### Video Transcript

In the figure below, the
intersection of line segment π΄πΆ and line segment π΅π· equals π. ππ· equals ππΆ. Line segment π΄π΅ and line segment
πΆπ· are parallel. And the measure of angle ππΆπ·
equals 40 degrees. Draw line segment ππΈ
perpendicular to line segment πΆπ·, and cut line segment πΆπ· at πΈ. Then, draw line segment ππΉ
perpendicular to line segment π΄π΅, and cut line segment π΄π΅ at πΉ. Find the measure of angle πΆππΈ
and the measure of angle π΅ππΉ.

Letβs begin by considering the
information that we are given, with the first piece of information being that the
two line segments π΄πΆ and π΅π· intersect at the point π. Next, we have two congruent line
segments, since ππ· is equal to ππΆ. Then, we are given that the line
segments π΄π΅ and πΆπ· are parallel. And the measure of angle ππΆπ· is
40 degrees.

Now we need to draw two new line
segments, with the first being the line segment ππΈ, which must be perpendicular to
line segment πΆπ· and with line segment ππΈ cutting line segment πΆπ· at point πΈ,
which would look like this on the figure. Similarly, we can draw the second
new line segment of ππΉ perpendicular to line segment π΄π΅, which cuts line segment
π΄π΅ at the point πΉ. We will need to determine the
measures of angles πΆππΈ and π΅ππΉ.

One way that we can find the
measure of the first angle πΆππΈ is by noticing that we have triangle πΆππΈ and we
know two of the angles within it. By recalling that the interior
angle measures in a triangle sum to 180 degrees, we can write that 40 degrees plus
the right angle of 90 degrees plus the measure of angle πΆππΈ equals 180
degrees. And simplifying the left-hand side
and then subtracting 130 degrees from both sides, we have that the measure of angle
πΆππΈ is 50 degrees. So, thatβs the first angle measure
calculated.

Next, letβs consider how we can
calculate the measure of angle π΅ππΉ. To do this, we can take a closer
look at triangle πΆππ·. Given that this triangle has two
congruent line segments, that means that by definition, triangle πΆππ· is an
isosceles triangle. We also drew a perpendicular line
from the vertex angle of this isosceles triangle to the base.

We can recall that one of the
corollaries of the isosceles triangle theorems states that the straight line that
passes through the vertex angle of an isosceles triangle and is perpendicular to the
base bisects the base and the vertex angle. The important part for this
question is that the line segment ππΈ that we drew bisects the vertex angle. The vertex angle would be the angle
πΆππ·. So both angles πΆππΈ and π·ππΈ
would have the same measure. They are both 50 degrees.

But we still need to find the
measure of angle π΅ππΉ, which is in a different triangle. However, we can observe that the
angles π·ππΈ and π΅ππΉ are vertically opposite angles. Therefore, they are both equal in
measure. They are both 50 degrees. We can then give the answers for
both the required angle measures. The measure of angle πΆππΈ is 50
degrees, and the measure of angle π΅ππΉ is 50 degrees.