### Video Transcript

Consider the figure shown. Fill in the blank with greater than, less than, or equal to: π΄πΆ what πΆπ·.

In this question, we are given a quadrilateral π΄π΅π·πΆ with a single pair of
parallel sides, so it is a trapezoid. We want to compare the lengths of two sides of this trapezoid by using the given
angle measures.

To answer this question, we can start by highlighting the two sides on the
diagram. It is worth noting that πΆπ· appears to be longer than π΄πΆ in the figure. However, this is not a mathematically rigorous argument, so we should try to prove
that this is the case. We can compare the lengths of the two sides by noting that these are both side
lengths in triangle π΄πΆπ·. This means that we can compare the side lengths by comparing the measures of the
angles opposite the sides in triangle π΄πΆπ·.

We are given the measure of the angle opposite side πΆπ· in triangle π΄πΆπ·. This means that we need to find the measure of the angle opposite side π΄πΆ. We can do this by noting that the diagonal π΄π· is a transversal of the parallel
lines in the trapezoid. We can then use the fact that alternate interior angles in a transversal of a pair of
parallel lines are congruent to note that the measure of angle π΄π·πΆ is 31
degrees.

We can now compare the lengths of these two sides by using the side comparison
theorem in triangles. This tells us that if one side in a triangle is opposite an angle of larger measure
than another side in the triangle, then it must be longer. In our triangle π΄πΆπ·, we can see that the angle opposite side πΆπ· has greater
measure than the angle opposite π΄πΆ. So πΆπ· is longer than π΄πΆ. We can reverse this inequality to get that π΄πΆ is less than πΆπ·, giving us the
answer of less than.