Question

Two complementary angles are in the ratio 4:5. Find the measure of the smaller angle.

Answer 1

Hint: We are given a question asking us to find the complementary angles and we are given that the ratio of the angles is 4:5. Firstly, we will open up the ratio and we will have the angles as 4x and 5x. Since, we know that the sum of complementary angles equals 90, so we will have the equation as \[4x+5x={{90}^{\circ }}\]. We will solve the above equation and get the value of ‘x’. We will then find the smaller angle from the given ratio. Hence, we will have the required angle.

Complete step-by-step answer:
According to the given question, we are given the ratio of two complementary angles and we are asked to find the measure of the smaller angle.
First of all, we will open up the ratio and we will get the angles as,
\[4x\] and \[5x\]
Since, we are given that these are the angles of a complementary angles so their sum would be equal to a right angle, that is, we have,
\[4x+5x={{90}^{\circ }}\]
Solving the above expression further, we have,
\[9x={{90}^{\circ }}\]
So, we get the value of ‘x’ as,
\[x={{10}^{\circ }}\]
The two complementary angles that we get are,
\[4x=4\times {{10}^{\circ }}={{40}^{\circ }}\]
and \[5x=5\times {{10}^{\circ }}={{50}^{\circ }}\]
We were asked to find the measure of the smaller angle and so we can say that the smaller angle is \[{{40}^{\circ }}\].

Note: The obtained complementary angles can be verified if they are complementary or not. We can simply add them, if their sum is equal to 90 then it is correct else the obtained angles are incorrect. We have,
\[{{40}^{\circ }}+{{50}^{\circ }}={{90}^{\circ }}\]
Therefore, we have the correct value of the complementary angles.