Triangle PQR is inscribed in a circle such that P, Q and R lie on the circumference of the circle. If PQ is the diameter of the circle and angle PQR = 40°, then what is the measure (in degrees) of angle QPR at vertex P?

Introduction / Context:
This problem uses basic circle geometry and properties of angles in a semicircle. A triangle PQR is inscribed in a circle and PQ is given as the diameter of the circle. The question asks for the remaining acute angle at P when the angle at Q is known. This combines the properties of a right triangle with the well known theorem that the angle in a semicircle is a right angle.

Given Data / Assumptions:

• P, Q and R are points on the circumference of a circle.
• Segment PQ is the diameter of the circle.
• Angle PQR at Q is given as 40°.
• We denote angle QPR at P as the unknown angle.
• The triangle is assumed to be non degenerate and standard Euclidean geometry applies.

Concept / Approach:
When a triangle is inscribed in a circle such that one of its sides is the diameter, the angle opposite the diameter is always a right angle. This result is known as Thales theorem. Therefore the angle at R, which is subtended by the diameter PQ, must be 90°. Once one angle of a triangle is known to be 90° and a second angle is given, the third angle can be found using the fact that the sum of internal angles of any triangle is 180°.

Step-by-Step Solution:
Step 1: From Thales theorem, since PQ is the diameter, angle PRQ = 90°. Step 2: The three angles of triangle PQR are angle PQR at Q, angle QPR at P, and angle PRQ at R. Step 3: Use the angle sum property of a triangle: angle PQR + angle QPR + angle PRQ = 180°. Step 4: Substitute the known values: 40° + angle QPR + 90° = 180°. Step 5: Simplify: 130° + angle QPR = 180° so angle QPR = 180° − 130° = 50°.

Verification / Alternative check:
We can quickly verify by checking that the three computed angles 40°, 50° and 90° add up to 180°, which they do. Also, it is reasonable that the angle opposite the diameter is the largest angle in the triangle (90°), and the other two angles are acute and sum to 90°. With angle PQR = 40°, the remaining acute angle must be 50°, which matches our calculation and the option chosen.

Why Other Options Are Wrong:
Option 40° would make the two acute angles equal, but then 40° + 40° + 90° = 170°, which violates the 180° rule. Option 45° would give the sum 40° + 45° + 90° = 175°, which is again incorrect. Option 55° would lead to 40° + 55° + 90° = 185°, too large. Option 60° produces 40° + 60° + 90° = 190°, which is impossible for a triangle. Only 50° satisfies the angle sum requirement while respecting Thales theorem.

Common Pitfalls:
Many students forget or misapply the property that the angle in a semicircle is a right angle, and they sometimes attempt to use more complicated circle theorems. Another common mistake is to subtract the given angle directly from 180° instead of accounting for the right angle at R. Always remember to use both facts together: first fix the right angle due to the diameter, then apply the angle sum of the triangle. Writing down all three angles explicitly helps avoid mental arithmetic errors.

Final Answer:
Thus, the measure of angle QPR is 50°.