In the given figure, which
angles must have smaller measures than the measure of angle one?
In this example, we need to
identify angles in the diagram whose measure is less than that of angle number
one. But we’re not given the
measures of any of the angles. So to solve this, we’ll need to
use the diagram and the relationships between angles in triangles.
In particular, we recall that
the measure of any exterior angle of a triangle 𝐴𝐵𝐶, for example, 𝐶𝐵𝐷 in
our diagram, is greater than the measure of either of the nonadjacent interior
angles of the triangle. If we label all the vertices in
the given diagram as shown, we see that angle one is an exterior angle in the
triangle 𝐴𝐶𝐷. Hence, its measure is greater
than those of the two nonadjacent interior angles in this triangle. And these are angles four and
five at 𝐴 and 𝐶, respectively.
Now we need to pay careful
attention to the wording of the question at this point. Note that the question asks
which angles must have smaller measures than that of angle one. So far, we’ve demonstrated that
this is true of angles four and five. And we can show via an example
that the remaining angles may not necessarily in certain circumstances have a
measure smaller than that of angle one.
Suppose our triangle 𝐴𝐶𝐷 is
an isosceles triangle and that the measure of angle one is 60 degrees. We can add any line we like
from 𝐷 to a point 𝐵 opposite 𝐷 to construct the missing angles. But since our triangle is
isosceles, we can use the fact that the angle bisector and perpendicular
bisector of the base are the same line. We see then that the measures
of angles six and seven in this scenario are both 90 degrees, which is greater
than 60 degrees. And that’s the measure of angle
one. So these two angles can have
measures that are not smaller than that of angle one.
Next, we know that the line
𝐷𝐵 bisects the angle at 𝐷 so that angles two and three have the same
measure. We see also that angles one,
two, and three combine to make a straight angle so that the sum of their
measures is 180 degrees.
Now, substituting 60 degrees
for the measure of angle one and recalling that angles two and three have the
same measure, we can solve this to find either of the measures of angle two or
three. Choosing angle two, we have its
measure equals 60 degrees. And of course, we know that
this is also the measure of angle three. So both of these are equal to
the measure of angle one. And we’ve therefore shown that
angles two, three, six, and seven need not have smaller measures than angle
one. Hence, only angles four and
five must have measures smaller than that of angle one.