Theory question 103
Angles and Polygons(a) The ratio of the interior angle to the exterior angle of a regular polygon is 5 : 2. Find the number of sides of the polygon.
(b) The diagram shows a circle PQRS with centre O, ∠UQR = 68°, ∠TPS = 74° and ∠QSR = 40°. Calculate the value of ∠PRS.
Show the model answer
Model answer
(a) Interior + exterior = 180°, so exterior angle = 2/7 × 180° = 360/7°
Number of sides = 360° ÷ 360/7° = 7
(b) TPQU is a straight line: ∠PQR = 180° − 68° = 112° and ∠SPQ = 180° − 74° = 106°
∠PSR = 180° − 112° = 68° (opposite angles of cyclic quadrilateral PQRS)
∠QPR = ∠QSR = 40° (same segment), so ∠SPR = 106° − 40° = 66°
In triangle PRS: ∠PRS = 180° − 68° − 66° = 46°
Explanation
Use opposite angles of a cyclic quadrilateral and angles in the same segment.