Angles formed between transversals and parallel lines

Angles of parallel lines

In this video we're going to
think a little bit about

parallel lines, and other lines
that intersect the parallel

lines, and we call
those transversals.

So first let's think about
what a parallel or what

parallel lines are.

So one definition we could use,
and I think that'll work well

for the purposes of this video,
are they're two lines that

sit in the same plane.

And when I talk about a plane,
I'm talking about a, you can

imagine a flat two-dimensional
surface like this screen --

this screen is a plane.

So two lines that sit in a
plane that never intersect.

So this line -- I'll try my
best to draw it -- and imagine

the line just keeps going in
that direction and that

direction -- let me do another
one in a different color --

and this line right
here are parallel.

They will never intersect.

If you assume that I drew it
straight enough and that

they're going in the exact
same direction, they

will never intersect.

And so if you think about what
types of lines are not

parallel, well, this green line
and this pink line

are not parallel.

They clearly intersect
at some point.

So these two guys are parallel
right over here, and sometimes

it's specified, sometimes
people will draw an arrow going

in the same direction to show
that those two lines

are parallel.

If there are multiple parallel
lines, they might do two arrows

and two arrows or whatever.

But you just have to say
OK, these lines will

never intersect.

What we want to think about
is what happens when

these parallel lines are
intersected by a third line.

Let me draw the
third line here.

So third line like this.

And we call that, right there,
the third line that intersects

the parallel lines we
call a transversal line.

Because it tranverses
the two parallel lines.

Now whenever you have a
transversal crossing parallel

lines, you have an interesting
relationship between

the angles form.

Now this shows up on a lot
of standardized tests.

It's kind of a core type
of a geometry problem.

So it's a good thing to really
get clear in our heads.

So the first thing to realize
is if these lines are parallel,

we're going to assume these
lines are parallel, then we

have corresponding angles
are going to be the same.

What I mean by corresponding
angles are I guess you could

think there are four angles
that get formed when this

purple line or this
magenta line intersects

this yellow line.

You have this angle up here
that I've specified in green,

you have -- let me do another
one in orange -- you have this

angle right here in orange, you
have this angle right here in

this other shade of green, and
then you have this angle

right here -- right there
that I've made in that

bluish-purplish color.

So those are the four angles.

So when we talk about
corresponding angles, we're

talking about, for example,
this top right angle in green

up here, that corresponds to
this top right angle in, what

I can draw it in that same
green, right over here.

These two angles
are corresponding.

These two are corresponding
angles and they're

going to be equal.

These are equal angles.

If this is -- I'll make up
a number -- if this is 70

degrees, then this angle
right here is also

going to be 70 degrees.

And if you just think about it,
or if you even play with

toothpicks or something, and
you keep changing the direction

of this transversal line,
you'll see that it actually

looks like they should
always be equal.

If I were to take -- let me
draw two other parallel

lines, let me show maybe
a more extreme example.

So if I have two other parallel
lines like that, and then let

me make a transversal that
forms a smaller -- it's even a

smaller angle here -- you see
that this angle right here

looks the same as that angle.

Those are corresponding angles
and they will be equivalent.

From this perspective it's kind
of the top right angle and each

intersection is the same.

Now the same is true of the
other corresponding angles.

This angle right here in this
example, it's the top left

angle will be the same as the
top left angle right over here.

This bottom left angle will
be the same down here.

If this right here is 70
degrees, then this down here

will also be 70 degrees.

And then finally, of course,
this angle and this angle

will also be the same.

So corresponding angles -- let
me write these -- these are

corresponding angles
are congruent.

Corresponding angles are equal.

And that and that are
corresponding, that and

that, that and that,
and that and that.

Now, the next set of equal
angles to realize are sometimes

they're called vertical angles,
sometimes they're called

opposite angles.

But if you take this angle
right here, the angle that is

vertical to it or is opposite
as you go right across the

point of intersection is this
angle right here, and that is

going to be the same thing.

So we could say opposite -- I
like opposite because it's not

always in the vertical
direction, sometimes it's in

the horizontal direction, but
sometimes they're referred

to as vertical angles.

Opposite or vertical
angles are also equal.

So if that's 70 degrees, then
this is also 70 degrees.

And if this is 70 degrees,
then this right here

is also 70 degrees.

So it's interesting, if that's
70 degrees and that's 70

degrees, and if this is 70
degrees and that is also 70

degrees, so no matter what this
is, this will also be the same

thing because this is
the same as that, that

is the same as that.

Now, the last one that you need
to I guess kind of realize are

the relationship between
this orange angle and this

green angle right there.

You can see that when you add
up the angles, you go halfway

around a circle, right?

If you start here you do
the green angle, then

you do the orange angle.

You go halfway around the
circle, and that'll give you,

it'll get you to 180 degrees.

So this green and orange angle
have to add up to 180 degrees

or they are supplementary.

And we've done other videos on
supplementary, but you just

have to realize they form the
same line or a half circle.

So if this right here is 70
degrees, then this orange angle

right here is 110 degrees,
because they add up to 180.

Now, if this character right
here is 110 degrees, what

do we know about this
character right here?

Well, this character is
opposite or vertical

to the 110 degrees so
it's also 110 degrees.

We also know since this angle
corresponds with this angle,

this angle will also
be 110 degrees.

Or we could have said that
look, because this is 70 and

this guy is supplementary,
these guys have to add up to

180 so you could have
gotten it that way.

And you could also figure out
that since this is 110, this

is a corresponding angle,
it is also going to be 110.

Or you could have said
this is opposite to

that so they're equal.

Or you could have said that
this is supplementary with

that angle, so 70 plus
110 have to be 180.

Or you could have said 70
plus this angle are 180.

So there's a bunch of ways
to come to figure out

which angle is which.

In the next video I'm just
going to do a bunch of examples

just to show that if you know
one of these angles, you

can really figure out
all of the angles.