### Video Transcript

Consider the shown figure. Fill in the blank using is greater than, is less than, or is equal to. π΄π΅ what πΆπ·.

In this question, we are given a figure containing information about various triangles. And we want to use this figure to compare two lengths in the figure: π΄π΅ and πΆπ·. To do this, we can start by noting that π΅π· is the same length as πΆπ·. If we then highlight π΄π΅ on the diagram, we can see that π΄π΅ and π·π΅ are both sides of triangle π΄π΅π·. This means that we can compare the lengths of π΄π΅ and πΆπ· by comparing the lengths of the sides in triangle π΄π΅π·.

We can compare the side lengths in a triangle 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, then it is longer. More formally, it tells us that if π₯π¦π§ is a triangle and the measure of angle π₯ is greater than the measure of angle π¦, then π¦π§ is longer than π₯π§. Therefore, we can compare the lengths of these two sides by comparing the measures of the angles opposite them in triangle π΄π΅π·. These are angles π΅π΄π· and π΄π·π΅, as shown.

We can find the measure of angle π΄π·π΅ by noting that it combines with angle π΅π·πΆ to make a straight angle. So, their measures sum to give 180 degrees. Therefore, the measure of angle π΄π·π΅ is equal to 180 degrees minus 92 degrees, which we can calculate is equal to 88 degrees. We can add this onto our diagram.

We now need to find the measure of angle π΅π΄π·. To do this, we can note that sides πΆπ· and π΅π· have the same length. So, triangle π΅πΆπ· is an isosceles triangle. And we know that in an isosceles triangle, the angles opposite the congruent sides have equal measure. So, angles π·πΆπ΅ and π·π΅πΆ have the same measure.

We can then use the fact that the sum of all of the measures of the internal angles in a triangle is 180 degrees to show that twice the measure of angle π·πΆπ΅ plus 92 degrees is equal to 180 degrees. We can then solve this equation for the measure of angle π·πΆπ΅. We subtract 92 degrees from both sides of the equation and divide through by two to get that the measure of angle π·πΆπ΅ is equal to 44 degrees. We can then add this angle measure onto the diagram to all of the angles congruent to angle π·πΆπ΅ as shown.

We now have two of the angle measures of triangle π΄π΅π·. So, we can use the sum of the internal angle measures in a triangle to find the measure of the third angle. We have that the measure of angle π΅π΄π· is equal to 180 degrees minus 88 degrees minus 44 degrees. We can calculate that this is equal to 48 degrees. And we can add this onto the diagram as shown.

We can now apply the side comparison theorem in triangles to triangle π΄π΅π·. We see that side π΄π΅ is opposite an angle with larger measure than side π·π΅. So, π΄π΅ must be longer than π΅π·. But remember, we are given in the diagram that π΅π· and πΆπ· are the same length. So, we can also say that π΄π΅ is greater than πΆπ·, which is our final answer.