Question Video: Determining Which Angle Has the Greatest Measure in a Triangle Mathematics

Video Transcript

In the given figure, which of the following angles has the greatest measure: angle one, angle two, or angle three?

It seems fairly clear from the figure that the angle with the greatest measure out of angles one, two, and three is angle one. However, it’s important that we prove that this is the case. We can do this using a known property of triangles involving the relationship between exterior and interior angles of any given triangle. This tells us that the measure of any exterior angle in a triangle is greater than the measure of either nonadjacent interior angles in that triangle.

In the triangle shown, since 𝑑 is an exterior angle and angles 𝑎 and 𝑏 are nonadjacent angles to 𝑑, this means that the measure of angle 𝑑 is greater than the measure of angle 𝑎 and the measure of angle 𝑑 is also greater than the measure of angle 𝑏.

So what does this mean for the measures of the three angles in the given figure? Well, we see that angle one is an exterior angle to the triangle containing the other two angles, angles two and three, and also that angles two and three are nonadjacent interior angles to this exterior angle. By the given property then, this means that the measure of angle one must be greater than the measure of angle two and also that the measure of angle one is greater than the measure of angle three. Hence, since the measure of angle one is greater than both of the other two measures, angle one is the angle in the given figure with the greatest measure.

It’s worth noting in fact that in any given triangle the measure of the exterior angle is equal to the sum of the measures of the two nonadjacent interior angles, which in our case means the measure of angle one is equal to the sum of the measures of angles two and three. This confirms our result that angle one must have the greatest measure of the three angles.