### Video Transcript

In this video, we will learn how to
apply the angle relationships between a pair of parallel lines and a transversal to
establish and use other relationships between parallel lines and transversals. There are many uses for the angle
relationships between parallel lines and transversals. The most common use is finding the
measures of other angles using results such as the alternate interior angles being
congruent. However, there are other uses, such
as proving relationships between transversals of lines.

To see this in action, we first
recall that a transversal of a pair of parallel lines will have congruent
corresponding angles. This is a useful result, since we
can use this to determine the angles between two lines using the angle between one
line and a parallel line. In particular, if we have a line
π΄π΅ that is perpendicular to another line πΈπΉ, then we know that any line parallel
to π΄π΅ must have a corresponding angle with a measure of 90 degrees. This gives us the following
result. If a line is perpendicular to line
π΄π΅, then it is perpendicular to any line parallel to line π΄π΅. This is not the only property we
can show using properties of transversals.

We recall that a transversal of a
pair of parallel lines will have congruent corresponding angles. We can also recall that if the
corresponding angles in a transversal of two lines are congruent, then the lines are
parallel. Therefore, if a transversal of two
lines is perpendicular to the two lines, then the lines must be parallel, since they
have congruent corresponding angles. This gives us the following
property. If two lines are perpendicular to
the same line, then they must be parallel. Letβs now see an example of
applying these properties to determine relationships between lines.

If line π΄π΅ is parallel to
line πΆπ· and line πΈπΉ is perpendicular to line πΆπ·, which of the following is
correct? (A) Line segment πΈπΊ bisects
line segment π΄π΅. (B) Line π΄π΅ is parallel to
line πΈπΊ. (C) Line π΄π΅ is perpendicular
to line πΆπ·. (D) Line π΄π΅ is perpendicular
to line πΈπΊ. (E) Line segment πΈπΉ bisects
line segment πΆπ·.

Letβs begin by adding the fact
that line πΈπΉ being perpendicular to line πΆπ· means the lines are at right
angles and the fact that lines π΄π΅ and πΆπ· are parallel to our diagram. We can then use corresponding
angles to note that line πΈπΉ is perpendicular to line π΄π΅. This shows that line π΄π΅ is
perpendicular to line πΈπΊ, which is answer (D). However, this is a particular
case of the fact that if a line is perpendicular to line πΏ, then it is
perpendicular to any line parallel to πΏ.

It is worth noting that
although the diagram looks like the transversal bisects the parallel lines, this
does not need to be, since we can translate the transversal and still have the
image being perpendicular to the parallel lines. We can see that line πΏ is
perpendicular to the parallel lines π΄π΅ and πΆπ· and that it is not a
perpendicular bisector of either line. Thus, we cannot conclude that
line segment πΈπΊ bisects line segment π΄π΅ or that it bisects line segment
πΆπ·. So answers (A) and (E) are
incorrect. Hence, line π΄π΅ is
perpendicular to line πΈπΊ, and the correct answer is option (D).

Before we move on to our next
example, there is another useful property we can show. Letβs consider a pair of distinct
parallel lines, π΄π΅ and πΆπ·, and another pair of distinct parallel lines, πΆπ· and
πΈπΉ. It appears as though all three of
the lines are parallel. We can prove this is the case by
sketching a transversal perpendicular to line πΈπΉ. Using corresponding angles, we can
show that all three lines are perpendicular to the transversal. Then, we can recall that if two
lines are perpendicular to the same line, then they must be parallel to each
other. Hence, all three lines are
parallel.

We have proven the following
result. If two distinct lines are parallel
to a third distinct line, then all three lines are parallel. Letβs now see an example of using
this property to determine the relationship between given lines.

Fill in the blank. If line π΄π΅ is parallel to
line πΆπ· and line π΄π΅ is parallel to line πΈπΉ, then line πΆπ· is what to line
πΈπΉ.

We first recall that if two
lines are parallel to the same line, then they must be parallel. Since both lines πΆπ· and πΈπΉ
are parallel to the same line, π΄π΅, we must have that the two lines are
parallel. Hence, the answer is that line
πΆπ· is parallel to line πΈπΉ, denoted as shown.

There is one final property that we
want to show about the transversals of parallel lines. And we will introduce this property
with an example.

Consider the following diagram,
where πΏ one, πΏ two, and πΏ three are all parallel and the lengths of the line
segments between the parallel lines are as shown. In the diagram, it appears that if
the lengths of the line segments of a transversal between three parallel lines are
equal, then the lengths of the line segments of any transversal between three
parallel lines will be equal. We can prove this by using
congruent triangles.

We can start with a perpendicular
transversal that is split into two sections of equal length by the parallel lines as
shown. We can note that the measure of
angle πππΏ is equal to the measure of angle πππ, as they are vertically
opposite angles. We now see that triangle πππ and
triangle πππΏ are congruent by the ASA criterion. In particular, this means that πΏπ
is equal to ππ. Hence, the other transversal is
also split into two sections of equal length. We can always add in a transversal
perpendicular to these lines. So the proof of this result in
general is very similar.

We have shown the following
property. If a set of parallel lines divide a
transversal into segments of equal length, then they divide any other transversal
into segments of equal length. Letβs now see some examples of
using this property to find the length of a transversal between parallel lines.

If lines π΄π΅, πΆπ·, and πΈπΉ
are all parallel and πΆπΈ equals two centimeters, find π΄πΈ.

We can see in the diagram that
we are given three parallel lines and two transversals. We can also see that π΅π· is
equal to π·πΉ. So, in this case, the parallel
lines divide one of the transversals into line segments of equal length. We can recall that if a set of
parallel lines divide a transversal into segments of equal length, then they
divide any other transversal into segments of equal length. Therefore, it will divide the
other transversal into sections of equal length. Thus, π΄πΆ is equal to πΆπΈ,
which is equal to two centimeters. We note that π΄πΈ is equal to
π΄πΆ plus πΆπΈ. So π΄πΈ is equal to two
centimeters plus two centimeters, which equals four centimeters.

In our final example, we will apply
multiple properties of the transversals of parallel lines to a triangle with a side
bisected by parallel lines.

Consider triangle π΄π΅πΆ and
lines π΄π and πΈπ·, which are parallel to line πΆπ΅. Find the length of the line
segment π΄π΅. Find the measure of angle
π΄π΅πΆ.

We are given three parallel
lines and two transversals of these lines. We can then recall that if a
set of parallel lines divide a transversal into segments of equal length, then
they divide any other transversal into segments of equal length. Since π΄πΈ is equal to πΈπΆ,
the segments of the other transversal must be equal in length. So π΄π· is equal to π·π΅, which
equals five millimeters. Since π΄π΅ is equal to π΄π·
plus π·π΅, then π΄π΅ equals five millimeters plus five millimeters, which equals
10 millimeters. π΄π΅ is equal to 10
millimeters.

It appears in the diagram that
triangle π΄π΅πΆ is a right triangle. However, we need to justify why
this is the case. We can do this by recalling
that if a line is perpendicular to line πΏ, then it is perpendicular to any line
parallel to πΏ. Since line πΈπ· is
perpendicular to line π΄πΆ and line πΈπ· is parallel to line π΅πΆ, we must have
that lines π΅πΆ and π΄πΆ are perpendicular. This means the angle at πΆ has
a measure of 90 degrees, so π΄π΅πΆ is a right triangle.

The sum of the measures of the
interior angles in a triangle is 180 degrees. So 180 degrees equals 35
degrees plus 90 degrees plus the measure of angle π΄π΅πΆ. Rearranging the equation, we
have the measure of angle π΄π΅πΆ equals 180 degrees minus 35 degrees minus 90
degrees, which equals 55 degrees. The answers to the two parts of
this question are 10 millimeters and 55 degrees.

Letβs finish by recapping some of
the most important points from this video. If a line is perpendicular to line
π΄π΅, then it is perpendicular to any line parallel to line π΄π΅. If two lines are perpendicular to
the same line, then they must be parallel. If two distinct lines are parallel
to a third distinct line, then all three lines are parallel. If a set of parallel lines divide a
transversal into segments of equal length, then they divide any other transversal
into segments of equal length.