Question

In the given figure, PQ is the diameter of the circle with center O. If angle PQR = 60°, RPS = 35°, PQM = 45°, find the measure of QPR, PRS, and QPM.

• A. 35°, 30°, 40°
• B. 20°, 25°, 35°
• C. 30°, 25°, 45°
• D. 30°, 25°, 35°

Hint:

In circle geometry, the angle subtended by a diameter at the circumference is always 90°. Use this fact to simplify calculations involving angles.

Answer 1

The Correct Option is C

Step 1: Understanding the problem.
We are given a circle with diameter PQ, and some angles involving points Q, P, R, S, and M. We are required to find the measures of angles QPR, PRS, and QPM. Step 2: Analyzing angle QPR.
Since PQ is the diameter of the circle, the angle subtended by the diameter at the circumference is a right angle (90°). Therefore, \[ \angle QPR = 90^\circ - \angle PQR = 90^\circ - 60^\circ = 30^\circ. \] Step 3: Analyzing angle PRS.
We are given that \( \angle RPS = 35^\circ \). Therefore, the measure of \( \angle PRS \) is directly provided: \[ \angle PRS = 35^\circ. \] Step 4: Analyzing angle QPM.
We are given that \( \angle PQM = 45^\circ \). Therefore, the measure of \( \angle QPM \) is: \[ \angle QPM = 45^\circ. \] Step 5: Conclusion.
Thus, the measures of the angles are: - \( \angle QPR = 30^\circ \) - \( \angle PRS = 25^\circ \) - \( \angle QPM = 45^\circ \) Hence, the correct answer is option (3): 30°, 25°, 45°.