Question

The adjacent angles of a parallelogram are as 2 : 3. Which of the following options is a measure of the angles?
A. ${{108}^{\circ }}$
B. ${{60}^{\circ }}$
C. ${{116}^{\circ }}$
D. ${{90}^{\circ }}$

Answer 1

Hint: We will assume the adjacent angles of the parallelogram to be 2x and 3x. So, we can form an equation as $2x+3x={{180}^{\circ }}$, as we know that the sum of adjacent angles of a parallelogram is ${{180}^{\circ }}$. We will solve this equation and get the desired answer.

Complete step by step answer:
To solve this question, let us first draw a parallelogram ABCD, such that the adjacent angles A and B are in the ratio of 2 : 3.

Let us consider $\angle A=2x$ and $\angle B=3x$. We know that the sum of adjacent angles of a parallelogram is ${{180}^{\circ }}$. So, that means that the sum of $\angle A$ and $\angle B$ would be equal to ${{180}^{\circ }}$. So, we can write it as,
$\begin{align}
  & \angle A+\angle B={{180}^{\circ }} \\
 & \Rightarrow 2x+3x={{180}^{\circ }} \\
 & \Rightarrow 5x={{180}^{\circ }} \\
 & \Rightarrow x=\dfrac{{{180}^{\circ }}}{5} \\
 & \Rightarrow x={{36}^{\circ }} \\
\end{align}$
So, we can now find the value of both the angles A and B. Therefore,
$\begin{align}
  & \angle A=2x\Rightarrow 2\times {{36}^{\circ }}={{72}^{\circ }} \\
 & \angle B=3x\Rightarrow 3\times {{36}^{\circ }}={{108}^{\circ }} \\
\end{align}$
We also know that the opposite angles of a parallelogram are equal. Therefore, we will have,
$\begin{align}
  & \angle C=\angle A={{72}^{\circ }} \\
 & \angle D=\angle B={{108}^{\circ }} \\
\end{align}$
So, we have the angles of the parallelogram as, $\angle A={{72}^{\circ }},\angle B={{108}^{\circ }},\angle C={{72}^{\circ }},\angle D={{108}^{\circ }}$Therefore, the measures of all the angles of the parallelogram are, ${{72}^{\circ }},{{108}^{\circ }},{{72}^{\circ }},{{108}^{\circ }}$ and only ${{108}^{\circ }}$ is present among the options.

So, the correct answer is “Option A”.

Note: We should also remember that the sum of all the angles of a parallelogram ABCD is equal to${{360}^{\circ }}$, that is, $\angle A+\angle B+\angle C+\angle D={{360}^{\circ }}$. We can cross check our answer by checking if the sum is equal to ${{360}^{\circ }}$, if we add the angles that we have obtained, that is,${{72}^{\circ }}+{{108}^{\circ }}+{{72}^{\circ }}+{{108}^{\circ }}={{360}^{\circ }}$. So, our answer is correct. We can also write parallelogram ABCD as $\parallel gm$ ABCD.