In the circle $PQR$ with center $O$, we are given that $\angle OPQ = 48^\circ$. We need to find the value of $m$, which represents the angle $\angle ROQ$.

1. Problem Description

In the circle $PQR$ with center $O$, we are given that $\angle OPQ = 48^\circ$. We need to find the value of $m$, which represents the angle $\angle ROQ$.

2. Solution Steps

Since $O$ is the center of the circle, $OP$ and $OQ$ are radii of the circle. Therefore, $OP = OQ$, which means that $\triangle OPQ$ is an isosceles triangle.

In an isosceles triangle, the angles opposite the equal sides are equal. So, $\angle OQP = \angle OPQ = 48^\circ$.

The sum of the angles in a triangle is $180^\circ$. Therefore, in $\triangle OPQ$,

$\angle POQ + \angle OPQ + \angle OQP = 180^\circ$

$\angle POQ + 48^\circ + 48^\circ = 180^\circ$

$\angle POQ + 96^\circ = 180^\circ$

$\angle POQ = 180^\circ - 96^\circ = 84^\circ$

Since $ROP$ is a straight line through the center of the circle, $\angle ROP = 180^\circ$.

Also, $\angle ROQ + \angle POQ = \angle ROP$.

Therefore, $\angle ROQ = 180^\circ - \angle POQ = 180^\circ - 84^\circ = 96^\circ$.

Thus, $m = 96^\circ$.

3. Final Answer

A. 96°