PK 2015 Q7
7. Martha draws some pentagons, and counts the number of right-angles in each of her pentagons. No two of her pentagons have the same number of right-angles. Which of the following is the complete list of possible numbers of right-angles that could occur in Martha's pentagons?
- A 1, 2, 3
- B 0, 1, 2, 3, 4
- C 0, 1, 2, 3
- D 0, 1, 2
- E 1, 2
Pink Kangaroo 2015, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 0, 1, 2, 3
The diagrams below show that it is possible to find pentagons with 0, 1, 2, 3 right angles. The angles in a pentagon add to 540°, so with 4 right angles, the fifth angle would be \(540° - 4 \times 90° = 180°\), which would make the shape flat at that vertex, thus creating a quadrilateral. Also, a pentagon with 5 right angles is not possible, because they wouldn't add up to 540°.
Solution © UK Mathematics Trust, from the PK 2015 solutions.