### Video Transcript

Is the line segment between the
midpoints of two sides of a triangle parallel to the other side?

To answer this question, letβs take
a triangle, which we can call triangle π΄π΅πΆ. We want to consider the line
segment between the midpoints of two sides of the triangle, which is also called a
midsegment. We can draw π· and πΈ at the
midpoints of line segments π΄π΅ and π΄πΆ. And as these are midpoints, we know
that there will be two pairs of congruent line segments, since π΄π· and π·π΅ are
congruent and π΄πΈ and πΈπΆ are congruent. We can join the midpoints π· and πΈ
to create the midsegment, which will be line segment π·πΈ.

We now need to determine if the
midsegment π·πΈ is parallel to the third side of the triangle, which is line segment
π΅πΆ. The important first step we take in
working out the answer is to draw a new line. We construct the line passing
through πΆ and a new point π such that this new line πΆπ is parallel to the line
segment π΄π΅. We can then extend the line segment
π·πΈ so that it meets the new line πΆπ at the point πΉ.

Now letβs consider some of the
sides and angles in the triangles π΄πΈπ· and πΆπΈπΉ. Firstly, we know that we have two
congruent sides. Since we set πΈ to be the midpoint
of line segment π΄πΆ, we can write that π΄πΈ equals πΆπΈ. Then, because we have a pair of
parallel lines and a transversal line segment π΄πΆ, we can identify a pair of
alternate interior angles. As these angles are congruent, we
can write that the measures of angles π·π΄πΈ and πΉπΆπΈ are equal. We can also observe that angles
π·πΈπ΄ and πΉπΈπΆ are vertically opposite angles. So their measures are also
equal. And so because we have two pairs of
congruent angles and an included pair of sides congruent, we can write that
triangles π΄πΈπ· and πΆπΈπΉ are congruent by the ASA criterion.

We can notice at this point that
even with these congruent triangles, we still havenβt determined if we have a pair
of parallel lines. But letβs think about what we can
determine from these congruent triangles. We can identify that the
corresponding sides π΄π· and πΆπΉ must be congruent. And of course we already created
the line segment π΄π΅ such that π· is the midpoint. So π΅π· is also congruent to
π΄π·. Now we know that in the figure,
there are three congruent line segments.

The final step is to consider the
polygon π΅πΆπΉπ·. π΅πΆπΉπ· is a quadrilateral, and we
have shown that the sides π΅π· and πΉπΆ are congruent. And we know that they are also
parallel because we constructed the line πΆπ to be parallel. This property is sufficient to
prove that the shape π΅πΆπΉπ· is a parallelogram. And we know that parallelograms
have two pairs of opposite sides parallel. So line segments π·πΉ and π΅πΆ are
also parallel.

We can therefore return to the
original question to see if the midsegment of a triangle is parallel to the third
side. And the answer is yes, because the
line segment π·πΈ is contained within the line segment π·πΉ. So either line segment π·πΈ or π·πΉ
can be said to be parallel to line segment π΅πΆ. In fact, this property that we have
just proved is one of the triangle midsegment theorems. It can be stated as the line
segment joining the midpoints of two sides of a triangle is parallel to the third
side. We can learn this property and
apply it directly in many different problems. Stating this theorem would be
sufficient for us to give the answer to the question as yes.