Question

find the value of ( x ).

Explanation:

Step1: Recall circle angle properties

The measure of a central angle equals its intercepted arc measure. The sum of all central angles in a circle is $360^\circ$.

Step2: Find arc PQ's central angle

Arc $PQ = 66^\circ$, so its central angle $\angle PRQ = 66^\circ$.

Step3: Set up equation for total angles

Let the central angle for arc $NM$ be $\angle NRM$. The sum of all central angles is:
$\angle PRQ + \angle QRM + \angle MRN + \angle NRP = 360^\circ$
We know $\angle MRN = 75^\circ$, arc $PQ=66^\circ$, and $\angle QRP = 66^\circ$. Also, $\angle QRM$ is vertical to $\angle NRP$, so $\angle QRM = \angle NRP$. Let $\angle NRP = y$, then:
$66^\circ + y + 75^\circ + y = 360^\circ$

Step4: Solve for y

Combine like terms:
$141^\circ + 2y = 360^\circ$
$2y = 360^\circ - 141^\circ = 219^\circ$
$y = \frac{219^\circ}{2} = 109.5^\circ$

Step5: Relate x to central angle

Arc $x$ is intercepted by central angle $\angle QRM = y = 109.5^\circ$, so $x = 109.5^\circ$.

Answer:

$109.5$