Points P, Q, and R lie on a circle such that angle PQR = 40° and angle QRP = 60°. What is the measure of the central angle subtended by arc QR (in degrees)?

Introduction / Context:
This circle geometry problem connects angles at the circumference with the central angle subtending the same arc. It also uses the angle sum property of a triangle formed by three points on the circle. These relationships are core results in circle theorems and frequently appear in competitive exams.

Given Data / Assumptions:

• P, Q, and R are points on the circumference of a circle.
• Angle PQR = 40°.
• Angle QRP = 60°.
• We must find the central angle (at the centre of the circle) subtended by arc QR.

Concept / Approach:
First, note that triangle PQR is an inscribed triangle in the circle, and its angles at Q and R are given. The third angle at P can be found using the triangle angle sum property. This angle at P is an inscribed angle that subtends arc QR. A basic circle theorem says that a central angle subtending a given arc is twice any inscribed angle that subtends the same arc. So, once we find angle QPR at P, we double it to get the central angle for arc QR.

Step-by-Step Solution:
Step 1: Use the triangle angle sum property for triangle PQR: angle PQR + angle QRP + angle QPR = 180°. Step 2: Substitute the given angles: 40° + 60° + angle QPR = 180°. Step 3: Add 40° and 60°: 100° + angle QPR = 180°. Step 4: Solve for angle QPR: angle QPR = 180° − 100° = 80°. Step 5: Angle QPR is an inscribed angle subtending arc QR. Step 6: The central angle subtending the same arc QR, call it angle QOR, satisfies angle QOR = 2 × angle QPR. Step 7: Therefore angle QOR = 2 × 80° = 160°.

Verification / Alternative Check:
Drawing a sketch of the circle with P, Q, and R on it and marking the angles helps visualise the situation. The large central angle QOR, opening over arc QR, must be more than 90° because angle QPR is already 80° at the circumference. Doubling 80° to 160° matches this intuition and fits neatly with the inscribed–central angle relationship, providing a strong visual check.

Why Other Options Are Wrong:
Option a, 80°, equals the inscribed angle at P and would only apply if P were at the centre, which it is not. Options b (120°) and c (140°) do not equal twice the inscribed angle 80° and therefore violate the central–inscribed angle theorem for the same arc. They cannot be correct in this configuration.

Common Pitfalls:
A common error is forgetting to compute the third angle in triangle PQR and using one of the given angles directly as the inscribed angle subtending arc QR. Another mistake is halving instead of doubling the inscribed angle when finding the central angle, mixing up which is larger. Always remember: central angle = 2 × inscribed angle for the same arc.

Final Answer:
The central angle subtended by arc QR is 160°.