### Video Transcript

In the figure, π΄π΅πΆπ· and
πΆπ΅π»π are parallelograms. Find the measure of angle
π΄π΅π».

In this question, we are told that
the two quadrilaterals formed here are parallelograms, which we can recall are
defined as quadrilaterals with both pairs of opposite sides parallel. That would mean that the line
segments π΄π· and π΅πΆ are parallel and line segments π΅πΆ and ππ» are
parallel. So, we have three parallel line
segments here. The other pair of opposite sides in
π΄π΅πΆπ·, thatβs line segments π·πΆ and π΄π΅, are parallel. And then in parallelogram πΆπ΅π»π,
line segments πΆπ and π΅π» are parallel.

Now, we are asked to find the
measure of angle π΄π΅π». And so, recalling some of the angle
properties of parallelograms will be helpful here. Firstly, we have that in a
parallelogram, opposite angles are equal in measure. And the sum of any two consecutive
interior angles is 180 degrees. So, letβs consider that we are
given the measure of angle π·π΄π΅. Angle π΄π΅πΆ would be a consecutive
interior angle to angle π·π΄π΅. So, the sum of their measures is
180 degrees.

Filling in the information that the
measure of angle π·π΄π΅ is 72 degrees, we have that 72 degrees plus the measure of
angle π΄π΅πΆ is 180 degrees. And by subtracting 72 degrees from
both sides, we have that the measure of angle π΄π΅πΆ is 108 degrees.

Next, knowing the measure of angle
πΆπ΅π» would be useful in helping us work out the required angle measure. We can identify that in
parallelogram πΆπ΅π»π one of the consecutive interior angles to angle πΆπ΅π» is
angle π΅πΆπ, whose measure we are given. Therefore, using the same property
as before, we know that these two angle measures must sum to 180 degrees. Filling in the measure of angle
π΅πΆπ as 51 degrees, we can determine that the measure of angle πΆπ΅π» is 129
degrees.

Now, we can consider the three
angle measures about the point π΅. We can recall that the sum of the
angle measures about a point is 360 degrees. So, the sum of the angle measures
of 108 degrees, 129 degrees, and the measure of angle π΄π΅π» is 360 degrees. We can simplify the left-hand side
to give 237 degrees plus the measure of angle π΄π΅π» equals 360 degrees. And subtracting 237 degrees from
both sides, we have the answer that the measure of angle π΄π΅π» is 123 degrees.