### Video Transcript

In this video, we will use
properties of parallel lines and transversals to find the missing length of a line
segment in a transversal line cut by parallel lines. To do that, letβs see how parallel
lines can produce proportionality.

A property of parallel lines is
that if three or more parallel lines intersect two transversals, then they cut off
the transversals proportionally. Letβs consider what this might look
like. We have three parallel lines with
two transversals, both of which cross all three parallel lines. Remember that a transversal is a
line that crosses at least two other lines. In this figure, weβll consider
these two transversals π and π since they are the transversals that cross all
three of the parallel lines. The three parallel lines cut by two
transversals in this figure creates four line segments.

Here, we have labeled those line
segments π, π, π, π. By this property, we can say that
the ratio π to π will be equal to the ratio of π to π. Another way to write this
proportion is π over π will be equal to π over π. Before we move on, itβs also worth
noting where we might see and use this property. This property is true and applies
inside of polygons. We can modify this figure to show
an example of this.

If we have the quadrilateral
π΄π΅πΆπ·, and itβs cut by the line segment πΈπΉ that is parallel to π΄π· and π΅πΆ,
notice that we have three parallel segments cut by two transversals. This means that the segments
created will be proportional. This means that segment π΄πΈ over
segment πΈπ΅ will be equal to segment π·πΉ over segment πΉπΆ.

As an extension of this, there is
one more property we need to consider, congruent segments on transversals. If three or more parallel lines cut
off congruent segments on one transversal, then they cut off congruent segments on
every transversal. Here are three parallel lines. And if we have a transversal line
π such that the two created segments are congruent to one another, then any
transversal that is also cut by these three parallel lines will have congruent
segments. On the transversal π, the two
segments will be congruent to each other. And on the transversal π, the two
created segments are congruent to one another.

But we need to pay close attention
to this property. Weβre saying that segment π is
equal to segment π or segment π is equal to segment π. The congruency is on the same
transversal line, not between lines. You could not say here that segment
π was equal to segment π. Now weβre ready to take these
properties and apply them to some examples.

Using the information in the
figure, determine the length of line segment πΈπΉ.

First, weβll identify line segment
πΈπΉ. And then weβll think about what we
already know based on the figure. We see in our figure that we have
three parallel lines. And so we can say that π΄π· is
parallel to πΈπ΅, which is parallel to πΉπΆ. Then we can say that the lines π·πΉ
and π΄πΆ are transversals of the three parallel lines. Because we have three parallel
lines being cut by two transversals, we know that the created segments are cut
proportionally, by the properties of parallel lines and transversals.

This means we can say that π·πΈ
over πΈπΉ will be equal to π΄π΅ over π΅πΆ. If we plug in the lengths we do
know, we have 48 over πΈπΉ is equal to 47 over 141. To solve, we cross multiply. 48 times 141 will be equal to 47
times πΈπΉ. Therefore, 6768 is equal to 47
times πΈπΉ. And dividing both sides by 47 gives
us 144 is equal to πΈπΉ. Since our segments are measured in
centimeters, we can say that πΈπΉ is equal to 144 centimeters.

In our next example, weβll look at
a case where we have two transversals that are cut by four parallel lines.

In the figure, lines πΏ one, πΏ
two, πΏ three, and πΏ four are parallel. Given that ππ equals 12, ππ
equals eight, π΄π΅ equals 10, and π΅πΆ equals five, what is the length of segment
πΆπ·?

First, letβs take the information
we were given and use it to label our figure. We know that πΏ one, πΏ two, πΏ
three, and πΏ four are parallel. We can add that to our figure. The line segment ππ equals 12,
and ππ equals eight. π΄π΅ equals 10, and π΅πΆ equals
five. The unknown length that weβre
trying to solve for is πΆπ·. We can also note that the lines π
and π prime are transversals of all four parallel lines. Because we have three or more
parallel lines that are cut by two transversals, we know that the created segments
on these transversals will be proportional. Therefore, by parallel lines in
transversal properties, we can set up a proportion to solve for the missing segment
πΆπ·.

ππ over πΆπ· is equal to ππ
over π΄πΆ. If we plug in what we know, weβll
have eight over πΆπ· is equal to 12 over 15. To find the distance from π΄ to πΆ,
we need to add 10 and five together, which gives us the 15. To solve, we cross multiply. Eight times 15 will be equal to 12
times πΆπ·. 120 equals 12 times πΆπ·. Dividing both sides by 12, we see
that πΆπ· will be equal to 10.

If we add this back to our figure,
we notice something interesting. The segment π΄π΅ is equal to the
segment πΆπ·. These are congruent segments. And because we know something about
congruent segments and transversals, this means we can also say that the segment
ππ is going to be congruent to the segment ππ. And therefore ππ equals eight and
ππ must be equal to four. However, this question was only
asking for the length of the line segment πΆπ·. And weβve shown that πΆπ· equals 10
length units.

Letβs consider another example.

Given that ππΏ equals nine
centimeters, find the length of line segment ππ.

First, letβs think about what we
see in this image. First of all, we have four parallel
lines: π΄π, π΅π, πΆπ, and π·πΏ. We can also see that the lines π΄π·
and ππΏ are transversals of the four parallel lines. We also see that on the line π΄π·,
the segments π΄π΅, π΅πΆ, and πΆπ· are congruent to one another. And when three or more lines cut
congruent segments on one transversal, we know it will cut congruent segments on all
transversals cut by those parallel lines. This is the congruent segments on
transversals property, which means the line segment ππ is congruent to the segment
ππ, which is congruent to the segment ππΏ.

We also know that the distance from
π to πΏ equals nine centimeters. And since the segment ππΏ is made
up of three congruent segments, we divide nine by three to find the length of each
of the segments. And then we see that ππ, ππ,
and ππΏ will all measure three centimeters. Weβre looking for the length of the
segment ππ, which is made up of two segments, ππ and ππ, both of which measure
three centimeters. And so we can say that segment ππ
is equal to six centimeters.

In our final example, we have three
parallel lines cut by two transversals, and weβre solving for two unknown
values.

In the given figure, find the
values of π₯ and π¦.

First, we see that we have three
parallel lines, line π½π, πΎπ, and πΏπ. We can also say that the line π½πΏ
and the line ππ are transversals of the three parallel lines. Additionally, line segment ππ and
line segment ππ are congruent to one another. Based on these three things, we can
say that we have congruent segments on transversals. Since three parallel lines cut
congruent segments on one transversal, they will cut congruent segments on the other
transversal.

This means that line segment π½πΎ
will be congruent to line segment πΎπΏ. And we can mark these congruent
segments on our figure. To find π₯ and π¦, we can then set
up two equations. We can say that ππ is equal to
ππ and π½πΎ is equal to πΎπΏ. Starting with ππ equals ππ, we
substitute five π¦ minus 25 in for ππ and three π¦ minus seven in for ππ. We add 25 to both sides of our
equation, which will give us five π¦ is equal to three π¦ plus 18. So we subtract three π¦ from both
sides to get two π¦ equals 18. And finally, dividing both sides by
two gives us π¦ equals nine.

Weβll follow a similar procedure to
solve for π₯. We substitute six π₯ minus 20 in
for π½πΎ and four π₯ minus eight in for πΎπΏ. By adding 20 to both sides, we find
six π₯ equals four π₯ plus 12. From there, we subtract four π₯
from both sides, which gives us two π₯ equals 12. Dividing both sides by two, we find
that π₯ equals six. We could take the values we found
for π₯ and π¦ and use them to find the length of each line segment. π½πΎ would be equal to six times
six minus 20, which is 16. And because we know that πΎπΏ is
congruent to π½πΎ, it would also be equal to 16. And then ππ would be equal to
five times nine minus 25, which is 20. And as ππ is congruent to ππ,
both of these lengths would be 20. However, here we were only asked
what the values of π₯ and π¦ were. And we can say that π₯ equals six
and π¦ equals nine.

Before we finish, letβs quickly
review the key points from this video. If three or more parallel lines
intersect two transversals, then they cut the transversals proportionally. This property is also true for
parallel lines and transversals in polygons. And finally, if three or more
parallel lines cut congruent segments on one transversal, then they cut congruent
segments on every transversal.