### Video Transcript

In this video, we will use
properties of parallel lines to find a missing length of a line segment in a
transversal cut by parallel lines. To do that, letβs consider
proportional parts of parallel lines. Itβs a property of parallel lines
that says if three or more parallel lines intersect two transversals, then they cut
off the transversals proportionally. Letβs see what this looks like. Here are three parallel lines, and
here are two transversals. Remember, a transversal just means
a line that crosses at least two other lines.

In our image, these two lines are
transversals. In the figure, we see that the
three parallel lines intersected by the two transversals create four different line
segments. And if the lengths of those line
segments β we have labeled here π, π, π, π β based on this principle, these
values will be proportional. And so we can say that π over π
will be equal to π over π. Or thereβs one other way we can
write this proportion. π over π is equal to π over
π. In both cases, when we take the
cross product, weβll have π times π is equal to π times π or π times π.

At this point, we should recognize
that this property will also be true inside of polygons. To see that, we can modify this
figure. Weβve modified our figure so that
it is now a trapezoid π΄π΅πΆπ·. And this trapezoid is cut by the
line segment πΈπΉ. And πΈπΉ is parallel to the line
segment of the trapezoid π΄π· and π΅πΆ. And so we can say that the line
segment πΈπΉ cuts this trapezoid into proportional line segments, where π΄πΈ is
proportional to πΈπ΅ and π·πΉ is proportional to πΉπΆ.

Letβs consider one more
polygon. If we have the larger triangle
π΄π΅πΆ being cut by two parallel lines πΈπΉ and πΊπ», where the parallel lines are
parallel to the side of the larger triangle πΆπ΅, we can say that π over π will be
equal to π over π, which would be equal to π over π. We have proportional side lengths
created by these parallel lines.

As an extension of this, thereβs
one more property we need to consider. And that is 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. Hereβs our three parallel lines,
and hereβs a transversal. If we know that the segments cut
off by these three parallel lines are congruent, then for any other transversal
between these three parallel lines, their segments cut off by the parallel lines
will be congruent to each other.

Be careful here though. Weβre saying that π will be
congruent to π and π will be congruent to π. This does not mean that this line
segment with a length π is equal to the line segment with a length π. Weβre saying that they will be
congruent on their transversal, not between transversals. Now weβre ready to look at some
examples to see how this plays out.

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

First of all, we can identify line
segment πΈπΉ. And then we need to think about
what we know based on the figure. In the figure, we have three
parallel lines. Line π΄π· is parallel to line πΈπ΅,
which is parallel to line πΉπΆ. We can also say that lines π·πΉ and
π΄πΆ are transversals of the three parallel lines. Based on this, we know that the
parallel lines are going to cut the transversals proportionally. This means that line segment π·πΈ
over line segment π΄π΅ will be equal to line segment πΈπΉ over line segment π΅πΆ
because of the parallel lines and transversal properties. Once we have this statement, we can
just plug in the values for the three line segments we know and use that information
to solve for the fourth line segment, which will look like this. 48 over 47 is equal to πΈπΉ over
141.

From there, we cross multiply. 141 times 48 must be equal to 47
times πΈπΉ. 6768 is equal to 47 πΈπΉ. And then we divide both sides of
this equation by 47, which tells us that 144 is equal to πΈπΉ. And so we can say that line segment
πΈπΉ must measure 144 centimeters.

In our next example, weβll be
dealing with two transversals, but this time theyβre cut by four parallel lines.

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

The first thing we can do here is
take the information given in our question and label the figure. First of all, we know that lines πΏ
one through πΏ four are all parallel. The line segment ππ measures 12,
the line segment ππ equals eight, the line segment π΄π΅ equals 10, and the line
segment π΅πΆ equals five. πΆπ· is the line segment weβre
interested in finding the length of. But thereβs one more thing we
should say about the figure. And that is that π prime and π
are lines that are transversals. Theyβre lines that cross all of the
parallel lines. Because weβre dealing with four
parallel lines and then we have transversals, we know that the segments of the
transversals will be proportional.

Using the parallel lines and
transversal properties, we can say that line segment ππ over line segment πΆπ·
will be equal to the length of line segment ππ. And this is where we need to be
careful. Weβre setting up a proportion for
π prime between πΏ one and πΏ three. And that means the corresponding
values from the transversal line π must also be from line one to line three. The corresponding proportional
segment would then be from π΄ to πΆ. And thatβs fine because we can add
the distance from π΄ to π΅ and the distance from π΅ to πΆ to find the distance from
π΄ to πΆ, which is 15.

When we plug in what we know, weβre
saying that eight over line segment πΆπ· will be equal to 12 over 15. To solve for πΆπ·, we cross
multiply. Eight times 15 is equal to 12 times
πΆπ·. 120 equals 12 times πΆπ·. And if we divide both sides by 12,
we see that πΆπ· must equal 10. Looking back on our figure, we can
see something interesting here, and that is this segment π΄π· has the same length as
the segment πΆπ·. And we know that if parallel lines
cut congruent segments of one transversal, then they will cut congruent segments for
every transversal. And that means we could say that
the segment between πΏ three and πΏ four of π prime will be equal to the segment
between πΏ one and πΏ two.

We actually could find out that
ππ equals eight. And therefore, ππ would equal
four. Additionally, we can show that the
distance between line one and line two is two times the distance between line two
and line three, which is also true of line three and line four. But back to the question in hand,
line segment πΆπ· had a measure of 10.

In our next example, weβll look at
parallel lines in a polygon, specifically in a triangle.

If πΆπΈ equals π₯ plus two
centimeters, what is π₯?

First of all, on our figure, we can
label πΆπΈ as π₯ plus two centimeters. And then we should think about what
else we know based on the figure. First of all, we see that line
segment πΈπ· is parallel to line segment πΆπ΅. And then we can say that line
segment π΄π΅ and line segment π΄πΆ are transversals of these two parallel lines. Based on these two facts, we can
draw some conclusions. We can say that the parallel lines
πΈπ· and πΆπ΅ cut this triangle proportionally. So we can say line segment π΄πΈ
over line segment π΄π· will be equal to line segment πΆπΈ over line segment π·π΅ by
parallel lines and transversal properties.

To solve then, we can just plug in
the values that we know for these line segments. Six over π₯ plus two is equal to
four over eight. The first way we could solve this
is by using cross multiplication. We can say six times eight is equal
to four times π₯ plus two. Therefore, 48 equals four times π₯
plus two. And if we divide both sides of the
equation by four, we see that 12 is equal to π₯ plus two. So we subtract two from both sides,
and we see that π₯ equals 10. Now, I said this is one way to
solve. And thatβs because if we think
about proportionality, and we know that the parallel lines cut these line segments
proportionally, we notice that line segment π·π΅ is two times line segment π΄π·.

And in order for things to be
proportional, that would mean that the same thing would have to be true on the other
side. This means that π₯ plus two must be
equal to six times two, which again shows us that side length πΆπΈ must be equal to
12 and therefore π₯ plus two must be equal to 12. So again, π₯ equals 10.

In our final example, we have three
parallel lines cut by two transversals, but we have two variables to solve for.

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

The first thing we wanna do is
identify what the figure tells us. First off, we see that line π½π is
parallel to line πΎπ, which is parallel to line πΏπ. And then we can say that the line
π½πΏ and the line ππ are transversals of those three parallel lines. We also notice that the length of
line segment ππ is equal to ππ. From this information, we can draw
a few conclusions. First, because these two
transversals are cut by parallel lines, the segments created on the transversals
will be proportional. And secondly, since we know that on
one transversal the segments are congruent, we can say that the segments on the
other transversal will also be congruent.

We can say that π½πΎ must also be
equal to πΎπΏ because theyβre segments that are cut by parallel lines. And this means we can set up two
equations. We can set up one equation for π½πΎ
equals πΎπΏ and another equation for ππ equals ππ. We can say six π₯ minus 20 equals
four π₯ minus eight and five π¦ minus 25 equals three π¦ minus seven. In our first equation, we subtract
four π₯ from both sides, and we get two π₯ minus 20 equals negative eight. We add 20 to both sides. Two π₯ then equals 12. And we divide both sides by two, to
see that π₯ is equal to six.

We can plug this value back into
our two expressions. Six π₯ minus 20 equals 16, and four
π₯ minus eight also equals 16. Following the same procedure to
solve for π¦, we subtract three π¦ from both sides of our equation. And two π¦ minus 25 equals negative
seven. We add 25 to both sides, and two π¦
equals 18. Dividing both sides by two, we see
that π¦ equals nine. If we plug those values back into
our expressions, five π¦ minus 25 equals 20, and three π¦ minus seven equals 20,
which means we can say the values that make these expressions true are π₯ equals six
and π¦ equals nine.

Before we finish, letβs take a look
at the key points from this video. The first property of parallel
lines we saw is that if three or more parallel lines intersect two transversals,
then they cut the transversals proportionally. This property is also true when
dealing with parallel lines and transversals in polygons. And finally, congruent segments on
transversals: if three or more parallel lines cut congruent segments on one
transversal, then they cut congruent segments on every transversal.