Question

Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. Which of the following is the measure of the larger angle.
A. ${{72}^{\circ }}$
B. ${{108}^{\circ }}$
C. ${{36}^{\circ }}$
D. ${{54}^{\circ }}$

Answer 1

Hint: Let us assume the measure of one angle to be x and the other to be ${{90}^{\circ }}-x$ . According to the given condition, we can write $2x=3\left( {{90}^{\circ }}-x \right)$ . We have to solve this to get the value of x. To get the other angle substitute the value of x in ${{90}^{\circ }}-x$ . The largest angle among these will be the required answer.

Complete step-by-step answer:
We know that two angles are Complementary when they add up to 90 degrees.
Let us assume the measure of one angle to be x. It is shown in the figure below:

Now, the other angle will be ${{90}^{\circ }}-x$ .
We are given that two times the measure of one angle is equal to three times the measure of the other.
$\Rightarrow 2x=3\left( {{90}^{\circ }}-x \right)$
Let us expand the RHS. We will get
$2x={{270}^{\circ }}-3x$
Let us collect the variables on one side. We will get
$2x+3x={{270}^{\circ }}$
On adding the LHS, we will get
$5x={{270}^{\circ }}$
Let us find x by taking 5 from LHS to RHS.
$x=\dfrac{{{270}^{\circ }}}{5}$
On solving the above expression, we will get
$x={{54}^{\circ }}$
Now, we have to find the measure of the other angle which is
\[{{90}^{\circ }}-x={{90}^{\circ }}-{{54}^{\circ }}={{36}^{\circ }}\]
We got the measure of angles as ${{54}^{\circ }}\text{ and }{{36}^{\circ }}$ .
We know that the largest angle is ${{54}^{\circ }}$ .
Hence, the correct option is D.

So, the correct answer is “Option D”.

Note: We can also write the expression according to the given condition as
$2\left( {{90}^{\circ }}-x \right)=3x$
Let us solve this.
${{180}^{\circ }}-2x=3x$
Let us take the variables on one side.
$\begin{align}
  & 3x+2x={{180}^{\circ }} \\
 & \Rightarrow 5x={{180}^{\circ }} \\
\end{align}$
On solving the above expression, we will get
$x=\dfrac{{{180}^{\circ }}}{5}={{36}^{\circ }}$
Hence, the other angle is ${{90}^{\circ }}-{{36}^{\circ }}={{54}^{\circ }}$