Angles (part 3)

Angles formed when a transversal intersects parallel lines.

Welcome back.

We're almost done learning all
the rules or laws of angles

that we need to start
playing the angle game.

So let's just teach
you a couple of more.

So let's say I have two
parallel lines, and you may not

know what a parallel line is
and I will explain

it to you now.

So I have one line like this --
you probably have an intuition

what a parallel line means.

That's one of my parallel
lines, and let me make the

green one the other
parallel line.

So parallel lines, and I'm
just drawing part of them.

We assume that they keep on
going forever because these are

abstract notions -- this light
blue line keeps going and going

on and on and on off the screen
and same for this green line.

And parallel lines are two
lines in the same plane.

And a plane is just kind of
you can kind of use like a

flat surface is a plane.

We won't go into
three-dimensional space

in geometry class.

But they're on the same plane
and you can view this plane as

the screen of your computer
right now or the piece of paper

you're working on that never
intersect each other and

they're two separate lines.

Obviously if they were drawn
on top of each other then

they intersect each
other everywhere.

So it's really just two
lines on a plane that never

intersect each other.

That's a parallel line.

If you've already learned your
algebra and you're familiar

with slope, parallel lines are
two lines that have the

same slope, right?

They kind of increase or
decrease at the same rate.

But they have different
y intercepts.

If you don't know what
I'm talking about,

don't worry about it.

I think you know what a
parallel line means.

You've seen this -- parallel
parking, what's parallel

parking is when you park a car
right next to another car

without having the two cars
intersect, because if the cars

did intersect you would have to
call your insurance company.

But anyway, so those
are parallel lines.

The blue and the green
lines are parallel.

And I will introduce you to
a new complicated geometry

term called a transversal.

All a transversal is is
another line that actually

intersects those two lines.

That's a transversal.

Fancy word for something
very simple, transversal.

Let me write it down just
to write something down.

Transversal.

54
00:02:18,69 --> 00:02:23,51
It crosses the other two lines.

I was thinking of pneumonics
for transversals, but I

probably was thinking of
things inappropriate.

58
00:02:31,71 --> 00:02:33,81
Going on with the geometry.

So we have a transversal
that intersects the

two parallel lines.

What we're going to do is think
of a bunch of -- and actually

if it intersects one
of them it's going to

intersect the other.

I'll let you think about that.

There's no way that I can draw
something that intersects one

parallel line that doesn't
intersect the other, as long as

this line keeps going forever.

I think that that might be
pretty obvious to you.

But what I want to do
is explore the angles

of a transversal.

So the first thing I'm
going to do is explore

the corresponding angles.

So let's say corresponding
angles are kind of the

same angle at each of
the parallel lines.

corresponding angles.

They kind of play the same
role where the transversal

intersects each of the lines.

As you can imagine, and as it
looks from my amazingly neat

drawing -- I'm normally not
this good -- that these are

going to be equal
to each other.

So if this is x, this
is also going to be x.

If we know that then we could
use, actually the rules that we

just learned to figure out
everything else about

all of these lines.

Because if this is x then what
is this going to be right here?

What is this angle going
to be in magenta?

90
00:03:58,97 --> 00:04:00,99
Well, these are opposite
angles, right?

They're on opposite
side of crossing lines

so this is also x.

94
00:04:06,94 --> 00:04:08,41
And similar we can do
the same thing here.

This is the opposite angle of
this angle, so this is also x.

97
00:04:18,58 --> 00:04:21,01
Let me pick a good color.

What is yellow?

What is this angle going to be?

Well, just like we
were doing before.

Look, we have this huge
angle here, right?

This angle, this whole
angle is 180 degrees.

So x and this yellow angle are
supplementary, so we could call

Well, if this angle is y, then
this angle is opposite to y.

So this angle is also y.

Fascinating.

And similarly, if we have x up
here and x is supplementary to

this angle as well, right?

So this is equal to 180 minus
x where it also equals y.

And then opposite angles,
this is also equal to y.

So there's all sorts of
geometry words and rules that

fall out of this, and I'll
review them real fast but

it's really nothing fancy.

All I did is I started
off with the notion of

corresponding angles.

I said well, this x
is equal to this x.

I said, oh well, if those are
equal to each other, well not

even if -- I mean if this is x
and this is also x because

they're opposite, and the
same thing for this.

Then, well, if this is x and
this is x and those equal

each other, as they should
because those are also

corresponding angles.

These two magenta angles
are playing the same role.

They're both kind of
the bottom left angle.

That's how I think about it.

We went around, we used
supplementary angles to kind

of derive well, these y
angles are also the same.

This y angle is equal to
this y angle because

it's corresponding.

So corresponding angles
are equal to each other.

It makes sense, they're kind
of playing the same role.

The bottom right, if you look
at the bottom right angle.

So corresponding
angles are equal.

139
00:06:22,87 --> 00:06:25,13
That's my shorthand notation.

And we've really just
derived everything already.

That's all you really
have to know.

But if you wanted to kind of
skip a step, you also know

the alternate interior
angles are equal.

So what do I mean by
alternate interior angles?

Well, the interior angles are
kind of the angles that are

closer to each other in the two
parallel lines, but they're on

opposite side of
the transversal.

That's a very complicated way
of saying this orange angle and

this magenta angle right here.

These are alternate interior
angles, and we've already

proved if this is
x then that is x.

So these are alternate
interior angles.

This x and then that x
are alternate interior.

And actually this y and this y
are also alternate interior,

and we already proved that
they equal each other.

Then the last term that you'll
see in geometry is alternate --

I'm not going to write the
whole thing -- alternate

exterior angle.

Alternate exterior
angles are also equal.

That's the angles on the kind
of further away from each other

on the parallel lines, but
they're still alternate.

So an example of that is this x
up here and this x down here,

right, because they're on the
outsides of the two parallel

of the transversal.

These are just fancy words,
but I think hopefully

you have the intuition.

Corresponding a angles make
the most sense to me.

Then everything else proves out
just through opposite angles

and supplementary angles.

But alternate exterior is
that angle and that angle.

Then the other alternate
exterior is this y and this y.

Those are also equal.

So if you know these, you know
pretty much everything you need

to know about parallel lines.

The last thing I'm going to
teach you in order to play the

geometry game with full force
is just that the angles in a

triangle add up to 180 degrees.

181
00:08:41,77 --> 00:08:45,58
So let me just draw a
triangle, a kind of

random looking triangle.

That's my random
looking triangle.

And if this is x, this
is y, and this is z.

We know that the angles of a
triangle -- x degrees plus y

degrees plus z degrees are
equal to 180 degrees.

So if I said that this is
equal to, I don't know, 30

degrees, this is equal to,
I don't know, 70 degrees.

Then what does z equal?

Well, we would say 30 plus 70
plus z is equal to 180, or

100 plus z is equal to 180.

Subtract 100 from both sides.

z would be equal to 80 degrees.

We'll see variations of this
where you get two of the angles

and you can use this property
to figure out the third.

With everything we've now
learned, I think we're

ready to kind of ease
into the angle game.

I'll see you in the next video.