore,

$\begin{array}{rcl}\alpha & =& 8\beta \\ & =& 8\left(10°\right)\\ & =& 80°\end{array}$

Hence, the measure of the given angle is $80°$.