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

Video Transcript

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

We’re not given the measures of any of the three angles we need to compare. And although angle two appears from the figure to be the largest, it’s tricky to see how we might prove that this is the case. In fact, we can do this using the diagram and the relationships between angles one, two, and three to compare their measures.

We note first that angle two is an exterior angle to the triangle containing angles one and three also that angles one and three are nonadjacent to angle two. Next, we recall that a particular property related to interior and exterior angles of a triangle is that the measure of any exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles in that triangle. So in the triangle shown, the measure of angle 𝑑 is greater than the measures of either angle 𝑎 or angle 𝑏.

So what does this mean for the given triangle with interior angles one and three? Well, as we noted previously, angle two is an exterior angle to the triangle, and angles one and three are not adjacent to angle two. So from our exterior angle property, we can say that the measure of angle two is greater than the measure of angle one and also that the measure of angle two is greater than the measure of angle three. Hence, since the measure of angle two is greater than that of both angles one and three, angle two must have the greatest measure out of the three.