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.