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To determine the relationship between \(\angle PSQ\) and \(\angle RSQ\), we can use the properties of angles in a circle.

### Analysis:
1. **\(\angle PSQ\):** This angle is subtended by the arc \(PQ\).
2. **\(\angle RSQ\):** This angle is subtended by the arc \(QR\).

### Using the Properties of Circle:
- In a circle, the 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 remaining part of the circle.
- Since \(\angle PSQ\) and \(\angle RSQ\) are subtended by arcs \(PQ\) and \(QR\) respectively, their measures relate to the corresponding arcs.

### Given Information:
- The angle subtended by the arc \(QR\) at point \(Q\) is \(2x\).
- The angle subtended by the arc \(PQ\) at point \(Q\) is \(x\).

### Conclusion:
- The relationship between \(\angle PSQ\) and \(\angle RSQ\) is that \(\angle PSQ = x\) and \(\angle RSQ = 2x\).
  
Thus, \(\angle RSQ\) is twice the size of \(\angle PSQ\).

Would you like more details or have any questions?

Here are five related questions:
1. What is the relationship between the central angle and the inscribed angle subtending the same arc?
2. How do you calculate angles subtended by the same arc in different parts of the circle?
3. Can you derive the relationship between arc length and the subtended angle at the center?
4. What is the significance of the angle subtended by a diameter?
5. How does the sum of angles in a cyclic quadrilateral relate to the circle's properties?

**Tip:** Remember that angles subtended by the same arc at the circumference are equal, which is useful in many geometric proofs involving circles.

What is the relationship between \(\angle PSQ\) and \(\angle RSQ\)?

Understand the relationship between angles \(\angle PSQ\) and \(\angle RSQ\) in a circle using properties of angles subtended by arcs. Learn how to use the Inscribed Angle Theorem to solve this problem. Suitable for students in Grades 8-10.

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