Common Core Geometry Unit 1 Lesson 4 (2024)

Learning the Common Core Geometry Unit 1 Lesson 4 - Complements and Supplements by EmathInstructions

Hello and welcome to another common core geometry lesson by E math instruction. My name is Kirk weiler, and today we're going to be doing unit one lesson number four complements and supplements. Now, in the last lesson, we looked extensively at different types of angles. Today, we're going to look at two very important angle relationships called complementary and supplementary angles. Let's get right into it. All right. Let's take a look at the definition of complements or complementary angles. Complementary angles. Two angles are called complementary, or complements of one another. If their measures add up to 90°.

When placed adjacent to each other, so that they share a common ray. They would form a right angle. So in this particular illustration, I've got two angles, one that's red, one that's blue. And if we measure them, what we would find out is that their measures added up to 90°. But what's kind of cool about complementary angles is that literally when you join them together by making them share a ray, the overall angle that's formed will appear to be a right angle. It in fact will be a right angle. Let's take a look at this. I'm going to just move the red angle over here. All right? And then I'm going to rotate it. Or I'm going to try to rotate it. Oh, that's the blue angle. Wow. Okay, red angle. Work with me here. Work with me. Almost a little bit farther. A little bit too far. Why I still need to work on my rotation ability.

There it goes, though. More or less, right. I don't want to spend too much time trying to get it perfect. But what we see is if we make them have a common ray, then these two angles form that classic 90° angle, that right angle. And therefore, these two angles are called complementary angles, sometimes referred to as complements. Now, we have a very, very similar relationship, which we're going to apparently see on the next slide. With supplementary angles. Let me read over this definition. Supplementary angles, two angles are called supplementary, or supplements of one another. If their measures add up to a 180°. When placed adjacent to each other, they would form a straight angle. Again, I really want to illustrate what it means for two angles to be adjacent, all right? Adjacent essentially means sitting by one another or touching one another.

So if I take this red angle now, and I move it over to the blue angle, maybe more successfully than before, maybe not. Then what I'll see is if they share a common ray, they should form a straight angle. Let's take a look at that. All right, I'm going to try to rotate this angle a little bit beforehand. A little bit farther. Bring it over. A little bit better than before. Maybe rotate it. Just a little more. And try to make those two rays overlap. Again, didn't quite get the job done. But if these two rays were the same, then what we'd see is we'd see that that formed a straight angle. Now it's again, it's easy enough to know if two angles are complementary or supplementary if we know their angle measurements. Because we can just see if they add up to 90° or add up to a 180°. But keep in mind that the measures of angles mean something.

They mean a rotation. So if I put those two angles together, like in the case of supplementary angles, I should get this 180° rotation. Okay? Let's get into some problems that involve complementary and supplementary angles. All right. Let's take a look at exercise one. Let me read it over for you. Exercise number one, our angle a and angle D shown below complementary supplementary or neither, support your answers with measurements. All right. So what I'd like you to do is take that protractor out if you haven't already. And measure both of these angles, then determine if they're complementary, IE they add up to 90, supplementary, they add up to one 80, or neither they don't need that add up to either 90 or one 80. Go ahead and take a little bit of time.

Okay, let's get into it. First, angle a and we can use the single angle naming system right now because there is only one angle a what we see is that the measure of angle a, which is an obtuse angle, is a 120°. Again, be careful when you're using a protractor to make sure that in this case, wow. That didn't work well. In this case, you're measuring it using this part of the scale, right? This inner scale. It's obviously an obtuse angle, so we wouldn't want to say that it's 60°, but that it's one 20. Let's now move over to this acute angle. That's going to be a harder one to measure. Let's take our protractor and take a look. I'm going to have to rotate it a bit. All right. I don't know what the best way to do it will be, but maybe we'll put it up like this.

Rotate it a little bit more. And what we see now, again, a little bit tricky, but we see that it measures 60°. We're now looking at the outside measurement for the acute angle. So I can say that the measure of angle D is 60°. Now it doesn't take too much work to add the two together. 120 plus 60 gives me a 180°. And because of that, we can say that the two angles are supplements or supplementary. Okay? Either way that you want to say it. All right? Take a moment and write down the two measurements and the fact that they're supplementary if you need to. And then we'll move on to exercise number two. Okay, let's go ahead. Exercise number two. Let me go ahead and read through this for you, and then we'll get into it.

Exercise number two, find the measure of angle BAC shown below. Then, using your protractor, construct angle D, which is complementary to BAC. Let me just circle that. Angle D is complementary to angle BAC. And angle E, which is supplementary to BAC. Mark the measure of each on the diagram. All right. Well, in order to do this problem, the first thing that we have to know is how big BAC is. So take a moment to measure BAC. All right, let's take a look. Real quick. Let me bring my protractor up. Put it down, maybe stretch it a little bit. So that we can read it off a little bit better. It looks like the measure of angle BAC is 36°. Bring this back down a little bit. Throw it over there. Now, we would like to construct an angle that is complementary to it.

To make that complementary angle, what we have to do is make one that when added to the 36, we'll give us 90°. A little bit of subtraction, right? If we just do 90 -36, what we'll end up getting is 54°. All right? So we have to create an angle that is 54°. Now this is where having a ruler would be awfully helpful. I'm going to use my straight line option on here to hopefully. Or not. No. Oh, the above. I don't think the above. I know that there's a way to do it. I just don't know how. Here we go. Here's my line. Wonderful. So let me just draw out a line. For you, you would just take a ruler and draw a nice straight line. Now I'm going to put my protractor up, okay? I need to make a 54° angle.

For you, you'll probably come up here and just sort of make a little mark at the 54, okay? Then throw your compass away. Well, don't throw it away. That would be a waste. All right, and do something like that. That's my 54° angle. What did we want to call it? We wanted to call it angle D right. So let's call this angle D, maybe I'll even put a little couple arrows on the end of it. So that's my complementary angle, because those two angles that 54° angle on that 36° angle add up to 90°. Now let's do a supplementary angle. Okay, for that, we're going to have to do 180° -36. Which is going to be a 144°. And then we're going to do exactly what we did before. Hopefully. We're going to grab our line. I got it this time. So much better.

The second time. That didn't work. That didn't turn out to be a line. Anyway, we're going to use it. I don't want to, and there's my line. I'll be darned. Okay, hold on. All right. That's why. Nice. Okay. Little by little, I'll understand how to use technology a bit better. But there it is. I need a 140° angle. All right. So that 144 is going to be right about there. Market. Move this out of the way. See now if I can get my line back. And there is my supplementary angle. Let me mark it. What were we supposed to do? Ah, the measure of angle E or E is supposed to be supplementary. So come back in. And we'll mark it. As angle E okay, well, besides some troubles that I had with straight lines, that could be an issue in geometry. This seemed to work out okay. The important thing here is not so much being able to draw an angle well. But understanding the terminology, right? If I know that I've got an angle that measures 36°, and I want an angle that is complementary to it, then I know that that must add up to 90 with the 36, and that's where I get this 54° angle.

Likewise, with supplementary, I need two angles that add up to one 80, so I can easily figure out that 144° angle. All right. Take a moment. Look at the board right down to anything you need to before we move on. Okay, let's do it. All right. Now, in exercise three, we're going to move on and we're going to take a look at some angles where we clearly have a right angle showing up. So let me read the problem for you. Okay, exercise number three. In the diagram below, points E, F and I are co linear. We remember what collinear means it means that they're all falling on a straight line. It says name one pair, one pair of complementary angles, and two pairs of supplementary angles in the diagram. All right, so I want to list two angles using the three letter system.

The one letter system is not going to work in this problem. Whatsoever, because all the angles, every single angle in this problem, has letter F as its vertex. So we can't say angle F that would be confusing. But we want to come up with one pair of angles writing them down that are complementary, and one pair of angles are two pairs of angles that are supplementary. Why don't you go ahead and see if you can do that? Okay, let's go through them. First things first, one pair of complementary angles. And I'm going to just abbreviate the COM PL, complementary. Well, complementary angles have to add up to 90°, right? We know that that's a right angle due to the right angle system, and therefore this angle, HFI, or IF, it doesn't matter. Angle, H, F, I, right? This angle must be complementary to angle G F H, that would be this angle. Those two angles must add up to 90° because they're adjacent angles that are forming a right angle.

Now the supplementary pairs. I've asked you to name two pairs of angles that are supplementary. Again, just a little abbreviation there. Supplements have to combine to form straight angles, or straight lines. And there are multiple pairs of them on here. But let's take a look at two such pairs. Whoops. Let me just delete that angle. Okay. And let's stick with this angle for a moment. Angle, H, F, I, right? Angle HFI, which I initially marked on the diagram, is supplementary to this angle. Because when put together, they form that straight angle. So what angle is that? Well, it's either HFE, or E, F, H it doesn't particularly matter. I'll go with what I first said, which is angle. H almost looks like an H, HFE. All right, get rid of that and actually have an H there. And how about something that looks like the letter E? There we go. And go HFE.

Now there's another nice pair of supplementary angles on here because there's really two right angles. Two right angles will always be supplementary with each other a 90 and a 90 half to add up to be a one 80, right? So let me get rid of these guys and let's just name the two right angles. One of the right angles is this one, which is angle G, F, I, and then the other right angle is this one, which is angle. We'll go with this. E F, G okay? So both of these two pairs are supplementary because when put together, they form this straight angle. All right, let me step out of the way for a moment. You write down these pairs if you need to, and then we'll move on to the next slide. All right, let's go ahead. Let's take a look at exercise four. Now what we'd like to do is we'd like these pieces of terminology complementary supplementary complement supplement to come up in problems and be able to use the terminology.

One of the things about geometry that's a bit different than algebra is that the terminology is extremely important. You have to memorize and internalize this terminology so that when it comes up, you know how to use it. Let me read through this problem for you. Exercise number four. It is given that angle a and angle B are complementary. All right? If you have no idea what the term complementary means, then you're out of luck in this problem. Or at least you're flying blind, right? You might guess correctly on what to do, but when you see the term complementary, you have to immediately say, oh, angle a and angle B they add up to 90°, right? Anyway, so angle a and angle beer complementary. If the measure of angle B is 15 more than twice the measure of angle a, determine the measure of both angles algebraically. Didn't you miss statements like 15 more than twice the measure? I bet all summer you were hoping that you could see problems like that. Anyway, take a moment and see if you can solve this problem algebraically. And then we'll go through the work together.

All right, let's do it. What do we know? We know that the measure of angle B is 15 more than twice the measure of angle a so I'm going to do a traditional let statement. I'm going to say let the measure of angle a be equal to X and I'm going to let the measure of angle B be equal to 15 more than twice the measure of angle a while twice the measure of angle a would be two X and 15 more than it would be two X plus 15. Okay. So that's easy enough. And then to figure out what the value of X is, we have to set up and solve an equation. Simple. Because the two angles are complementary, when I add their two measures together, X plus two X plus 15, I must get 90. Watch out. Don't put down a 180. Of course, now I'm in the realm of algebra, three X plus 15 is equal to 90. I can subtract 15 from both sides. That's going to give me three X is equal to 75. If I've done my subtraction right, which is always questionable.

Divide both sides by three, and I get X is 25. Now keep in mind, again, that is only the value of X it is not the answer to the problem. The problem asks me to find the measure of both angles algebraically. Of course, I have found the measure of one of them because the measure of a is X so X must be 25°. On the other hand, the measure of angle B will be two times X plus 15, that's going to be 50 plus 15. And that's going to leave me with the measure of angle B being 75°. Not 75°, 50 plus 15 isn't 75°. What in the world? It's 65°. There we go. Now, by the way, do you know what happened there? Not how did I make a mistake? That happened because I added 50 to 15 and I got 75 or some reason. I'm not a number sky. Anyway, no, what happened and what I'm referring to is why did I catch my mistake? I caught my mistake because after writing down the 75, I then very quickly mentally said, 75 plus 25, and I said, oh, that's a hundred. Wait a second.

They're complimentary. They must add up to 90. And then that made me realize I had made a mistake here. So whenever you can, try to check your answers. I wish I could claim I made the mistake on purpose just so I could go through that little piece. But let's face the fact. I just added 50 and 15 wrong. Anyway, take a look at the work. Right now what you need to, and then we'll move on to another problem. Okay, let's do it. Exercise number 5. Let's see what this one says. It is given that the measure of angle BAE is three X plus 20 is three X plus 5. And the measure of angle QR P is 5 X -25. If BAE and QR P are supplementary, determine the value of X algebraically. Now this is very, very similar to the last problem except here we're working with two angles that are supplementary instead of complementary. So take a moment, set up your algebra and solve for the measure of both angles. Or actually, no. Here, we only have to figure out the measure or the value of X why don't you go ahead and try that. Okay, let's go through the problem. So because the two angles are supplementary.

What I know is that they're going to have to add up to a 180°. You could argue that this problem is actually simpler than the last problem, because we don't have to do any let statements. We simply take the measure of the first angle, three X plus 5. The measure of the second angle, 5 X -25, add them together and set them equal to a 180°. Here we don't particularly need that degree marker, and I'll drop it from here on out. But here I'll have 8 X when I combine the three X and the 5 X, watch out here. We have a 5 and a negative 25, which will combine to be a negative 20. And I'll get a 180. So let's just add 20 to both sides. Okay. We get 8 X is equal to 200. Divide both sides by 8. Okay. And we're going to get X is equal to 25. Apparently, I love the answer 25. Just the way it is. Simple enough on the algebra. The key here is really being able to use that piece of terminology.

That piece of terminology supplementary to know that these two angles must add up to a 180°. Take a moment now if you need to to write down the work on this problem. And then we'll move on to the next problem. All right, let's do it. All right, exercise number 6. Callista insists that if two angles are supplementary to the same angle, then they must have the same measure. Is this true? Support your answer with examples. All right, so if two angles are supplementary to the same angle, then they must have the same measure. Is that true? Well, think about it for a minute and try to come up with some examples that either show that this is true, that callista is right. Or try to come up with a counter example that shows that callista is wrong. All right. Well, callista is in fact right. And it's simple enough, right?

Let's say we take the angle 30°. It says if two angles are supplementary to the same angle, then they must have the same measure. Well, if our angle is 30°, I just can't seem to actually get that off of there. If our angle is 30°, then what we know is that any angle that supplementary to it. Any angle that supplementary to 30° must have a measure of a 150°. So all angles that are supplementary to 30 must be a 150°. So in fact, callista is right. If I knew that two angles were supplementary to an angle that was 30°, I wouldn't really even have to know that they were both one 50, I would know that they would have to be equal. And that's just one example. We could come up with any example you wanted.

For instance, if we have the angle of 60°, then any angle that's supplementary to it would have to be one 20. The same is true also of complementary angles. If I knew an angle was complementary to another one. So if I had a 20° angle, then any angle complementary to it would have to be 70°. And so complements of equal angles would also have to be equal. We'll be using that line of reasoning in geometric proof, eventually. And thankfully, quite a few lessons down the road. All right. Anyway, let's wrap the lesson up. Today, we saw two very important pieces of terminology. Supplementary angles and complementary angles, or angles that are supplements and angles that are complements. Remember complementary angles are complements, add up to 90° or a right angle. And supplementary angles add up to a 180° or a straight angle.

It'll be very important for you to know both pieces of terminology as we move forward. For now, though, I'd like to thank you for joining me for another common chord geometry lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking and keep solving problems.

Common Core Geometry Unit 1 Lesson 4 (2024)


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