Question

In the adjoining figure, O is the centre of the circle. If <SPQ = 45° and ∠POT = 150°, find the measures of: 

1. ∠PST 
2. ∠PUT 
3. ∠QTR 
4. ∠QRT

Answer 1

Given:

∠SPQ = 45°
∠POT = 150°
O is the centre of the circle.

(i) To find ∠PST:

∠POT = 360° − 150° = 210°

The angle subtended by a major arc at the centre is twice the angle subtended at any point on the circle.

Therefore, ∠PST = `1/2 × ∠POT = 1/2` × 210° = 105°

(ii) To find ∠PUT:

SPQT is a cyclic quadrilateral (all vertices on the circle).

Opposite angles of a cyclic quadrilateral add up to 180°.

∠PUT + ∠PST = 180°

∠PUT + 105° = 180° → ∠PUT = 180° − 105° = 75°

(iii) To find ∠QTR:

The exterior angle of a cyclic quadrilateral equals the opposite interior angle.

∠QTR = ∠SPQ = 45°

(iv) To find ∠QRT:

∠TQR = ∠PST = 105° (angles subtended by the same arc in a circle are equal)

The sum of angles in triangle QRT is 180°.

∠QRT = 180° − (∠QTR + ∠TQR) = 180° − (45° + 105°) = 30°