### Video Transcript

In the figure, the ray from π΅
through π· bisects the angle π΄π΅πΈ. What is the measure of angle
π΄π΅π·?

In this question we are given a
figure and asked to determine the measure of a given angle using a bisector and a
given angle measure.

To answer this question, we can
begin by recalling that an angle bisector bisects an angle into two angles with
equal measure. In this case, we are told that the
ray from π΅ through π· bisects angle π΄π΅πΈ. So we must have that the measure of
angle π·π΅πΈ is equal to the measure of angle π΄π΅π·. We can add this information onto
the figure.

We can then recall that the measure
of a straight angle is 180 degrees. And we can see in the figure that
angle π΄π΅πΆ is a straight angle. Hence, the sum of the angle
measures that make the straight angle is 180 degrees. We have that 180 degrees is equal
to 54 degrees plus the measure of angle π·π΅πΈ plus the measure of angle π΄π΅π·.

Since the measure of angle π·π΅πΈ
is equal to the measure of angle π΄π΅π·, we can replace its measure with that of the
measure of angle π΄π΅π· to obtain two times the measure of angle π΄π΅π·. We can then subtract 54 degrees
from both sides of the equation and evaluate to get that 126 degrees is equal to the
two times the measure of angle π΄π΅π·. Finally, we can divide both sides
of the equation by two to find that the measure of angle π΄π΅π· is 63 degrees.