GK 2019 Q9
9. In the diagram, \(PQ = PR = QS\) and \(\angle QPR = 20°\). What is \(\angle RQS\)?
- A 50°
- B 60°
- C 65°
- D 70°
- E 75°
Grey Kangaroo 2019, Question 9. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 60°
Since \(PQ = QS\), triangle \(PSQ\) is isosceles and hence \(\angle PSQ = 20°\). Since the angles in a triangle add to 180°, we have \(20° + 20° + \angle SQP = 180°\) and hence \(\angle SQP = 140°\). Since \(PQ = PR\), triangle \(PRQ\) is isosceles and hence \(\angle PRQ = \angle RQP\). Also \(\angle PRQ + \angle RQP + 20° = 180°\) and hence \(\angle RQP = 80°\). Since \(\angle RQS = \angle SQP - \angle RQP\), the size of \(\angle RQS\) is \(140° - 80° = 60°\).
Solution © UK Mathematics Trust, from the GK 2019 solutions.