### Video Transcript

In the given figure, if π΄πΆ is
parallel to π·πΈ, the measure of angle π΄π΅πΈ equals 55 degrees, and the measure of
angle πΆ equals 75 degrees, find the measure of angle π΄π΅πΆ.

The first piece of information that
we are given here is that π΄πΆ is parallel to π·πΈ. We can also fill in the two angle
measures that we are given that the measure of angle π΄π΅πΈ is 55 degrees and the
measure of angle πΆ is equal to 75 degrees. We can then establish that the
angle that we wish to calculate is here, the measure of angle π΄π΅πΆ.

So there are a couple of ways in
which we could find the measure of this unknown angle. But both methods will use the fact
that we have these parallel line segments. By using the parallel lines π΄πΆ
and πΈπ· and the transversal π΅πΆ, we can identify a pair of alternate interior
angles. And since alternate interior angles
are congruent, we can say that the measure of angle π·π΅πΆ is equal to the measure
of angle πΆ. These will both be 75 degrees.

We can notice then that these three
angles made at the vertex π΅ all lie on a straight line. We recall that the angle measures
on a straight line sum to 180 degrees. Therefore, we can write that the
measure of angle π΄π΅πΈ plus the measure of angle π΄π΅πΆ plus the measure of angle
π·π΅πΆ must be equal to 180 degrees. We can then simply fill in the
angle information that we know. Then, by adding 55 degrees and 75
degrees, we get 130 degrees. We can then simplify this equation
by subtracting 130 degrees from both sides, leaving us with the measure of angle
π΄π΅πΆ is equal to 50 degrees. And so we have found the value of
this unknown angle.