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
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.