### Video Transcript

In the following figure, π΄πΆ
equals π΄π΅, line segment π΄πΆ is parallel to line segment π·πΉ, and line segment
π΄π΅ is parallel to line segment π·πΈ. If the measure of angle π΄π΅πΆ
equals 30 degrees, find the measure of angle πΈπ·πΉ.

Letβs begin this question by noting
that we have two congruent line segments, as the line segments π΄πΆ and π΄π΅ are
congruent. In fact, this also means that the
larger triangle, triangle π΄π΅πΆ, is an isosceles triangle. So, given the information that the
measure of angle π΄π΅πΆ is 30 degrees, then we know that the measure of angle π΄πΆπ΅
is 30 degrees, since, by the isosceles triangle theorem, we know that an isosceles
triangle has two congruent angles.

Now, we need to find the measure of
angle πΈπ·πΉ, which is in the smaller triangle. Although we donβt yet have any
angle measures for this triangle, we can calculate some of them by using the
properties of parallel lines. Using the two parallel line
segments π΄π΅ and π·πΈ and the transversal of line segment πΆπ΅, we can determine
that the measure of angle π·πΈπΉ is also 30 degrees, as these angles are
corresponding. Then, using the other pair of
parallel lines segments, π΄πΆ and π·πΉ, with the same transversal, we can determine
that the measure of angle π·πΉπΈ is 30 degrees.

Finally, we can observe that the
two angles we have calculated along with the angle πΈπ·πΉ, whose measure we need to
calculate, are all contained in the triangle πΈπ·πΉ. And since the interior angle
measures in a triangle sum to 180 degrees, we know that these three angle measures
sum to 180 degrees. We can simplify this equation and
then subtract 60 degrees from both sides to give us the answer that the measure of
angle πΈπ·πΉ is 120 degrees.