### Video Transcript

From the figure, how do π΄π΅ and
π΅πΆ compare?

In this question, we are given a
figure containing a triangle π΄π΅πΆ, with two of its sides extended, and the measure
of two external angles at these sides. We want to use this information to
compare the lengths of two of the sides in the triangle.

To do this, we can start by
highlighting the two line segments whose lengths we want to compare. We can see that these are sides of
a triangle. So we can compare the lengths of
these sides using the side comparison theorem in triangles. We can recall that this tells us
that in a triangle, if one side of the triangle is opposite an angle of larger
measure than another side of the triangle, then it must be the longer side. Therefore, we can determine which
side is longer by comparing the measures of the angles opposite each side in
triangle π΄π΅πΆ.

We can find the measure of the
internal angle at π΅ by noting that it combines with the angle of measure 128
degrees to make a straight angle. So its measure is 180 degrees minus
128 degrees, which we can calculate is 52 degrees. We can follow the same process for
the other angle. We see that its measure is given by
180 degrees minus 108 degrees, which we can calculate is 72 degrees.

We can find the measure of angle π΄
by recalling that the sum of the measures of the internal angles in a triangle is
180 degrees. So we have 52 degrees plus 72
degrees plus the measure of angle π΄ equals 180 degrees. We can then subtract 72 degrees and
52 degrees from both sides of the equation to obtain that the measure of angle π΄ is
180 degrees minus 72 degrees minus 52 degrees, which we can calculate is 56
degrees.

Finally, we can see that the
measure of the angle opposite π΄π΅ has larger measure than the angle opposite π΅πΆ
in triangle π΄π΅πΆ. So by the side comparison theorem,
side π΄π΅ is longer than side π΅πΆ. Hence, π΄π΅ is greater than
π΅πΆ.