Question

The larger of two supplementary angles exceed the smaller by 24 degrees. Then the angles are:
(a) ${{112}^{0}},{{88}^{0}}$
(b) ${{102}^{0}},{{78}^{0}}$
(c) ${{92}^{0}},{{68}^{0}}$
(d) ${{122}^{0}},{{98}^{0}}$

Answer 1

Hint: For solving this question we should know that a pair of angles whose sum is ${{180}^{0}}$ are called supplementary angles. First, we will assume the two unknown angles in terms of two variables and then we will form two linear equations in terms of two assumed variables as per the given data. After that, we will solve the two equations to find the correct answer.

Complete step-by-step answer:
Given:
Two supplementary angles such that their difference is 24 degrees.
Let $\alpha $ and $\beta $ are two supplementary angles where $\alpha $ is greater than $\beta $ .
Supplementary angles are the pair of angles such that their sum is ${{180}^{0}}$ .
Then, the sum of $\alpha $ and $\beta $ will be ${{180}^{0}}$ .
$\alpha +\beta ={{180}^{0}}.............\left( 1 \right)$
Now, it is given to us that the larger angle exceeds the smaller angle by ${{24}^{0}}$ . Then,
$\alpha -\beta ={{24}^{0}}.............\left( 2 \right)$
Now, we will find the value of $\alpha $ and $\beta $ from the equation (1) and (2).
Adding (1) and (2). Then,
$\begin{align}
  & \alpha +\beta +\alpha -\beta ={{180}^{0}}+{{24}^{0}} \\
 & \Rightarrow 2\alpha ={{204}^{0}} \\
 & \Rightarrow \alpha ={{102}^{0}} \\
\end{align}$
Now, put $\alpha ={{102}^{0}}$ in equation (1) and find the value of $\beta $ . Then,
\[\begin{align}
  & \alpha +\beta ={{180}^{0}} \\
 & \Rightarrow {{102}^{0}}+\beta ={{180}^{0}} \\
 & \Rightarrow \beta ={{180}^{0}}-{{102}^{0}} \\
 & \Rightarrow \beta ={{78}^{0}} \\
\end{align}\]
Thus, the required supplementary angles are ${{102}^{0}}$ and ${{78}^{0}}$ .
Hence, (b) is the correct option.

Note: This problem was very easy to solve if the concept of supplementary angles is clear and then solve the linear equations correctly to get the answer correctly. There can be one short method to get the correct option and that is one can check the options in which option sum of the angles is ${{180}^{0}}$ . Then, only option (b) satisfies that condition and we will mark the correct option more quickly.