Called complementary or supplementary angles, these are terms you may have heard at some point in a math class, but we'll be working a lot with them when it comes to angles and triangles. I want to make sure you have this down really well. So, I'm going to show you in this video that these are just two fancy words to describe the idea that angles have to add up to two special numbers. Complementary angles add up to 90 degrees, and supplementary angles add up to 180. So, these are just fancy words to mean 90 and 180. We'll do a bunch of examples together. Let's get started here.

Complementary angles form 90 degrees. We know from the coordinate system that it's 0 and this is 90. So, if I have some given angle, like 30 degrees, how do I find the complement of that angle? Well, basically, this just says that two angles, θ_{1} and θ_{2}, have to equal 90. Now, don't worry about the θ_{1} and θ_{2} that much because, usually, in these problems, one of the angles is already going to be given to you. So, what do I have to add to 30 to get to 90? I just have to add 60. Another easy way to do this is 90 minus 30 will give you 60 degrees. That's sort of like a shortcut way of using this equation. That's really all there is to it.

For supplementary angles, it's a very similar idea, except now we have to add not to 80, not to 90, but to 180. If I go from 0 to 90, that's like halfway of a circle. But if I go another quarter, like 90, then this means that the straight line is going to be 180 degrees. So, if I have an angle like 30, what's the supplement of that angle? I just have to figure out another angle that adds to it to get me to 180. So 180 minus 30 gives me the supplement of that angle, which is 150 degrees. So those two angles, 150 and 30, are supplementary. That's really all there is to it.

Now there's a sort of mnemonic that I use to help remember this, which is that complementary angles form corners, so 'c' with 'c', and supplementary angles form straight lines, so 's' with 's'. Complementary corners, supplementary straight. That's really all there is to it. Let's go ahead and get some examples.

So we have the we're gonna find the complements and supplements of these angles here that are given to us. We've got 20 degrees. So we've got our equation over here. Again, don't pay attention to the θ_{1}, θ_{2}, because all you have to do is just think about what angle do I have to add to 20 to get to 90. So, if you wanted to sort of set this up, if you couldn't do this in your head, you would just say θ plus 20 equals 90, and then you would just, knock off 20 from both sides. So, knock off 20, knock off 20, and you're gonna get that θ is equal to 70. So the complement of 20 degrees is 70 degrees.

Very similar with a supplement, except now you're gonna have to add up to 180. So θ plus 20 equals 180. Subtract 20 from both sides over here, and you'll see that θ is equal to 160. So that's the supplement of this angle.

Let's take a look now at the second example over here, 100 degrees. What do I have to add to 100 degrees to get me to 90? Well, that's kind of weird because if you were to sort of visualize this, 100 degrees is gonna look something like that. So I'd have to go backwards and add a negative angle to get to 90. And that's kind of weird. So what you need to remember here is that complementary angles and supplementary angles are always assumed to be positive. So what that means here is that 100 degrees actually has no complement, because you'd have to add a negative angle. However, it does have a supplement because, remember, what do I have to add to 100 to get me to 180? I just have to add 80 degrees. That's all there is to it.

Pretty straightforward concept. Let me know if you have any questions. Thanks for watching.