PQRS is a cyclic quadrilateral in a circle and PQ is a diameter of the circle. If angle RPQ = 38 degrees, what is the measure of angle PSR?

Introduction / Context:
This question tests properties of cyclic quadrilaterals and angles subtended by diameters in a circle. Recognising that a diameter subtends a right angle is crucial, and so is the property that opposite angles of a cyclic quadrilateral sum to 180 degrees.

Given Data / Assumptions:
• PQRS is a cyclic quadrilateral, so all four vertices lie on a circle. • PQ is a diameter of the circle. • Angle RPQ, at P between PR and PQ, is 38 degrees. • We are asked to find angle PSR.

Concept / Approach:
First, use the fact that an angle subtended by a diameter at any point on the circle is a right angle. So angle PRQ is 90 degrees because it subtends chord PQ which is a diameter. Then in triangle PQR, we can find angle PQR using the angle sum property. Once we know angle PQR, we use the property of a cyclic quadrilateral: opposite angles of a cyclic quadrilateral sum to 180 degrees. Here, angle PSR is opposite angle PQR.

Step-by-Step Solution:
1. Since PQ is a diameter, angle PRQ is a right angle, so angle PRQ = 90 degrees. 2. In triangle PQR, we know angle RPQ = 38 degrees and angle PRQ = 90 degrees. 3. Use the triangle angle sum: angle RPQ + angle PRQ + angle PQR = 180 degrees. 4. Substitute: 38 + 90 + angle PQR = 180. 5. Combine known angles: 128 + angle PQR = 180. 6. So angle PQR = 180 - 128 = 52 degrees. 7. In cyclic quadrilateral PQRS, opposite angles are supplementary, so angle PQR + angle PSR = 180 degrees. 8. Substitute angle PQR = 52 degrees: 52 + angle PSR = 180. 9. Thus angle PSR = 180 - 52 = 128 degrees.

Verification / Alternative check:
You can sketch a circle, mark PQ as a diameter and place R and S on the circle. By roughly measuring angles in your diagram, you will see that angle PSR is obtuse and larger than 120 degrees, which makes 128 degrees a reasonable value. The calculations use standard circle theorems and triangle angle sums, so the result is well supported.

Why Other Options Are Wrong:
• 52°: This equals angle PQR itself and would make the opposite angles sum to only 104 degrees instead of 180 degrees. • 77°: When added to 52 degrees, the sum is 129 degrees, not 180 degrees, so it violates the cyclic quadrilateral property. • 142°: This would make the sum with 52 degrees equal to 194 degrees, which is impossible for opposite angles. • 90°: This would imply angle PQR is 90 degrees, contradicting the triangle angle sum we already used.

Common Pitfalls:
Some students forget that a diameter subtends a right angle, or they apply the 180 degree sum property to adjacent instead of opposite angles in the cyclic quadrilateral. Another pitfall is to mix up which angle is at P versus at Q or R. Carefully labelling the diagram and writing down known values helps to avoid these errors.

Final Answer:
The measure of angle PSR is 128°.