Theory question 104
Circle TheoremsIn the diagram, P, Q, R and S are points on the circle centre K. KR is a bisector of ∠SRQ, ∠KSP = 41° and ∠SKR = 80°. Find: (a) ∠RQP; (b) ∠SPQ
Show the model answer
Model answer
KS = KR (radii), so ∠KSR = ∠KRS = (180° − 80°)/2 = 50°
(a) ∠RSP = 50° + 41° = 91°; PQRS is cyclic, so ∠RQP = 180° − 91° = 89°
(b) KR bisects ∠SRQ, so ∠SRQ = 2 × 50° = 100°; opposite angles of the cyclic quadrilateral give ∠SPQ = 180° − 100° = 80°
Explanation
Use the isosceles triangle KSR first, then opposite angles of a cyclic quadrilateral.