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Question

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

Solution

**Complementary Angles :**

When the sum of the measures of two angles is $90\xb0$, 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 $\alpha +\beta =90\xb0$.

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

$\begin{array}{rcl}\alpha +\beta & =& 90\xb0\\ & \Rightarrow & 8\beta +\beta =90\xb0\left[\because \alpha =8\beta \right]\\ & \Rightarrow & 9\beta =90\xb0\\ & \Rightarrow & \beta =10\xb0\end{array}$

Therefore,

$\begin{array}{rcl}\alpha & =& 8\beta \\ & =& 8\left(10\xb0\right)\\ & =& 80\xb0\end{array}$

**Hence, the measure of the given angle is **$80\xb0$**.**

Mathematics

Standard IX

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