Complementary and Supplementary Angles - Online Tutor, Practice Problems & Exam Prep

Complementary angles add up to 90 degrees, while supplementary angles total 180 degrees. To find the complement of an angle, subtract it from 90, and for the supplement, subtract it from 180. In right triangles, the two non-right angles are complementary, summing to 90 degrees. For example, if one angle is represented as x and the other as x + 10, the equation $x+x+10=90$ can be solved to find the angles.

Intro to Complementary & Supplementary Angles

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, θ₁ and θ₂, have to equal 90. Now, don't worry about the θ₁ and θ₂ 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 θ₁, θ₂, 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.

Find the complement & supplement of a $45°$ angle.

Complement: ____

Supplement: ____

• A. $Complement=135\degree;Supplement=45\degree$
• B. $Complement=45\degree;Supplement=135\degree$
• C. $Complement=315\degree;Supplement=135\degree$
• D. $Complement=45\degree;Supplement=315\degree$

Solving Problems with Complementary & Supplementary Angles

Everyone, welcome back. So in a previous video, we learned complementary angles add up to 90 degrees, and supplementary angles add up to 180 degrees. But some problems, like the one we're going to work out down here, will throw at you some shapes or diagrams or even triangles that will have the angles written in terms of variables like x. They won't give you an angle and ask you for the complement or supplement. They'll give you something like this. Now it might seem like we have to use a different method to solve these types of problems, but we actually use what we know about complementary and supplementary angles to solve them. Let me just go ahead and show you using this example over here. We'll walk through it together, and I'll show you how this works. Alright?

So this is a right triangle over here. We have a right triangle with this little square in the corner. That means that it's 90 degrees. So how do we solve this problem here? Up until now, whenever we have triangles, we usually know what 2 of the angles are, and we could solve for the other one, because we know all the angles in a triangle add up to 180 degrees. Well, the whole idea here is, if this is fixed at 90 degrees in a right triangle, because one angle is fixed at 90 degrees, the other two angles over here, these two have to also be 90 degrees. So those two angles are also going to be complementary. So, how can we use this to help us solve this problem? Well, basically, we know that whatever these two angles are, they have to add up to 90 degrees. So we can set it up just like we set up an equation for complementary angles. So x plus x plus 10, those two things, if I put them together, they have to add up to 90 degrees.

Now what we can do is we can just solve this like any other linear equation. I can combine like terms. This ends up being 2x+10=90. Subtract 10 from both sides over here. We're going to get 2x=80. When you divide by 2 for both sides, what you'll see is that x is equal to 40 degrees. Alright?

So x is equal to 40 degrees. Are we done here? Is that just the answer? Well, not quite, because the question asks us to find what each of the angles is in the triangle below. It's not enough to find just what x is. Now you actually have to plug it back into those angles to solve for those. Alright? Now for this over here, just because it's x, we actually know that this is already going to be 40 degrees. And then what we can do is we'll say, well, what's x plus 10? That's just going to be 40 plus 10, so in other words, this is 50 degrees. We know these two angles are complementary, 40 and 50. They both add up to 90, so now that's our whole triangle. Alright? So these are your angles: 50 degrees, 40 degrees, and 90 degrees. Alright? All the angles have to add up to 180 degrees, and we can use complementary angles to solve for this. Hopefully, that makes sense. Thanks for watching.

Example 1

Everyone, welcome back. So, let's take a look at this example over here because we're given a straight line with some angles, and these angles are actually given to us in terms of variables like x. So, we're going to use what we know about complementary and supplementary angles to solve. This isn't a triangle, but we can still use those same ideas to solve this problem here. We're going to solve for each of these angles below. Alright? How do I do that? Well, I've got a straight line like this, and I've got these two angles over here. And what we know about supplementary angles is that they will always add up to 180 degrees. So, in other words, if I take this angle, whatever it is, and I add it to this angle, whatever that is, those things have to add up to 180. So, I can set up an equation here, which is I'm literally going to take those two things and add them together and set them to 180. So, 50-2x+(17x-20). When I combine those two things, I should get 180 degrees. Alright?

So how do we solve these types of equations? This is really just a linear equation. We've dealt with these before in previous videos. The idea here is that you just combine like terms. So the 50 combines with the negative twenty. The negative 2 combines with the 17. So what's 50 and then +-twenty? When you combine those two things over here, you'll just get that that's 30. And then what happens to the negative 2 and the 17x, that thing just combines to 15x. So when you add those two things, you should get 180. And again, we just subtract 30 from both sides, to get x by itself. So subtract 30 over here. And what you'll see here is that 15x=150. So how do I solve for this? Just divide by 15 on both sides. So divide by 15, and you'll see that x=10. Alright. So is that it? Is x equals 10 and I'm just done? No, not quite. Because what you have to do is you have to plug these numbers back into or you have to plug this x equals 10 back into these equations to solve for those angles. It's not enough to just figure out what x is, you might think that you're just done, but you actually have to go back and actually solve for those angles. Alright?

So then how do we do this? Well, 17x minus 10 minus 20 is going to be 17 times 10 minus 20. And if you work this out, that's 170 minus 20. So in other words, that's going to be 150 degrees. Alright? That's what this angle ends up being. So I'm going to sort of highlight this in blue like this. So this blue angle over here is going to be 150 degrees. Alright. So now you can go ahead and plug in x equals 10 into this equation, or what you can do is you can just say, well, if this is 180 degrees, then that means that this has to be 30 degrees over here. Right? Because 150 and 30 have to add up to 180 degrees. Alright? So you really actually have to plug this back into just one of the equations, and then you can solve for the other one. However, if you were to plug this back into 50 - 2x, you would see that this actually gets you 30. Alright. So let me know if you have any questions, but this is the correct answer. These two angles over here are 150 and 30. Let me know if you have any questions. Thanks for watching.

Here’s what students ask on this topic:

What are complementary and supplementary angles?

Complementary angles are two angles whose measures add up to 90 degrees. For example, if one angle is 30 degrees, its complement is 60 degrees because 30 + 60 = 90. Supplementary angles are two angles whose measures add up to 180 degrees. For instance, if one angle is 110 degrees, its supplement is 70 degrees because 110 + 70 = 180. These concepts are essential in geometry, especially when dealing with triangles and other polygons.

How do you find the complement of an angle?

To find the complement of an angle, subtract the given angle from 90 degrees. For example, if you have an angle of 25 degrees, the complement is calculated as 90 - 25 = 65 degrees. This means the complement of a 25-degree angle is 65 degrees. This method works for any angle less than 90 degrees.

How do you find the supplement of an angle?

To find the supplement of an angle, subtract the given angle from 180 degrees. For example, if you have an angle of 45 degrees, the supplement is calculated as 180 - 45 = 135 degrees. This means the supplement of a 45-degree angle is 135 degrees. This method works for any angle less than 180 degrees.

What is the relationship between the angles in a right triangle?

In a right triangle, one of the angles is always 90 degrees. The other two angles are complementary, meaning they add up to 90 degrees. For example, if one of the non-right angles is represented as x and the other as x + 10, you can set up the equation x + (x + 10) = 90 to find the values of the angles. Solving this equation, you get 2x + 10 = 90, which simplifies to x = 40. Therefore, the angles are 40 degrees and 50 degrees, adding up to 90 degrees.

Can an angle have both a complement and a supplement?

Yes, an angle can have both a complement and a supplement as long as it is less than 90 degrees. For example, a 30-degree angle has a complement of 60 degrees (90 - 30) and a supplement of 150 degrees (180 - 30). However, if an angle is 90 degrees or more, it cannot have a complement because complementary angles must add up to 90 degrees.