Question

find the value of ( x ).

Explanation:

Step1: Sum of central angles is 360°

The total of all angles around center $R$ equals $360^\circ$. Let the unknown central angle for arc $QM$ be $y$.
$$90^\circ + 116.5^\circ + x^\circ + y^\circ = 360^\circ$$

Step2: Recognize vertical angles

$\angle PRQ$ and $\angle NRM$ are vertical angles, so $y = 90^\circ$. Substitute $y$:
$$90 + 116.5 + x + 90 = 360$$

Step3: Calculate sum of known angles

Add the given values:
$$90 + 116.5 + 90 = 296.5$$

Step4: Solve for $x$

Subtract the sum from $360^\circ$:
$$x = 360 - 296.5$$
$$x = 63.5$$

Answer:

$63.5$