Common Core Geometry Unit 1 Lesson 3 Types of Angles

Now there's a lot of text up on the screen right now. And I'm going to turn around and I'm going to read it for you just to make sure that we've got all the different types of angles down and we can use them in future work. So let's go through them. All right, an acute angle, an acute angle is an angle that has a measure between zero and 90°. Acute literally comes from the Latin word for sharpening. Acute angles are going to look sharp or they're going to look pointy. Now in contrast, we have obtuse angles, and obtuse angle has a measure between 90° and a 180°. These are opposites of cute angles, and they're going to look blunt or a bit dull. Very, very important. Are the right angles. A right angle is one that has a measure of exactly 90°. It represents exactly one quarter of a full rotation. It is perhaps the most important angle of all in mathematics, because it indicates two directions that are completely independent of one another. 

Think of that in terms of sort of like north and east, right? They're at 90° with respect to each other. A straight angle, a straight angle is an angle that has a measure of a 180°. A straight angle represents exactly one half of a full rotation. It would look indistinguishable from a straight line. And finally, probably the least important of all angles are reflex angles. A reflex angle has a measure between 180° and 360°. Reflex angles are important in mathematics, but won't be important in our initial studies. All of these types of angles are important to understand and important to know their names. With the exception of maybe the reflex angle. We're not going to use that much in algebra two. You'll use reflex angles quite a bit. Angles, they become greater than a 180°. In fact, in common core algebra two, you'll even start talking about angles with negative measurements. But let's not go there right now. In fact, let's get right into the exercises. Okay. 

Exercise number one. Let me read it off for you. Exercise number one, each type of angle is shown below. Judging by side alone, classify each as acute obtuse right, straight or reflex. All right. Now, most of the time, angles won't have these sort of arc rotations on them. All right? But I really wanted to make sure that you understand what the rotation was, especially for the reflex angle. Take a moment, look back at those definitions and classify each one of these 5 angles as one of those types. Take a moment now. All right, let's go through them. This first angle, to me, is clearly larger than a 90° angle, but not larger than a 180. So this is an obtuse angle. All right. On the other hand, the second angle is so large that it goes past a 180° turn. And therefore, it is a reflex angle. All right, this very pointy sharp one over here is smaller than a 90° angle. And therefore, it is acute, all right? Our angle down here in the left hand corner. Well, this basically looks like a straight line. And therefore it's what we call a straight angle. All right. And finally, this last one. And this is the toughest one of all without a protractor, but this is our right angle. Okay. 

Now, the vast majority of times we're not going to want to judge angles just based on site alone. We're going to want to have protractors or some kind of other information that allows us to determine whether they're obtuse, reflex, straight, right, or acute. Again, the reflex not so, so important. But the other four terms very important. All right, let me step out of the way for a moment, write down anything you need to, then we'll move on to the next slide. Okay, let's do it. Exercise number two, hopefully you have your protractors handy, if not, get them out now. There's going to be a lot of measurement in this course. Let me read through exercise two for you. Find the measure of each of the following angles. Unless otherwise stated, we will always assume we are measuring the non reflex angle. All right? So always assuming we're measuring the non reflex angle. Then, classify it as acute obtuse or right. Name the angle using the three letter combination. Each one of these could be named with just a single letter, but I'd like you to use the three letter combination. 

Now, we got some work measuring angles in the last lesson. So take your protractor out now, measure each of these three angles and then classify them as either right acute or obtuse. Take a moment. All right, let's go through the measurement. We've got our protractor. Let's bring it down and put the center of the protractor right at point V all right, taking a look, I can make mine. Oops, I can make mine a little bit bigger, which will help out some. And it looks like the measure of this angle is 108° again, maybe shrink it down a little bit. Yeah, a 108°. Let me use the naming system. All right? Measure of angle, T, V, M, is 108°. If you said it was the measure of angle and the T that would be okay as well. Quite frankly, it would also be okay to call it the measure of angle V, it's just in this particular problem I asked you to use the three letter combination to get used to it. Okay? Let's take a look at the next angle. All right, and this one I've got to do a little bit of work with my protractor. That was fun. Get the center right there. Ah, there we go. A nice 90° angle. That was probably pretty obvious by just looking at the picture. But again, it's important to use our protractor in this case. 

Let me get rid of that thing. And let's call it the measure of angle Y O U and how big were you? 90°. Okay? Obviously, that means that this is a right angle. I forgot to do one part of the problem, which was to classify this. That's a right angle. And our first one was an obtuse angle. I may elated one of the primary directives when it comes to answering a math problem, which is do the whole thing, not just the part that you think you're supposed to do. Anyway, let's measure that last angle. Again, this one's a little bit tricky, because we're going to have to rotate the protractor. Let me rotate it up like this. Bring it so it's centers in point a, to rotate it a little more. And it looks like its measure is just a bit more than 50°. I'll call it 51°. Yep, that looks about right. I'm just trying to get rid of that. Oh, that looks interesting. I'll make it a little bit smaller. And we'll say the measure of angle CAT. Is 51°. And that makes it acute. It's a very, very cute angle. Not funny. Math is typically not very funny. I'm typically not very funny. All right, anyway, we've got the three angles. Let me pause, get out of the way, so you can take a look compared to your measurements in your classifications. All right, let's move on to the next exercise. 

Exercise number three, got a little algebra involved in here. You can tell that immediately when X shows up in the problem, let me read it for you. Given WRX solve for the value of X and find the numerical values of measure of angle VR X and measure of angle V. Let's first talk about that. What does that mean? Given WRX. Well, what it means is that points W are and X are co linear, right? They all fall on the same line segment. That's what that given means. If you ever see more than two points with a line segment or a line symbol itself over it, what it means is that all those points are collinear. Why is that important in this problem? Well, it's important because we know that angle measures add, which means this angle, plus the measure of that angle must be equal to a 180° because this itself is a straight angle. Straight line, straight angle, essentially the same thing. Now, because I know that they add to become a 180°, I can solve for the value of X why don't you take a moment and go ahead and do that. All right, let's go through the algebra. Simple enough here. We can say that the angle whose measure is three X plus 22. 

When I add that to two X -12, those two angles together must sum to be a straight angle or a 180°. That means I'm going to get 5 X watch out 22 -12 is going to be ten. And that's going to be a 180. Subtract ten from both sides. And we're going to get 5 X is equal to 170. That doesn't really look like an equal to. That looks like an equal to. Now I'll divide both sides by 5. And X will be three, 15, 30, four. Boy, this is where I'm going to calculate it would be nice, wouldn't it? Let's say 15, yeah. 20 and four. So X is 34. Now that doesn't mean that we're done with the problem. In fact, in order to finish out the problem, we still have to figure out what the numerical values of VR X and WR VR, but that's as simple as taking the 34 and substituting them both into these two expressions. In fact, let's figure out what the measure of angle, let's see VR X's. The measure of angle VR X is going to be this measure. Is two X -12. Which will be two times 34 -12. Or 68 -12. Which is 56°. Right? Now, we can certainly figure out the measure of wrv by substituting 34 into there. Let's do that. 

The measure of angle W RV is three times 34. Plus 22. Which is going to be a hundred and two. Yep. Plus 22. So I have to think about my multiplication in my head. And that's going to be a 124°. That's all we've got those two angles. And we're done with the problem. I will say, though, that there's a very nice check in this problem, right? I've got those two angles. And what I can now see is if I add them together, will I get a straight angle? Will I get that 180° angle? Well, let's take a look at that really quick. Let's do it over here on the side of our board, a hundred whoops. That's not right. Not a 126. It's a 124. Let's add to that, the 56. Zero carry the one. And yeah, that's a 180°. A 180° to make sure I'm not blocking the answer. All right, so what we're doing here, we probably could have done even in the last lesson. But just as a reminder, we saw something new, which was this given, the idea of that three letters with the bar over it, meaning that the points were collinear. And then using the fact that angles add to set up an algebraic equation that allows us to solve for X and find the measure of the individual angles. Take a moment now and copy down any of this that you need to before we go on to the next problem. Okay, let's move on. Exercise number four, okay. 

Let me read this one to you. It says given that angle N MQ is right. Answer the following questions. Let array if measure of angle PMQ is 40°, find the measure of. And letter B totally separate if the measure of angle is twice the measure of angle PMQ, find the measure of angle. Okay. Now, a couple of different things before I have you jump in and work on this problem. First things first, we're told that this angle and MQ is a right angle. And notice the symbol for right angle. I know you've probably seen that in other courses, you know, in middle school and even earlier grades. But we always use a little kind of square symbol to show a right angle. If we've been told it. Now I didn't necessarily have to put that into the diagram for you. But it's a good way to introduce it. Okay? So why don't you take a moment, pause the video and see if you can work out the answer to letter a we'll take a look at that, then we'll move on to B. Okay, let's work through it. First, let me get rid of this. No need to have it. What do we really know? In this problem, what we know is that this angle has a numerical value of 90°. We were then told that the measure of angle PMQ, which is this angle. Is 42°, and we're asked for the measure of angle. Which is this angle. Well, because angles add, we can say that the measure of angle must be that overall angle, 90°, minus the 42°, which gives us 48°. All right? We don't really have to set up an algebraic equation for that. 

We know that the whole is 90°, that almost looked like the hole, that the hole was 90°. We know part of it is that 42°, we can subtract it away to find that other part 48°. And of course, it's an easy then check to say 48 plus 42 is 90. Now, importantly, in this problem, letter B has nothing to do with letter a. We don't want to take those results and carry them over. So pause the video for a minute if you need to to write down the work from letter a, because then I'm going to erase at least the work that's over here on the diagram. Okay. Let's take a look at B B involves a little bit more algebra than a did. I mean, really, a just involved a little bit of subtraction. All right, let me read letter B again. If is twice p.m. Q, find the measure of angle. All right? Now, you've seen this kind of wording before. If this is twice this, and then we have to set up an equation. Okay? So let's make sure that we get it. Is twice p.m. Q I'm going to do a little let's statement. Yeah, I know. Let's statements. Let measure of angle P M and I'll let that equal X and then because is twice that we have really no choice. It's going to be two X, right? We know is twice p.m. Q so now, because we know that this is X and this is two X and this is 90. We can set up an exceptionally easy equation to solve. 

We can simply say X plus two X is equal to 90. Of course that gives us three X is equal to 90. Divide both sides by three and X is equal to 30. Now, just to make sure it asks us to find the measure of angle, is actually the one that's two X, not the one that's X so now we do what's called back substituting. Whoops, not quite. Let's try that again. The measure of angle and MP then will be two times 30 or 60 degrees. All right, simple enough. But we had to interpret that statement about this angle being twice that angle, that allowed us to solve an algebraic equation to find the measure of both angles. Take a moment, write down anything you need to, and then we'll move on. Okay, let's do it. So what did we learn today? Well, what we reviewed were the different types of angles, the acute angle, the obtuse angle, the right angle, the straight angle, and even the reflex angle. All of these are going to become important. Probably the single most important piece of terminology is the right angle. Because we'll tell you that they're right angles, and then you have to know what that means. 

Immediately know, that means that we have a 90° angle. Sometimes it'll be important to know whether angles are acute or obtuse, less than 90, or greater than 90. And sometimes we'll need to know that angles add up to form straight angles to form a 180° rotations. As I mentioned, numerous times, the reflex angle probably is not going to be as important as the other ones. But you'll see that in later courses. For now, let me just thank you for joining me for another common core geometry lesson from E math instruction. My name is Kirk Weiler, and until next time, keep thinking and keep solving problems.