Question

The sum of two complementary angles is $$90$$ degrees. If one of the angle measures $$9$$ degrees more than twice the other. Find the measure of the smaller angle?

Answer 1

Hint: In this question, we need to find the measure of the smaller angle. Let us assume the measure of the smaller angle to be $$x$$ . Geometrically, two angles are said to be complementary angles if their sum is $${90}^o$$ . In some cases, they form a right angled triangle. Also given that the larger angle measures $$9$$ degrees more than twice the smaller one . First, we need to form the expression according to the given question. Then we can find the measure of the smaller angle.

Complete step by step answer:
We know that two angles are said to be complementary angles if their sum is $${90}^0$$ .
Let us consider the measure of the smaller angle to be $$x$$
Given that the other angle is $$9$$ degrees more than twice the smaller one .
$$\Rightarrow 9+2x$$
Now we know that the sum is $${90}^o$$
$$\Rightarrow x+9+2x={90}^o$$
On adding,
We get,
$$\Rightarrow 3x+9={90}^o$$
By subtracting $$9$$ on both sides,
We get,
$$\Rightarrow 3x={81}^o$$
On dividing $$3$$ on both sides,
We get,
$$\Rightarrow x={27}^o$$
Thus the smaller angle is $${27}^o$$
Now we can also find another angle.
$$\Rightarrow 9+2x$$
By substituting the value of $$x$$,
$$\Rightarrow 9+2({27}^o)$$
On simplifying,
We get,
$$\Rightarrow {63}^o$$
Thus the two complementary angles are $${63}^o$$ and $${27}^o$$ .
Therefore, the smaller angle is $${27}^o$$ .

Note:
Complementary angles are nothing but a pair of angles with the sum of $${90}^o$$ and always have positive measures . One angle is said to be the complement of the other angle. It is composed of two acute angles which are mostly less than $$90$$ degrees. Similarly, two angles are said to be supplementary angles if their sum is $${180}^o$$ . A simple example for supplementary angles are $${160}^o$$ and $${20}^o$$. So students should not be confused between the words supplementary and complementary.