What is the relationship between an exterior and interior angle in a triangle?
What is one of the exterior angles is acute and one of them is a right angle?
See solution.
Practice makes perfect
We want to explain why a triangle cannot have an obtuse, acute, and a right exterior angle. To do so we will begin by analyzing the relationship between the exterior and interior angle in a triangle.
As we can see, ∠ 1 is the interior angle of △ ABC and ∠ 2 is the exterior angle. Notice that they form a linear pair. If two angles form a linear pair, then they are supplementary and hence the sum of their measures is equal to 180.
m ∠ 1 + m ∠ 2 = 180
What if the exterior angle ∠ 2 is acute? This means that m ∠ 2 <90. If m ∠ 2 is less than 90, then m ∠ 1 has to be greater than 90 so that they add to 180. Therefore, the interior angle is obtuse.
m ∠ 1 >90
Now, what if another exterior angle ∠ 4 in this triangle was a right angle? Then it would have the measure of 90, and hence the interior angle would also have a measure of 90.
m ∠ 4 = m ∠ 3 =90
Suppose one of the interior angles has a measure that is greater than 90, m ∠ 1 >90. Then imagine that another one is a right angle, m ∠ 3 =90. If this is the case, then their sum is already greater than 180.
m ∠ 1 + m ∠ 3 >180
We get a contradiction with the Triangle Angle-Sum Theorem. Therefore, a triangle cannot have an obtuse, acute, and a right exterior angle.
