### Video Transcript

In this video, we will learn how to
use angle relationships formed by parallel lines and transversals to solve problems
involving algebraic expressions and equations. First of all, letβs remember some
properties of parallel lines and transversals. First of all, when any two lines
intersect, we know that the vertically opposite angles will be congruent to one
another. Opposite angles are two angles
between secant lines that share a vertex, like we see here. The measure of angle π΄ would be
equal to the measure of angle πΆ, and the measure of angle π΅ would be equal to the
measure of angle π·.

When any two secant lines
intersect, we can also say that the adjacent angles will sum to 180 degrees. In this case, the measure of angle
π΄ plus the measure of angle π· must be equal to 180 degrees. We could also say the measure of
angle π· plus the measure of angle πΆ must be 180 degrees. In fact, any pair of adjacent
angles in this image would sum to 180 degrees.

But now we want to extend this to
see what happens when two parallel lines are cut by the same transversal. Here, line two and line three are
parallel, and theyβre both being cut by the transversal line one. If weβre just looking at line one
and line two, weβve already shown that the vertically opposite angles are congruent
to one another. But when we extend that to parallel
lines cut by a transversal, we say if two or more parallel lines are cut by the same
transversal, corresponding angles are congruent.

Looking at the intersection of line
one and line two, we could label the angles created as position one, two, three, and
four. And then, since we know that line
three is parallel to line two and is also being intersected by line one, when we
label them in the same direction β top left, top right, bottom right, bottom left β
the corresponding angles will be congruent, which means angles in position one are
congruent to one another, angles in position two are congruent to one another, and
so on.

There are a few other things we
should note. When two parallel lines are cut by
a transversal, alternate angles are congruent. These are angles on opposite sides
of the transversal. The one shown here are alternate
interior angles, as they are on either side of the transversal but in between the
two parallel lines. When two parallel lines are
intersected by a transversal, there are two pairs of alternate interior angles. There are also two pairs of
alternate exterior angles that are congruent. These are on either side of the
transversal and on the outsides of the two parallel lines.

Letβs look at one final angle pair
relationship. Consecutive interior angles, or
sometimes called cointerior angles, are angles on the same side of the transversal
and in between the two parallel lines. These angles sum to 180
degrees. Itβs also worth noting that all of
those properties extend to multiple parallel lines, not just two. In this image, we have three
parallel lines, and for all three of the intersections, the corresponding angles
will be congruent. But what if we have an additional
transversal? For this other transversal, itβs of
course true that the corresponding angles will be congruent. However, just because we know
something about the angles of one of the transversals does not mean we can say
anything about the angle relationships in the other transversal. So we have to remember to apply
these properties to one transversal at a time. Letβs look at some examples.

Work out the value of π₯ in the
figure.

First of all, letβs think about the
type of lines weβre seeing in the figure. We have two parallel lines that are
cut by a transversal, which makes the 61-degree angle and the π₯-angle alternate
interior angles. And when two parallel lines are cut
by a transversal, alternate interior angles are congruent. This means both of these angles are
equal in measure and π₯ must be equal to 61.

We can look at another example with
a different set of angle pairs.

Work out the value of π₯ in the
figure.

First of all, letβs think about
what we can say about these lines. We have two parallel lines cut by a
transversal, and here are the two angles weβre considering. Theyβre on the same side of the
transversal and theyβre in the same position at either intersection, which means we
call them corresponding angles. And when two parallel lines are cut
by a transversal, corresponding angles are congruent, and therefore π₯ must be equal
to 72.

In our next example, weβll not only
have two parallel lines, but weβll also have two transversals.

Answer the questions for the given
figure. Find the value of π₯ and find the
value of π¦.

First of all, letβs think about
what we know about these lines. We have two parallel lines. Letβs call them line one and line
two. And then we have a transversal,
line three, that intersects both line one and line two. And we have a second transversal we
can call line four that also intersects line one and line two. First, letβs just consider line
one, two, and three, the left side of the image. Since line three is a transversal
that cuts line one and line two, the angle π₯ and the angle 60 degrees are
consecutive interior angles, sometimes called cointerior angles. And we know that these angles must
sum to 180 degrees, which means π₯ plus 60 must be equal to 180.

If we subtract 60 from both sides
of this equation, we see that π₯ must be equal to 120. To solve for π¦, we can look at
line one, two, and four. We have two parallel lines cut by a
transversal, and the angle pair relationship of π¦ and 110 would be corresponding
angles. Theyβre on the same side of the
transversal and on the same side of their parallel lines, respectively. Theyβre in the same position at
both intersections, and corresponding angles are congruent to one another, which
means π¦ must be equal to 110.

Now that we have all of this
information, it would be possible to find this fourth angle inside this
quadrilateral. This angle is a consecutive
interior angle with 110 degrees, which means 110 plus this angle must equal 180. 70 plus 110 equals 180. If we wanted to perform one final
check that weβve calculated everything correctly, we could sum the four interior
angles created by these lines. We know that the four interior
angles inside a quadrilateral must sum to 360 degrees. In this case, they do, and that
confirms that π₯ equals 120 and π¦ equals 110.

Weβre ready to look at another
example.

The given figure shows a pair of
parallel lines and two transversals, one of which crosses at right angles. Write an expression for π in terms
of π. Using this expression for π, find
a fully simplified expression for π in terms of π.

Letβs see what we have. We have two parallel lines. We could call them line one and
line two, and then we have two transversals, which we could call line three and line
four. The first thing we wanna do is
write an expression for π in terms of π. π and π occur at the intersection
of line one and line three. They are adjacent angles on the
same side of line one. This makes them supplementary
angles. They must add up to 180
degrees. If we know that π degrees plus π
degrees must equal 180 degrees and we want π in terms of π, this means we want to
try to get π by itself and have π on the other side of the equation. To do that, we can subtract π
degrees from both sides of the equation, which means π degrees will be equal to 180
degrees minus π degrees, and therefore π will be equal to 180 minus π. This completes part one, our first
expression.

Now, using this expression for π,
we need to find a fully simplified expression for π in terms of π, which means we
first need to look at the relationship between π degrees and π degrees. When it comes to π and π, they
are in between the two parallel lines and on the same side of the transversal. They are consecutive interior
angles, sometimes called cointerior angles. And when two parallel lines are
crossed by a transversal, the consecutive interior angles are supplementary. They sum to 180. And that means π degrees plus π
degrees must equal 180 degrees.

For this expression, we want to
express π in terms of π, and that means weβll need to get π by itself. First, we subtract π from both
sides, which gives us π degrees equals 180 degrees minus π degrees or π equals
180 minus π. But remember, we want to substitute
our expression weβve already found for π. We know that π equals 180 minus
π. Remember though, weβre subtracting
all of π, so we need to keep that in parentheses and then distribute that
subtraction, which means subtract 180. But subtracting negative π would
be adding π, and that means π equals 180 minus 180 plus π. 180 minus 180 equals zero, and zero
plus π equals π. A fully simplified expression for
π in terms of π is that π is equal to π.

In fact, if we take a closer look
at π and π, we see that the relationship between these two angles are alternate
interior angles, and we know that alternate interior angles will be congruent. So π in terms of π would be π
equals 180 minus π, and π in terms of π would be π equals π.

Weβre now ready to look at our
final example.

In the following figure, π§ equals
two π₯ minus 69 and π€ equals two π¦ minus 59. Find π₯ and π¦.

First of all, letβs see what we
have. We have two parallel lines. If we extend them a little further,
we can call them line one and line two. And then we have two transversals,
which we can call line three and line four. At first, it seems like we have a
lot of variables and not as much information. But when we look at the
relationship of π€ degrees and π§ degrees, we can say that π§ degrees plus π€
degrees must equal 180 degrees. In addition to that, line three
cuts through line one and line two. And that means this fourth interior
angle would also be equal to π§ degrees since these two angles are
corresponding. And this means our two transversals
and our two parallel lines have given us a quadrilateral. And the interior angles of a
quadrilateral must sum to 360 degrees. So we can say that π§ degrees plus
π€ degrees plus 83 degrees plus π₯ must equal 360 degrees.

But we already know what π§ plus π€
equals. Since π§ plus π€ equals 180
degrees, we can say 180 plus 83 plus π₯ must equal 360. So 263 degrees plus π₯ degrees
equals 360 degrees. If we subtract 263 degrees from
both sides, π₯ degrees equals 97 degrees and π₯ must equal 97. And because we know that π§ equals
two π₯ minus 69, we can plug in 97 for π₯ and we find that π§ equals 125. Putting π§ equals 125 in our figure
and π₯ equals 97, we can use this information to find π€. We know that π§ plus π€ equals
180. We plug in 125. And when we subtract 125 from both
sides, we find out π€ equals 55.

But remember, our goal is to find
π₯ and π¦, and that means we need to plug in what we know about π€ to find π¦. Since π€ equals two π¦ minus 59, we
can say 55 equals two π¦ minus 59. Add 59 to both sides, and we get
114 equals two π¦. Divide both sides by two; 114
divided by two is 57. So we can say that here π¦ must
equal 57. In this figure under these
conditions, π₯ is 97 and π¦ is 57.

Before we finish, letβs review our
key points. When two or more parallel lines are
cut by a transversal, corresponding angles are congruent, alternate interior angles
are congruent, alternate exterior angles are congruent, and consecutive interior
angles or cointerior angles are supplementary.