A quadrilateral PQRS is inscribed in a circle of centre O, such that PQ is a diameter and ∠PSR = 120°. Find the value of ∠QPR.

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Accepted Answer

Given: PQRS is a quadrilateral inscribed in a circle with center O, and PQ is a diameter. We know that an angle subtended by a diameter on the circle is a right angle (90°). Therefore, ∠PSQ = 90° since PQ is the diameter and S lies on the circle.

Also given: ∠PSR = 120°.

In quadrilateral PQRS, the sum of opposite angles is 180° because it is cyclic.

Therefore, ∠PSR + ∠PQR = 180°  
=> 120° + ∠PQR = 180°  
=> ∠PQR = 60°

Now, consider triangle PQR:  
- ∠PSQ = 90° (angle subtended by diameter)  
- ∠PQR = 60° (from above)

Sum of angles in triangle PQR = 180°  
=> ∠QPR + ∠PQR + ∠PSQ = 180°  
=> ∠QPR + 60° + 90° = 180°  
=> ∠QPR = 180° - 150° = 30°

Hence, the value of ∠QPR is 30°.

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