IMC 2020 Q22
22. The diagram shows a regular pentagon and an irregular quadrilateral.
What is the sum of the three marked angles?
- A 72°
- B 90°
- C 108°
- D 126°
- E 144°
Intermediate Mathematical Challenge 2020, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 108°
The exterior angle of a regular pentagon is \(\dfrac{360°}{5} = 72°\).
Therefore the interior angle of a regular pentagon, in degrees, is 180 − 72 = 108. The angles at a point sum to 360°, so the reflex angle in the irregular quadrilateral, in degrees, is 360 − 108 = 252. Finally the interior angles of a quadrilateral sum to 360°, so the sum of the marked angles, in degrees, is 360 − 252 = 108.
(Note that the sum of the three marked angles equals the interior angle of the pentagon.)
Solution © UK Mathematics Trust, from the IMC 2020 solutions. Longer versions with investigations: Solutions and investigations (PDF).