IMC 2005 Q7
7. In the diagram, what is the sum of the marked angles?
- A 180°
- B 360°
- C 450°
- D 540°
- E 720°
Intermediate Mathematical Challenge 2005, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 360°
Let the sizes of the interior angles of the quadrilateral at the centre of the figure be \(a\)°, \(b\)°, \(c\)° and \(d\)°. Then \(a + b + c + d = 360\). Each of these angles is vertically opposite to one of the unmarked angles in the four triangles, as shown. So the sum of the marked angles plus the four angles of the quadrilateral is equal to the sum of the interior angles of four triangles, that is 720°. Hence the sum of the marked angles is 720° − 360° = 360°.
Solution © UK Mathematics Trust, from the IMC 2005 solutions.