Question

In figure, $\angle POR:\angle ROQ = 5:7$. The value of $\angle POR$ is:

Answer 1

Hint: First assume the measurement of the angles according to the given ratio as $5x$ and $7x$ and then use the property of pair of linear angles to find the value of $x$. Then use the value of $x$ to find the desired measurement of the angle.

Complete answer:
We have given the ratio of the angles $\angle POR$ and $\angle ROQ$ as:
$\angle POR:\angle ROQ = 5:7$.
The goal of the problem is to find the measurement of the angle $\angle POR$ using the give data:
First assume that $x$ be any number such that:
$\angle POR = 5x$ and $\angle ROQ = 7x$
We can see that $\angle POR$ and $\angle ROQ$ are the pair of linear angles, therefore the sum of these angles is $180^\circ $. That is,
$\angle POR + \angle ROQ = 180^\circ $
Now, substitute the values $\angle POR = 5x$ and $\angle ROQ = 7x$ in the above equation, then we have
$5x + 7x = 180^\circ $
Simplify the above equation for the value of$x$:
$ \Rightarrow 12x = 180$
$ \Rightarrow x = \dfrac{{180}}{{12}}$
$ \Rightarrow x = 15$
So, the value of $x$ is $15$. Now, we use the value of $x$ to find the measurement of the angles.
We have an assumption that:
$\angle POR = 5x$
Substitute the value $x = 15$ in the above angle, we have
$\angle POR = 5\left( {15} \right)$
$\angle POR = 75^\circ $
Therefore, the measurement of the $\angle POR$ is $75^\circ $.
Similarly, we have an assumption that:
$\angle ROQ = 7x$
Substitute the value$x = 15$ in the above angle, we have
$\angle ROQ = 7\left( {15} \right)$
$\angle ROQ = 105^\circ $
Therefore, the measurement of the angle $\angle ROQ$ is $105^\circ $.
The problem is asking for the measurement of the angle $\angle POR$.

Therefore, the measurement of the angle $\angle POR$ is $75^\circ $.

Note: We have to notice from the figure that $\angle POR$ and $\angle ROQ$ are the pair of linear angles, therefore the sum of these angles is $180^\circ $. That is,
$\angle POR + \angle ROQ = 180^\circ $
We can use this property in the solution to find the required result.