We are given that $\angle ROS = 66^\circ$ and $\angle POQ = 3x$. We are also told that some construction lines are shown. We are asked to calculate the value of $x$. Since OP bisects $\angle ROS$, $\angle POS = \angle POR = \frac{1}{2} \angle ROS$.

1. Problem Description

We are given that $\angle ROS = 66^\circ$ and $\angle POQ = 3x$. We are also told that some construction lines are shown. We are asked to calculate the value of $x$. Since OP bisects $\angle ROS$, $\angle POS = \angle POR = \frac{1}{2} \angle ROS$.

2. Solution Steps

Since $OP$ bisects $\angle ROS$, we have

$\angle POS = \angle POR = \frac{1}{2} \angle ROS$

We are given that $\angle ROS = 66^\circ$, so

$\angle POS = \frac{1}{2}(66^\circ) = 33^\circ$.

We are given that $\angle POQ = 3x$. Since $OQ$ bisects $\angle POS$, $\angle POQ = \angle QOS = \frac{1}{2} \angle POS$. Therefore,

$\angle POQ = \frac{1}{2} \angle POS = \frac{1}{2} (33^\circ) = 33^\circ$

$\angle QOS = \frac{1}{2} \angle POS = \frac{1}{2} (33^\circ) = 33^\circ$

However, $\angle POQ = 3x$. Therefore,

$3x = 33^{\circ}$.

Dividing both sides by $3$, we have

$x = 11^{\circ}$.

3. Final Answer

$x = 11^{\circ}$

The answer is B.