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The measure of an angle which is $9$ times its supplement.
A. $162$
B. $81$
C. $90$
D. $10$

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Last updated date: 24th Feb 2024
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IVSAT 2024
Answer
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Hint: We will assume the supplement angles as $x$ and $y$. We have the relation between supplement angles as $x+y=180{}^\circ $. In the problem they have mentioned the angle $x$ is $9$ times its supplement, then we will get a relation between $x$ and $y$. From these two relations we can find the values of both $x$ and $y$.

Complete step by step answer:
Let $x$ and $y$ be the supplement angles.
We know that the sum of the supplement angles is equal to $180{}^\circ $.
$\therefore x+y=180{}^\circ ...\left( \text{i} \right)$
In the problem, we have given that the one angle is $9$ times its supplement, then we will get
$x=9y$ or $y=9x$

If $x=9y$, from equation $\left( \text{i} \right)$ the value of $y$ is given by
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$\begin{align}
  & x+y=180{}^\circ \\
 & \Rightarrow 9y+y=180{}^\circ \\
 & \Rightarrow 10y=180{}^\circ \\
\end{align}$
Dividing with $10$ on both sides of the above equation, then we will have
$\begin{align}
  & \Rightarrow \dfrac{10y}{10}=\dfrac{180{}^\circ }{10} \\
 & \Rightarrow y=18{}^\circ \\
\end{align}$
Now the value of $x$ is $9y=9\times 18{}^\circ =162{}^\circ $.

So, the correct answer is “Option A”.

Note: In this problem they have mentioned that the angles are supplementary. So, we have taken the sum of the angles as $180{}^\circ $. If they have mentioned that the angles are complementary, then we need to take the sum of the angles as $90{}^\circ $ and follow the same procedure to find the angles.