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Welcome back.

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

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that we need to start
playing the angle game.

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

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

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know what a parallel line is
and I will explain

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it to you now.

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

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what a parallel line means.

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

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green one the other
parallel line.

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

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We assume that they keep on
going forever because these are

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

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

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And parallel lines are two
lines in the same plane.

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

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flat surface is a plane.

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We won't go into
three-dimensional space

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in geometry class.

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

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

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you're working on that never
intersect each other and

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they're two separate lines.

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Obviously if they were drawn
on top of each other then

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they intersect each
other everywhere.

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

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intersect each other.

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That's a parallel line.

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

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with slope, parallel lines are
two lines that have the

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same slope, right?

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They kind of increase or
decrease at the same rate.

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But they have different
y intercepts.

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

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don't worry about it.

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I think you know what a
parallel line means.

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

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

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without having the two cars
intersect, because if the cars

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did intersect you would have to
call your insurance company.

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But anyway, so those
are parallel lines.

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The blue and the green
lines are parallel.

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And I will introduce you to
a new complicated geometry

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term called a transversal.

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All a transversal is is
another line that actually

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intersects those two lines.

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That's a transversal.

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Fancy word for something
very simple, transversal.

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Let me write it down just
to write something down.

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Transversal.

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It crosses the other two lines.

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I was thinking of pneumonics
for transversals, but I

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probably was thinking of
things inappropriate.

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Going on with the geometry.

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So we have a transversal
that intersects the

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two parallel lines.

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

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if it intersects one
of them it's going to

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intersect the other.

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I'll let you think about that.

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

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

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this line keeps going forever.

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I think that that might be
pretty obvious to you.

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But what I want to do
is explore the angles

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of a transversal.

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

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the corresponding angles.

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So let's say corresponding
angles are kind of the

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same angle at each of
the parallel lines.

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corresponding angles.

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They kind of play the same
role where the transversal

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intersects each of the lines.

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

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

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going to be equal
to each other.

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

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

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just learned to figure out
everything else about

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all of these lines.

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

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What is this angle going
to be in magenta?

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Well, these are opposite
angles, right?

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They're on opposite
side of crossing lines

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so this is also x.

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And similar we can do
the same thing here.

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

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Let me pick a good color.

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What is yellow?

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What is this angle going to be?

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Well, just like we
were doing before.

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Look, we have this huge
angle here, right?

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This angle, this whole
angle is 180 degrees.

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

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

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So this angle is also y.

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Fascinating.

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

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this angle as well, right?

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

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

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

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

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it's really nothing fancy.

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All I did is I started
off with the notion of

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corresponding angles.

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

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

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

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

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

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each other, as they should
because those are also

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corresponding angles.

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These two magenta angles
are playing the same role.

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They're both kind of
the bottom left angle.

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That's how I think about it.

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We went around, we used
supplementary angles to kind

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

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This y angle is equal to
this y angle because

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it's corresponding.

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So corresponding angles
are equal to each other.

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

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

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So corresponding
angles are equal.

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That's my shorthand notation.

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And we've really just
derived everything already.

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That's all you really
have to know.

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

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the alternate interior
angles are equal.

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So what do I mean by
alternate interior angles?

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

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

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opposite side of
the transversal.

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

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this magenta angle right here.

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These are alternate interior
angles, and we've already

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proved if this is
x then that is x.

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So these are alternate
interior angles.

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This x and then that x
are alternate interior.

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

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and we already proved that
they equal each other.

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

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

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exterior angle.

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Alternate exterior
angles are also equal.

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

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

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

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

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of the transversal.

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These are just fancy words,
but I think hopefully

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you have the intuition.

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Corresponding a angles make
the most sense to me.

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Then everything else proves out
just through opposite angles

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and supplementary angles.

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But alternate exterior is
that angle and that angle.

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

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Those are also equal.

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

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to know about parallel lines.

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

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

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triangle add up to 180 degrees.

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So let me just draw a
triangle, a kind of

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random looking triangle.

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That's my random
looking triangle.

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

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

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degrees plus z degrees are
equal to 180 degrees.

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

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

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Then what does z equal?

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

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100 plus z is equal to 180.

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Subtract 100 from both sides.

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z would be equal to 80 degrees.

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

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

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

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ready to kind of ease
into the angle game.

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I'll see you in the next video.