Descending angles
Given four of the angles in two triangles, can you find the smallest angle overall?
Age
11 to 14
Challenge level
Problem
The six angles of two different triangles are listed in decreasing order.
The list starts: 115 °, 85 °, 75 °, 35 °
What is the smallest angle in the list?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Since the three angles in each triangle must add up to 180 degrees, 115 cannot be in the same triangle as 85 or 75, so 85 and 75 must be in the same triangle. 75+85=160 so the remaining angle in this triangle is 180 - 160 = 20 degrees.
This leaves the other two angles in the list together (115 and 35) which sum to give 150, so the other angle not included in the list is 180 - 150 = 30 degrees.
Therefore the complete list is 115, 85, 75, 35, 30, 20, so the smallest angle is 20 degrees.