### Video Transcript

In the given figure, what is the
length of line segment πΉπΈ?

Letβs begin by considering the
information shown in the figure.

We notice that we have two pairs of
parallel line segments, with the first being that line segments π΄π· and π΅πΈ are
parallel. The second pair of parallel line
segments are π΄π΅ and π·πΆ. So, by definition, that means that
the quadrilateral π΄π΅πΆπ· is a parallelogram, since these are quadrilaterals with
both pairs of opposite sides parallel.

One of the properties of
parallelograms that will be useful in this problem is that opposite sides are equal
in measure or congruent. So, this line segment π΅πΆ, which
is already marked as congruent to line segment πΈπΆ, is also congruent to the line
segment π΄π·, which is opposite it in the parallelogram. So, we can mark it with the same
two lines as the other two congruent line segments.

We are asked to find the length of
the line segment πΉπΈ. It doesnβt appear as though we have
enough information, as we only have the length of one line segment on the
diagram. So, a good approach would be to
check if perhaps the two triangles π΄π·πΉ and πΆπΈπΉ have a mathematical
relationship. For example, we can check if they
are congruent triangles.

As we have the pair of parallel
line segments π΄π· and π΅πΈ, we can work out information about some of the angle
measures in these triangles. Using these parallel line segments
and the transversal π΄πΈ, we can note that angles π·π΄πΉ and πΆπΈπΉ are congruent,
as these are alternate angles. Similarly, angles π΄π·πΉ and πΈπΆπΉ
are also alternate angles and so are congruent.

Therefore, we have determined that
we have two pairs of congruent angles. And we have already established
that the pair of included sides, π΄π· and πΈπΆ, between these angles are
congruent. So, this proves that triangles
π΄π·πΉ and πΈπΆπΉ are congruent by using the ASA, or angle-side-angle, congruence
criterion. This will allow us to find the
length of line segment πΉπΈ.

In the congruent triangles, the
side which is corresponding to line segment πΉπΈ is line segment πΉπ΄. As these are corresponding sides,
they are congruent. And the length of line segment πΉπ΄
is six centimeters.

So, we have found the answer of six
centimeters for the length of line segment πΉπΈ by first using the properties of
parallelograms and then proving that there is a pair of congruent triangles.