### Video Transcript

Use less than, equal to, or greater
than to fill in the blank to compare side length π΅πΆ to side length π΄πΆ.

First, we can identify which side
lengths weβre trying to compare. We want to compare side length π΅πΆ
to side length π΄πΆ. One strategy we can try to use is
the angle-side inequality in a triangle. To do that, weβd need to compare
the angles opposite the two side lengths weβre interested in. Now you might be thinking we donβt
have any information about the angles. However, we can use some properties
of triangles to find out some information about the angle measures. In a triangle, if two sides have
the same length, then their opposite angles are the same, which means here angle
π΄π΅πΉ will be equal in measure to angle πΉπ΄π΅ and angle π΄π·πΉ will be equal to
angle π·π΄πΉ.

Both triangle π΄π΅πΉ and triangle
π΄π·πΉ are isosceles triangles, and they have two equal sides. This means we can say that segment
π΄π· is going to be equal in length to segment π΄π΅ as these two triangles are
congruent. It also means angle π΄πΉπ΅ will be
equal to angle π΄πΉπ·. This means we found that side
length π΄π΅ must be smaller than side length π΄πΆ. But how can we say something about
side length π΅πΆ? For this, weβre going to think
about the line segment π΄πΉ. Weβve seen that the line segment
π΄πΉ bisects the segment π΅π·. The line segment π΄πΉ is a bisector
of the isosceles triangle π΄π΅π·. And when that happens, it is a
perpendicular bisector, which means both of these angles must measure 90
degrees. And if thatβs the case, in the
smaller isosceles triangles, the smaller blue angles must be equal to each other and
must therefore be equal 45 degrees each.

If all of these smaller angles
measure 45 degrees, the big angle we were looking for, angle π΅π΄πΆ, is a right
angle. And since line segment π΅πΆ is
opposite that right angle, itβs the hypotenuse of the larger triangle π΄π΅πΆ, and it
is therefore the longest side length of this triangle. The order of the side lengths for
the larger triangle π΄π΅πΆ must be π΄π΅ is smaller than π΄πΆ, which is smaller than
π΅πΆ. Since π΅πΆ is the hypotenuse and is
the longest side length in this triangle, we can say π΅πΆ is greater than π΄πΆ.