In this problem, the goal is to find the value of angle $x$ in the given circle diagram. The circle contains points $P$, $Q$, $R$, and $S$, with given angles $\angle PRQ = 21^\circ$ and $\angle QSR = 24^\circ$.

Key observations:

1. Angle at the Center vs. Angle at the Circumference: In a circle, an angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference.
2. Inscribed Angles: The angles $\angle PRQ$ and $\angle QSR$ are subtended by the same arc, $PQ$, and the angle subtended by the same arc at different points is equal.
3. Inscribed Quadrilateral: Since points $P$, $Q$, $R$, and $S$ lie on the circumference of the circle, angles subtended by opposite sides in a cyclic quadrilateral are supplementary (i.e., they sum to $180^\circ$).

Solution process:

The two angles $\angle PRQ = 21^\circ$ and $\angle QSR = 24^\circ$ are angles subtended by the same arc, and thus $\angle PRQ = \angle PQS$. Therefore, $x$, the angle at $QSR$, is equal to the angle subtended by the arc $PR$.

Thus, we can now solve this as follows:

$x = \angle PRQ + \angle QSR = 21^\circ + 24^\circ = 45^\circ$

So, the value of $x$ is:

$\boxed{45^\circ}$

Would you like more detailed steps or have any questions?

Related Questions:

1. What is the rule for angles subtended by the same arc?
2. How can you use the inscribed angle theorem to find angles in other circle problems?
3. How do we prove that opposite angles in a cyclic quadrilateral sum to $180^\circ$?
4. What other types of angle properties can be derived from a circle's geometry?
5. How does the angle at the center compare to angles on the circumference subtended by the same arc?

Tip:

Always remember that angles subtended by the same arc in a circle are equal!