Geometry (all content)
More on complementary and supplementary angles. Introduction to opposite angles. Created by Sal Khan.
So let's review everything that we know so far, because it's good to keep reviewing. Because these are things you should never forget the rest of your life. So if I have a line and if I draw an angle that goes-- let's say this is the pivot point, right? If I go all the way around the line, or in a circle, that's 360 degrees. We learned that there are 360 degrees in a circle. Right? We also learned that if I have lines like this. If I have two angles-- let me draw it like that. And this is angle x. This is angle y. x and y are supplementary. And that just means that they add up to 180 degrees. x plus y is equal to 180 degrees. And why does that make sense? Because look, if we add up x plus y we have gone halfway around the circle. So that's 180 degrees, right? So this is part of the way, and this is the rest of the way. So x plus y are going to equal 180 degrees. So hopefully we have learned that. And then let me switch colors for the sake of variety. Let me use my line tool. If I have-- let's see, I'm going to draw perpendicular lines. If I have that line, and then I have that line. And they are perpendicular. And then I have another line. Let's say it goes like that. And then I say that this is angle x. Woops. This is angle x. And this is angle y. Well, I said this line and this line are perpendicular, right? So that means that they intersect at a 90 degree angle. So we know that this whole thing is 90 degrees. And so what do we know about x plus y? Well, x plus y is going to equal 90 degrees. Or we could say that x and y are complementary. And I always get confused between supplementary and complementary. You just got to memorize it. I don't know if there's any-- let's see, is there any easy way? 180, supplementary. You could say that 180-- 100 starts with an O, which supplementary does not start with. So there. There's your mnemonic. Complementary. And 90 starts with an N, and complementary does not start with an N. That's your other mnemonic. Complementary. I don't know if I'm spelling it right. Who cares? Let's move on. So let's learn some more stuff about angles. And what I'm going to do is I'm going to give you an arsenal, and then once you have that arsenal you can just tackle these beastly problems that I'm going to throw at you. So just take these for granted right now, and then in a few videos, probably, we're going to tackle some beastly problems. And you know, I'm using variables here. And if you're not familiar with variables you can put numbers here. If x was 30 degrees, then y is going to be 60 degrees. Right? Or in this case, if x is, I don't know, 45 degrees, then y is going to be 135 degrees. That other way. Let me draw another property of angles of intersecting lines. So if I have two angles, two lines that intersect like this. So a couple of interesting things. So first, I'm going to teach you about opposite angles. Let me switch colors. Let me switch to yellow. So if this is x degrees, then it turns out that the angle opposite to it is also equal to x degrees. And you don't believe me? Well let me prove it to you. Let's say we call this, I don't know, let's call this y degrees. Right? And I'm going to prove to you that the x and the y are the same. Well what do we know already? Let's call this other angle-- and I'm doing this to confuse you-- angle z. Well what do we know about angle x and angle z? It may not be obvious to you because I've drawn it slightly different, but I'll give you a small hint with an appropriately interesting color. So what angle is this whole thing right here? Well I'm just going along a line, right? That's halfway around a circle. So what angle is that? Well that's 180 degrees. So what does x plus z equal? Well, x plus z is going to equal that larger angle. x plus purple z is going to equal-- I think I'll switch to the blue; maybe it's taking too much time for me to switch-- is equal to 180 degrees. Or x and z are supplementary. I've run out of space. So what do we know about z? Well z is equal to 180 minus x. Right? Because x plus z is 180. Fine. Now, what's the relationship between z and y? Well, z and y are also supplementary. Because look, if I drew this angle here. Look at this big angle. What angle is that? Well once again I'm still going halfway around the circle. Right? But now I'm using this line right here. So that's 180 degrees. So we know that angle z plus angle y is also equal to 180 degrees. Right? Or, I don't want to keep writing it, but z and y are also supplementary. But we just figured out that z is 180 minus x. Right? So let's just substitute that back in here. So we get 180 minus x plus y is equal to 180 degrees. Why don't we subtract 180 degrees from both sides of this equation. That cancels out, and we get minus x plus y is equal to 0. And then add x to both sides of this equation, and we get y is equal to x. That was a very long way of showing you something that is fairly simple-- that opposite angles are equal to each other. So x is equal to y. And if you've played around with this, if you just drew a bunch of straight lines and they intersected at different angles, I think when you eyeball it it would make sense. And then similarly, if that's z then the other opposite angle here is also z degrees. So what do we know now? The total angles in a circle, 360 degrees. When two angles kind of combine, go halfway around the circle-- or they combine, kind of form a line. There's different ways you can think about it. We know they're supplementary. They add up to 180 degrees. x plus y is 180 degrees. If they add up to 90 it's complementary. x plus y is 90. And then opposite angles are equal to each other. Right? This angle is equal to this angle. And then this angle is going to be equal to this angle for the same reason-- because it's opposite. In the next video I'm going to show you about parallel lines and transversals. More fancy words for what I think are fairly straightforward concepts. I'll see in the next video.