To solve for $x$ in the given diagram, we will apply some basic theorems related to circles and angles.

Key Observations:

• $\angle PRQ = 22^\circ$
• $\angle QSR = 28^\circ$
• $x$ is the angle at $Q$.

Strategy:

1. Cyclic Quadrilateral Theorem: In a cyclic quadrilateral (where all vertices lie on the circle), the sum of opposite angles is $180^\circ$.

   This means that: $\angle PRQ + \angle PSQ = 180^\circ$ Given that $\angle PRQ = 22^\circ$, we can find $\angle PSQ$.

2. Angles in the Same Segment: Angles subtended by the same arc (or chord) at the circumference of a circle are equal.

Thus, using the theorem and relations, we can proceed:

Step-by-Step Solution:

1. Use the Cyclic Quadrilateral Theorem to find $\angle PSQ$: $\angle PRQ + \angle PSQ = 180^\circ$ $22^\circ + \angle PSQ = 180^\circ$ $\angle PSQ = 180^\circ - 22^\circ = 158^\circ$

2. Now, notice that $x$ and $28^\circ$ are on the same arc $PQ$. Since the angles subtended by the same arc are equal, we can set: $x = 28^\circ$

Thus, the value of $x$ is $\boxed{28^\circ}$.

Let me know if you'd like any further clarification!

Related Questions:

1. What is the Cyclic Quadrilateral Theorem and how is it used in circle geometry?
2. How do you find the measure of an angle subtended by the same arc?
3. What are the properties of angles in a cyclic quadrilateral?
4. Can we apply the Inscribed Angle Theorem to solve other circle geometry problems?
5. How can angle chasing be useful in solving complex circle geometry problems?

Tip:

Remember that angles subtended by the same arc in a circle are always equal, which is a powerful tool in circle geometry.