Theory question 104
Angles and Polygons(a) In the diagram, PQRST is a quadrilateral. PT//QS, ∠PTQ = 42°, ∠TSQ = 38° and ∠QSR = 30°. If ∠QTS = x and ∠PQT = y, find: (i) x; (ii) y.
(b) In the diagram, PQRS is a circle centre O. If PÔQ = 150°, ∠QSR = 40° and ∠SQP = 45°, calculate ∠RQS.
Show the model answer
Model answer
(a)(i) PT ∥ QS, so ∠TQS = ∠PTQ = 42° (alternate). In ΔQTS: x = 180° − 42° − 38° = 100°.
(ii) In ΔQRS (right angle at R): ∠RQS = 180° − 90° − 30° = 60°. P, Q, R are on a straight line, so y = 180° − 60° − 42° = 78°.
(b) ∠PSQ = ½ × 150° = 75° (angle at centre). ∠PSR = 75° + 40° = 115°.
Opposite angles of cyclic PQRS: ∠PQR = 180° − 115° = 65°.
∠RQS = 65° − 45° = 20°.
Explanation
The angle at the centre is twice the angle at the circumference.