IMC 2016 Q10
10. The angles of a quadrilateral taken in order are \(x\)°, \(5x\)°, \(2x\)° and \(4x\)°. Which of the following is the quadrilateral?
- A kite
- B parallelogram
- C rhombus
- D arrowhead
- E trapezium
Intermediate Mathematical Challenge 2016, Question 10. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E trapezium
The sum of the interior angles of a quadrilateral is 360°, so \(x + 5x + 2x + 4x = 360\), that is \(12x = 360\). Therefore \(x = 30\) and the angles of the quadrilateral, taken in order, are 30°, 150°, 60° and 120°. The diagram shows the shape of the quadrilateral.
Since 30 + 150 = 180, we see that \(AB\) and \(DC\) are parallel. Since it has no equal angles, it is not a rhombus or a parallelogram so it is a trapezium.
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).