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Angles formed between transversals and parallel lines

Angles of parallel lines. Created by Sal Khan.

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Video transcript

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.