Common Core Geometry Unit 1 Lesson 2 Lines, Rays, and Angles
Math
Learning Common Core Geometry Unit 1 Lesson 2 Lines, Rays, and Angles by eMathInstruction
Hello and welcome to common core geometry by E math instruction. My name is Kirk Weiler, and today we'll be doing unit one lesson number two, lines raise and angles. Now in these introductory lessons on common core geometry, we're going to be trying to build up the tools that you'll need to do more challenging work later on. Last lesson, we looked at points and distances and segments. In this lesson, we'll look at related topics of lines raise and angles. So let's get right into it. All right.
Let's first talk about what a line is. Okay? Let me read off its definition. A line is an infinite set of points that extends forever in two directions. Since lines are uniquely determined by any two points on the line, we symbolize them by a double arrow over two points. So in this one example, what I've got is a, B, those are two points with that arrow that goes both directions. Sometimes they're referred to by lower case letters such as L N and N and we'll see both of them quite often. Remember, if that double arrow is not over the two letters, then technically, we're talking about the distance of a line segment. The length of the line segment between point a and B when we've got the double arrow over it, that means we're talking about an infinitely long line. In fact, let's take a look at this in a little geo gebra widget. All right? If I simply, within this, plot a point, okay. And plot another point, those two points are going to uniquely determine a line. In fact, let me draw that line in. I'm going to choose my line.
Choose point a and choose point B and notice now what happens, right? We have this line that extends infinitely in two directions. Based on just two points. And I can move those two points around or I can draw maybe another completely random line. And that's kind of nice too. But if I move that second point, B, or even that third point C, right? Then what I'm getting are these lines, these collections have an infinite number of points that go on forever in two directions. Now, that's a line, right? And again, a line is symbolized. By those double letters, double capital letters with the double ended arrow on both ends. On the other hand, let's talk about what array is. Rays are very, very similar to lines, right? Array is an infinite set of points that extends forever in one direction just one direction from a starting point. Like lines rays are symbolized by using two points, but only a single arrow, right? And we kind of obscure this a little bit here, but there's that point with the single letter.
Let me get rid of this just for a second. All right, there it is. Let's take a look at array down here. Here we go. Again, in order to draw a ray, what I need are two points, right? I need a starting point. Like point a and then I need some point that kind of shows the direction of the ray. So we'll take point B but now if I ask my program to draw a ray, right? And I started at a and go on to B, then what we notice is that the collection of points is only going in one direction. Only going in that direction. All right? And if I move then, point B around, right? B is really just sort of setting the direction of the ray. A is telling us where it starts, that second point then tells us the direction of the ray. And again, important distinction. There's really four things now that could be a little bit confusing, right? You've got a line segment with a single bar, no arrows. You've got a line, same thing, two letters, but then a double error, and then you've got a ray. Right? Same thing, two letters, but a single ended arrow. And finally, to make matters even more confusing, you've also got just the single two letters single two letters.
You've got the two letters that have nothing above them whatsoever. And keep in mind, when you have that, when you've got this situation, again, what this means is that the length of the line segment between point a and point B literally a length. All right? Okay. Wow, we haven't even done an exercise yet. Let's jump into exercise number one. Exercise number one asks us, why is it possible to talk about the measure very important word we're going to see the measure a lot or length of line segment AB, but it's not possible to talk about the measure of line AB or ray AB. So think about this for a minute. Why can we talk about the measure of this line segment, but not of this line nor of that ray? Take a moment. Well, it's probably pretty obvious. I even have a little ruler here. But if I took my ruler up, rotated it. I could easily figure out then what the length of line segment AB is. So I could say, oh, a, B, no symbol over the top of it.
In this case, would be about 73 millimeters. On the other hand, the problem with line AB gets into, where do I put the zero mark? Right? I can't just put it at a because the line extends forever in two directions. Now, at least with a ray, I could say, well, maybe I'll put the zero mark at point a, but then my issue is, where does the ray end? And the answer is it doesn't end anywhere, right? We think of space as extending infinitely. Now don't get me wrong, I understand that you've got the idea of a finite universe and all of that. But in geometry, the idea is that this ray goes on forever and the same thing with this line. You can probably summarize all of this by just simply saying, raise. And lines. Have infinite. Infinite lengths. Meaning that those lengths are so large, we can not even assign a number to them. Okay? Let's move on to the second exercise. Maybe. Maybe not. All right, exercise number two. In the diagram below, all points shown are collinear. So EF and a all lie on the same line, which you might say to yourself, well, that's obvious. You've got a line drawn there, and you've got three points on it. But the key is with geometry, we want to make sure that we don't just look at a picture and take something for granted that we don't really know.
For all we know, there could be a slight bend at point F here, right? It could be just slightly bent so that they're not collinear. So I need to tell you that they are. Now, exercise letter a says give three different ways to name this line. Three different ways to name this line. Let me give you one way to name it. All right? And there's actually many, many different ways. But remember, lines are dictated by any two points that lie on the line. So one thing I could say is I could say that this is line. EF, right? Both E and F lie on the line, and then I put that double letter over it. That double ended arrow over it. I keep messing that one up. Or at least messing up what I'm saying or how I'm referring to it. What I'd like you to do is take a moment and give me two other ways you can name this line. And try to use at least one of them that doesn't involve the two capital letters. Okay. Well, again, we can pick any two points that lie on this line.
For instance, you could say, ah. Well, we have line, FA, right? And it doesn't matter what order you put these points in. So it could also be line. AF. We could even have line EA. I know I'm listing way more than three. And finally, and this is a little bit different, but you'll see it many times in this course. Sometimes you'll see a little lower case letter as we do right over here. You'll see a little lower case letter to name the line. That's going to be done more often when we're not really concerned about points that lie on the line, but we're concerned about the line overall. So it would be completely acceptable to also call this line M literally line N let's take a look at letter B letter B says give two ways to name the ray that starts at point a very important, it starts at point a and extends in the direction of point F all right. Why don't you give this a shot on your own? Okay, let's give two ways to name this ray. Well, if it starts at point a, then there is no getting around it. You've got to have the a come first. But then you could say that it's ray AF. Or since these three points are collinear, you could also say that it's ray AE. All right? Both of those are the same ray. And that's very, very important to understand. I may have named them differently, but the plain fact is ray AF and ray AE, both start at point a and then they go infinitely in the direction of F okay? Take a moment now if you need to and write down anything you have to on the screen. And then we'll move on to the next exercise.
All right. Let's go ahead. Let's talk about angles. Angles are going to be very, very important in geometry. Okay? So first, let's understand the definition of an angle. An angle. The geometric object created by two rays with a common starting point. Angles are symbolized by a single point, for instance, angle a, or a triple of points, such as angle, BAC, where the second letter is the vertex of the angle. All right? So for instance, here, we've got this angle, we could name it as angle a, we could name it angle CAB. We could name it angle BAC. We couldn't name it angle ABC. All right? We would never want to call this thing angle ABC. Simply because that second letter in our three letter naming system has to be what's known as the vertex of the angle, where the two rays come together very, very critical. All right? Now let's talk about the measure of an angle. Okay? The measure of an angle is the amount of rotation needed to rotate one of the rays about their shared point, so that it lies on top of the other ray. The measure of an angle is symbolized by putting a little M in front of the angle symbol. And angle a, and angle BAC. Now, once again, get into measuring angles a little bit more, because you've been doing this for so many years. So, so many years. And yet, just make sure you really get it.
The idea of measuring an angle is pretty simple, but it's based on the idea of a circle. Here, we've got a protractor, right? And I'm going to make that protractor into the full circle that it truly is. And the idea is very, very simple. I'm going to make it much bigger in just a moment. There we go. Right? The idea of a protractor or of an angle measurement is very simple. If we put the center of the circle at the vertex of the angle, okay? And then we divide the circle up into 360, evenly divided spaces. Okay? Then all we're doing is measuring that many spaces. So for instance, this one happens to be a 40° angle. All right? It takes 40° or 40 of these marks to rotate up the one ray so that it lies on top of the other ring. And it doesn't matter how big the angle is, in other words, how long the rays are. Because we can adjust the size of the circle and we will always always always divide that circle up into 360 evenly divided markings. Which we call degrees. We'll learn about other ways to measure angles later on in this course. But we're going to stick with degrees mostly. All right? So let's get into a little bit more angle work. All right, exercise number three. Consider the angle shown below.
Letter a, give three acceptable names for this angle and letter B, use your protractor to find the measure of this angle. All right, now I know we haven't done a lot of angle work yet in this course. But you've done angle work before another courses. So what I'd like you to do is pause the video now, try to give three different names for this angle, and also try to measure it. And then we'll go through all the answers. All right, let's go through it. First, let's talk about the naming. Now, it may seem that this is somewhat trivial. Ah, how do you name an angle? But you would be surprised at how much this presents a difficulty for students. Time and time again, especially when we give you the name of an angle. Students will get confused by it. So the first step is simply knowing how to name them. Let's take a look. All right. The easiest way to name this angle is simply by calling it angle D and you might even ask yourself, why don't I always do that? We're going to take a look why a little bit later on. But that's one of the ways of naming it angle D we could also name it angle F D E and we could name it angle E oops. D F the key in this three letter naming system again is that the D lies as the second letter. It's at the vertex.
So you can kind of see the two rays ray ED and ray DF. All right? Now let's take a look at the measurement of this angle. Now this can always be a little bit tricky for students and again, just like the measurements of length that we saw in the first lesson. If it's not perfect, that's okay. I'm probably not going to always get it perfect. But what I want to do is I want to put the vertex of the angle at the center of my protractor. And then I want to make sure I'm reading the right scale. But what I do is I look up on here and it appears that it is at, let me maybe make this even a bit bigger so that you can really see it. It appears that it's at 38°. Again, watch out. You don't want to think it's at a 142°, but hopefully everybody remembers that 90° is a right angle, and we're nowhere near a right angle yet. For me, I've got this fancy schmancy way of being able to rotate this thing up to here. You probably don't have that though on your protractor. Regardless, though, what we see is that the measurement of angle D. Is 38°.
Don't forget to put that degree marker there. It's important. Whether your teacher takes off for it or not on a test, that's up to them. But for me, that degree marker is pretty important. All right, let me get rid of our compass, shrink it down a bit. Throw it out of here. I love the fact that it balances off the side of my board. All right, let me get out of the way for a second. Pause the video if you need to to write down any of this and then we'll move on to the next slide. Okay, let's move on. Exercise number four. Answer the following questions based on the diagram shown below. Note that the race symbols do not need to be attached to the line segments to still discuss angles. Array extends forever, whether an arrow is drawn or not. Let me explain what I mean by that, right? I don't have little arrows at the end of raid DG or DF or DE. Because the plain fact is, the ray is there, whether it's drawn or not, right? There is a set of points that extends from D through G forever and likewise on these other two. I'm not going to always have those arrow marks there. Sometimes I will. But let's take a look at the rest of this particular problem.
Letter a says find the measure of angle FD, letter B says find the measure of angle GDF, right? And then we'll talk about the other ones. But first, take a moment, use your protractor to find the measure of those two angles. All right, let's go through. Bring it back into the screen, the measure of angle F, D, E all right, I bring my protractor down. Okay, maybe make it a little. I don't want to tilt it. Still don't want to tilt it. There it is. Right? So the measure of angle appears to be 36°. 36°. Now, the measure of the second angle, GDF, that's a little bit trickier because one of the rays isn't horizontal. So just like you had to do, I'm going to have to rotate my protractor in order to measure it. All right? And it's actually probably a little bit harder for me than it is for you. It doesn't really matter which one I use in the zero mark. But again, I keep the center of my protractor, the center of my circle right there at the vertex. And now I can see that it's past a 90° angle. In fact, it's at an angle of. Now I can maybe shrink this thing down. I can't believe I just rotated it again. There we go. This is something you're not going to be able to do, but I can and it's a nice feature. I'm now at a 108° angle. 108°. And let's throw that away.
All right, so hopefully you got your 36° angle and your 108° angle. Okay? Let's take a look though at letter C and D in this problem. Letter C asks, why can neither of these angles measured in parts a and B be labeled as angle D? All right? Finally, it says find the measure of angle GDE. How does its measure compared to those seen in a and B? Well, first, let's talk about letter C because it is remarkably important. Why can neither angle FD E nor angle GDF be simply called angle D we all like to save time. It's nice when we can refer to an angle by a single letter as opposed to three letters, but why can't we do that? Think about that for a moment. All right, let's talk about it. The plain fact is the reason that you can't call either of those angles angle D is because you wouldn't know what angle we were referring to. Unfortunately, D point D itself is the vertex. Of more than one angle. And that's the key. And because it's the vertex of more than one angle. If we called anything angle D, we wouldn't know which angle we were referring to. In fact, it's not just the vertex of two angles, both of these two, it's actually the vertex of three angles. It's the vertex of angle FD. Of angle GDF and of the final one that we're going to be measuring today, which is angle G, D, E take a moment right now, and go ahead and measure that angle if you haven't done it already.
All right, let's get the measure of angle GDE. Bring my protractor back down. Excellent. GDE is going to be all the way over here. And it's going to be at one 40 somewhat around one 43, one 44. I'm going to call it 144°. All right. Let's bring this back down and get out of it out of there. So what's this question about, how does it compare to these two angles? So we saw that fde was 36. We saw that GDF was a 108. How does that one 44 compare? Well, if it's not obvious, if we add the one O 8 to the 36, we get a 144. And of course, that should make all the sense in the world. Because what we're doing is we're measuring rotation and the plain fact is if we're going to rotate 36° to get this ray up to here, rotate a 108° to get this ray over to here. Then to get this rate of rotate all the way over to here, we have a 144°. Angle measures add just as segment measures also added in the last lesson. And that's going to be important. And we're going to use it a lot. It's probably fairly obvious to most of you. But we always want to make sure that you got that. In fact, the last problem we're going to do is going to emphasize that. Let me get out of the way just for a moment though, in case you need to pause the video and write any of this down.
All right, let's move on. And do the last exercise. Let's take a look. Exercise number 5 says the diagram shown is not drawn to scale. This is a pretty important piece of terminology. You should more or less assume that diagrams are not drawn to scale. Unless told otherwise or unless unless you're told that you should be physically measuring something on the diagram. Obviously, if the diagram is not drawn to scale and we're asking you to measure something on the diagram that would seem a little unfair. All right, but we're telling you this diagram is not drawn to scale, which means you shouldn't take out your protractor and try to answer the question using that. If the measure of angle CBA is X plus 7, CBA is X plus 7. The measure of angle DBC is 48, and the measure of angle DBA is four X -20, then find the numerical value of the measure of angle DBA. All right, now one thing that's rather nice about this problem is all those different angle measurements have been drawn and marked on the diagram. If they hadn't, we'd have to really kind of fish through what all of these things mean.
All right? The question is, how do we find the measure of angle DBA? Now obviously this is going to involve a little bit of algebra in some of you will already understand how to do it. So I'd like to pause for a minute and see who can work out this problem on their own. All right, let's work through it. Now, this really emphasizes what we were looking in at in the last problem, which is the idea that angle measurements will add. In other words, the measurement, let me just do it this way and do nice formal kind of work here. The measure of angle CBA, all right, that's this angle. Plus the measure of angle DBC must be equal to the measure of angle DBA. All right. What this is going to allow us to do now, it's going to allow us to set up an equation where we can solve for the variable X the measure of angle CBA, well, that's X plus 7. The measure of angle DBC is 48. And the measure of angle DBA is four X -20. In a certain sense, this now ceases to be a geometry problem and becomes more or less purely algebraic. So let's go ahead and finish solving this equation pretty quickly. What we're going to have is we're going to have X plus 55.
That almost looks like a plus, but not really. So let me get rid of it so you don't think it says X 455. Where's my plus? There it is. X plus 55 is four X -20. All right, we can subtract an X from both sides. And since you're in geometry not now and not algebra one, I think we'll do two steps at once. So we're going to get 75 is equal to 5 X divide both sides by 5. And X is going to be 15. Now, wait a second. I had four X minus X is 5 X that is, that's a little bit silly. Let me get rid of that. I'm pretty sure that I've four oranges, and I subtract an orange. I don't have 5 oranges, although that would be kind of awesome. I have three. So X isn't 15, X is 25. All right, two important lessons here. Important lesson number one, everybody makes mistakes. Important lesson number two, though, is make sure you finish the problem. It's very satisfying when we solve an equation we get the value of X but in this problem, they didn't say what is the value of X, they wanted the numerical value of the measure of DBA. Well, that's easy enough because the measure of DBA. Is going to be four X -20. All right. Since we now know what the value of X is, we can substitute it in. X is 25, so we'll have four times 25 -20. Four times 25 is a hundred. And a hundred -20 all works out nicely. The measure of angle DBA is 80°.
All right. And that's it. Let me step aside, take a look at the work, pause the video if you need to, and then we'll wrap it up. Okay, let's move on to the last slide. In the lesson today, we looked at the ideas of lines, rays, and how rays come together in order to form angles. Angles are going to become increasingly important in this course. We're going to see them. They're going to be just as important as line segments. We saw that angles will add together in ways just like the lengths of segments will add together. We also saw how to name angles and specifically and very importantly how to name them using three letters so we know exactly which angle we're talking about. Make sure even though this is a relatively simple lesson that you internalize this that you practice with naming angles with understanding what the measure of angles represent. Before moving on to more challenging work in geometry. For now, though, I'd like to thank you for joining me for another common core geometry lesson by E math instruction. My name is Kirk Weiler, and until next time, keep thinking and keep solving problems.