### Video Transcript

Triangles π΄π·πΈ and π΄π΅πΆ in the
given figure are similar. What, if anything, must be true of
the lines π·πΈ and π΅πΆ? Option (A) they are parallel, or
option (B) they are perpendicular.

In this question, we are told that
there are two similar triangles. They are triangle π΄π·πΈ, which is
the smaller triangle, and triangle π΄π΅πΆ, which is the larger triangle. We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion.

Now, we are asked what must be true
of the two lines π·πΈ and π΅πΆ. But we are arenβt given any
information about the lengths of any sides in this figure. So, letβs see what we can work out
by using the angle properties of these similar triangles. Since the corresponding angles are
congruent, we know that the measure of angle π΄π·πΈ is equal to the measure of angle
π΄π΅πΆ. And, in the same way, the
corresponding angles π΄πΈπ· and π΄πΆπ΅ must be of equal measure.

The third pair of angles in each
triangle is the common angle at vertex π΄, which we could refer to as angle π·π΄πΈ
in triangle π΄π·πΈ and angle π΅π΄πΆ in triangle π΄π΅πΆ. But we can consider the information
from the first pair of angles. These angles are constructed
between the lines π·πΈ and π΅πΆ and the line π΄π΅. And we know that these angles are
congruent.

We know that if we have a pair of
parallel lines and a transversal, then the corresponding angles are congruent. And remember that the converse of
this is also true. That is, if corresponding angles in
a transversal of two lines are congruent, then the lines are parallel. And this is the situation that we
have here. The two lines are π·πΈ and
π΅πΆ. The transversal is the line
π΄π΅. And corresponding angles are
congruent. Therefore, the lines π·πΈ and π΅πΆ
are parallel. So, the answer to the question is
that given in option (A). We can say that the lines π·πΈ and
π΅πΆ are parallel.

Itβs worth noting that we could
have proved the same property using the second pair of angles that we found. The only difference in using the
corresponding congruent angles π΄πΈπ· and π΄πΆπ΅ would be that the transversal would
instead be the line π΄πΆ. But this would still prove that
lines π·πΈ and π΅πΆ are parallel.