IMC 2007 Q15
15. The diagram shows a regular pentagon \(PQRST\). The lines \(QS\) and \(RT\) meet at \(U\). What is the size of angle \(PUR\)?
- A 108°
- B 112°
- C 116°
- D 126°
- E 132°
Intermediate Mathematical Challenge 2007, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 126°
Each interior angle of a regular pentagon is 108°, so \(\angle SRQ\) = 108°. As \(SR = QR\), triangle \(SRQ\) is isosceles with \(\angle RQS = \angle RSQ = \angle 36\)°. Similarly, \(\angle SRT = \angle STR\) = 36°. So \(\angle SUR\) = (180 − 2 × 36)° = 108°.
From the symmetry of the figure, \(\angle PUR = \angle PUS\) = (360° − 108°) ÷ 2 = 126°.
Solution © UK Mathematics Trust, from the IMC 2007 solutions.