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.