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## Geometry (all content)

# Angles (part 3)

Angles formed when a transversal intersects parallel lines. Created by Sal Khan.

## Want to join the conversation?

- what if at1:46the transversal line is on top of one of thelines(7 votes)
- Well try it, if a line is above the parallel lines then they will eventually intersect.(2 votes)

- What are interior angles and exterior angles?(4 votes)
- Interior angles are just the angles inside the polygon. A triangle has 3 of them for example.

Exterior angles are a bit more tricky. But basically, you can find an exterior angle by taking one of the lines on the polygon and extending it out further, beyond one of the other lines it intersects. The new angle you get is called an exterior angle, because it is on the outside of the polygon. It's kind of hard to explain, so here are a few diagrams that should help:

http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i5/s6egintext.gif

http://image.wistatutor.com/content/feed/tvcs/interior-exterior-angles.gif

https://upload.wikimedia.org/wikipedia/commons/thumb/5/51/ExternalAngles.svg/300px-ExternalAngles.svg.png

For more info, I'd suggest you search "exterior angles" on khan academy. There are a few videos that should help give you some intuition. I hope this helps.(3 votes)

- Why is this video so pixelated?

the whole time(2 votes)- It's because it's an old video which does not have very high quality. That's why it's under "Sal's old angle videos."(4 votes)

- Does a transversal have to be a line segment or a line or can it be both ?(3 votes)
- I believe it can be a line segment or a line. Although all of the definitions I've found (Wikipedia and Wolfram Mathworld) - say "Line" and never "Line Segement".(2 votes)

- at4:11are the red "x"'s and the orange "x"'s vertical angles?(1 vote)
- He explained what transversal is... But I still don't get it. Can anyone explain please?(1 vote)
- In my example here, there are two parallel lines (shown by the >) and a third line going straight, and intersecting them both. The third line is called the transversal.

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The video after this one is really helpful for explaining this.(3 votes)

- On that diagram how do we get cointerior angles?(2 votes)
- The angle marked in yellow between the green and purple lines is cointerior with the angle marked in magenta between the cyan and purple lines.

The angle marked in yellow between the cyan and purple lines is cointerior with the angle marked in orange between the green and purple lines.

Cointerior angles are the angles formed between the transversal and the parallel lines, which are on the same side of the transversal line. This means that there are two pairs of cointerior angles.(1 vote)

- can you alter a triangle to equal more than 180 degrees?(1 vote)
- No. You need to make the sum of the interior angles equal to 180 or else it wouldn't be a triangle.(3 votes)

- how does a triangle add up to 180 degrees(1 vote)
- Draw a square. Draw a diagonal line across the square, from one corner to the other. See how you made two triangles, that are each half of the square? If the square is 360, then half must be 180.(4 votes)

- What is the angle of a circle with a diameter of 10?(1 vote)
- 360 degrees. The number of degrees never changes depending on the diameter. It really depends on what the shape is.(3 votes)

## Video transcript

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. It crosses the other two lines. I was thinking of mnemonics
for transversals, but I probably was thinking of
things inappropriate. But anyway 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. That's how I think of them. So this angle, this angle,
and this angle are 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? Well, these are opposite
angles, right? They're on opposite
side of crossing lines so this is also x. And similar we can do
the same thing here. This is the opposite angle of
this angle, so this is also x. 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 this angle y, and this is
equal to 180 minus x, right? And we're just using
supplementary angles here. 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 they're corresponding. 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. 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 lines, right -- one's on
top, one's on bottom. Then they're on opposite
sides, on alternate sides, 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. 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.