# Question Video: Determining Which Angles in a Triangle Must be Less Than a Particular External Angle Mathematics

In the given figure, which angles must have smaller measures than 𝑚∠1?

02:45

### Video Transcript

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.