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Angles (part 3)

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

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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.