Common Core Geometry Unit 1 Lesson 7 Additional Geometric Terminology
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Learning the Common Core Geometry Unit 1 Lesson 7 Additional Geometric Terminology by eMathInstruction
Hello and welcome to common core geometry by E math instruction. My name is Kirk Weiler, and today we're going to be doing unit one lesson number 7 creatively entitled. Additional geometric terminology. Geometry is full of terminology. And let me be honest with you. The way that math teachers basically look at things, in general, is when we give you a piece of terminology. When we say, hey, this is just the name that we're giving to something. Then the idea is that you'll learn it. You know, and you'll learn it either by reading it over and over again by memorizing it by making flashcards or whatever, but you'll learn it. And that's important because as the course goes on, we won't have a ton of ton of terminology, but probably more than you had in algebra.
So it's important to commit these things to memory, many of them have names that make sense, at least in my own they make sense. And so that they'll help you remember them. But we're going to work with them today and some problem solving context to make sure that you've got some of this terminology down. So let's get right into it. We're going to start with a huge slide where we talk about the different terminology. We're just going to go through each one sort of little by little and I'll point some things out to you. All right? But let me go through the first one. All right, midpoint. The point on the line segment that divides it into two segments of equal length. So this is simple enough in the diagram I've got up here. I've got line segment a, B, if it was let's say 20 inches long, then that point M it's midpoint would be the one and only point that would divide it into ten inches and ten inches. That's simple enough. All right?
Another piece of terminology that's very similar to midpoint. Segment bisector align ray or segment doesn't matter which that passes through the midpoint of another segment. So in other words, in this particular diagram, we've got line T and it's passing through midpoint M and therefore we would say that T bisects bisects line segment a, B think about this for a moment. You can't bisect bisect means to cut in half by meaning two sect, meaning, I guess, got in half. The point is, you can't cut a line or a ray in half, but you can cut a line segment in half. And that's exactly what's going on here. With this line, it's acting as a knife. Let's take a look at some more terminology. If we can cut a line segment in half, we can also cut an angle in half. And that's what we get with an angle bisector. A line segment align segment or ray that divides an angle into two congruent angles. So in this particular picture, we've got angle BAC. An angle BAC is being bisected by ray AD. It's cutting it into these two angles, each of which are 35°.
All right, simple enough. Our final piece of terminology for right now is the word perpendicular. Perpendicular. Any two lines segments or rays that intersect to form strictly 90° or right angles. We say, for example, that AB is perpendicular to CD. Now again, this little piece of symbolism for perpendicular is amazingly important. Sometimes we won't use the word, will only use the symbol. And you'll have to know what it means. So a, B, and CD or perpendicular, if the two lines AB and CD form right angles. Now, again, this is kind of important because it's unique to a right angle. If you've got one right angle, when two lines intersect, they're all right angles. And again, it should make some sense just from simple math. If this is 90°, and of course, the entire rotation here would be a 180 because this is a line, then a 180 -90 would make this 90 and likewise this 90, and that one 90. All of them are 90° angles, though we'll tend to only mark one because quite frankly, if we marked all four 90° angles, this is the way it would look. Right? That's not a great drawing.
But the point is, if I do that, then it just looks like I've got two lines crossing and covering them up with a square. Anyway, the point isn't that. The point is when you hear the term perpendicular, or you see the symbol for perpendicular up here, then you have to know what that implies is something about 90° or right angles. Okay. Now, what we're going to be doing in the rest of this lesson. Is simply using these four pieces of terminology to solve a variety of different problems. And the point of the problems aren't the problems themselves, but to read through the problems, interpret what you know, also we'll examine what we don't know in certain problems. And then use those to solve for certain quantities. All right, let's get right into it. Let's see what the first problem asks us to do. I'll read it for you. Exercise number one in the following exercise you will work with the definition of a midpoint. Letter a in the following diagram, use measurements to determine if a, H is the midpoint of FG. All right. Well, this is probably the only time that you'll need your ruler today, but use your ruler to figure out whether H is the midpoint of line segment FG.
Take a moment and pause the video now and use whatever time you need. All right, let's go through it. Well, what I hope is that you found that, in fact, wasn't the midpoint. If I bring my ruler up here and I measure FH in millimeters, what I'd find is that FH is just about 35 millimeters. On the other hand, if I measure GH, I would find that it is more like 48 millimeters. So the plane fact is the answer here is no. And for the reason, is we would just simply say, because FH doesn't equal GH. The midpoint, the midpoint is going to be that one and only point probably somewhere right around here, on the line segment that would make those two measurements equal. Let's take a look at letter B letter B says using measurements, see I already lied to you, you're going to have to use your ruler again here. Using measurement locate point C on AB at its midpoint. What equation can be written about AC and BC. All right. Well, take that ruler of yours and do your best in terms of locating the midpoint of that line segment. Take a moment. Are you ready? Let's find it. Okay. So let me take my ruler up.
Let's figure out how long AB is. All right. Lost a little bit there. It looks like AB. AB is I'm going to say 76 millimeters. All right. Now, this is probably pretty obvious to you. But what that means is that wherever the midpoint is, it's going to have to make it so that the two smaller line segments are exactly half of that 76 millimeters. So if I take 76 and I divide it by two, I'll find that each one of those line segments would have to be 38 millimeters. I can then take my ruler, come up, find the 38 marking. Bring my ruler down. And locate the point. And then maybe clean it up a bit. That's going to be the midpoint 38 millimeters. 38 millimeters. That's point C and then the question asks me to write down what equation I could state about a C and B C and equation is a statement about equality. And again, we can't really make a statement about equality with geometric objects, but we can make a statement about equality with geometric measurements.
In other words, I can write down that AC is equal to BC. Now, it's not likely not likely that your geometry teacher would take off for writing something like AC is equal to BC with those line segments there. But technically speaking, this is a very, very strange statement. Again, because literally when we have AC with that little bar above it, we're not talking about 38 millimeters. We're talking about the physical line segment itself. And so it's a little bit weird to say that a line segment is equal to a line segment, as opposed to this, which says the length of AC is equal to the length of BC. It's a little bit of a technical point, but you do want to make sure you understand how your teacher would like to see it done. All right, let's move on and do another problem that involves more terminology. All right, exercise number two. This one's kind of a fun one. Let me read it over for you. Exercise number two. In the following diagram, C is the midpoint of AB. D is the midpoint of BC and E is the midpoint of CD. If ED is equal to 5, then determine the length of AB. All right.
Well, if you've got a good sense for what midpoints are at this point, you can probably do this problem without any problem. Or whatever. That seemed a little redundant. Anyway, what I'd like you to do is take a little bit of time and see if you can figure out what the length of AB is. Are you ready to go through it? Let's do it. Remember to always pause the video for as long as you need to. When you have that little bit of time to work. Don't think that that small pause I give you is the amount of time I think it'll take. For some of you it may be. But not likely. Okay, what do we know? We know ED is equal to 5. All right? But we know that E is the midpoint of CD. And of course that means that CE must also be 5. Okay? But we know that D is the midpoint of BC. And since this is quite clearly now ten, this then must also be ten due to the fact that D is the midpoint of CB. That, of course, makes CB in total 20 units long. Which would make a C also 20 units long given that C is the midpoint of AB. That then leads us to finally be able to say that AB has a length of 40 units. Yep. And we don't have any units like inches or centimeters in this problem. It's a unitless problem. But notice how we use the midpoint definition repeatedly to keep doubling these lengths to get finally the one that we want. All right.
Let's go ahead and move on to the next problem. See what other kind of terminology we can work with. All right. Let's read through this problem. Exercise number three. In the diagram below, segments AB and DE intersect at point C are either segment bisectors. Evaluate whether each of the following statements is true or false and give a reason based on measurement that supports your answers. So remember that a segment bisector has to be another segment a line or a ray. It's a straight, straight object. And a segment bisector has to cut another segment in half, okay? So the real question in this picture is, is DE cutting a B in half is AB cutting DE in half. So we want to rank both of these statements in a and B as true or false. DE bisects AB AB bisects DE justify. Take that ruler and see what you can find. Should we go through it? Let's do it. Now what I'm going to do is I'm going to do all the measurements I need right away. Okay? So I'm going to bring this down.
I think I'm going to just immediately figure out how long a C is and I find that AC is 35 millimeters. I like doing these measurements in millimeters because to me, it's sort of about the most accurate I can get on BC is also 35 millimeters. Great. Oh, that was interesting. All right. CE is 25 or 26. I'm going to call it 25 millimeters. And finally, because I love reading things upside down. I'm going to find that CD looks like oops. Looks like it's 45 millimeters. Or something like that. All right. Let's step back. Let's talk about true and false. Letter a DE bisects AB. Again, in plain English, DE cuts AB in half. Well, was AB cut in half, where DE sliced it. You bet. It was cut into a 35 and a 35. So I'm going to say that this is true. And justify, that's because AC equals BC. Now our other true false question. AB bisects DE. Well, does AB if AB's the knife does it slice DE in half? And the answer is no. 45 and 25 aren't the same. So we would call that a false statement and we would say the following. D.C. doesn't equal EC. Keep in mind that if you say CD and CE, that's perfectly okay.
It really doesn't matter the order of the two letters. At least in this case. All right, segment bisector is pretty easy. Let's do a little bit more work. Angle bisector is great. I think I lied a lot right up front. I said, oh, you're just going to do a little bit of measurement. And here we are. We're using our rulers and our protractors. Again and again. Let's take a look at this problem. All right? Exercise number four. In the following exercise you will work with the concept of angle bisectors. Good to know. Letter a does ray AG bisect angle BAD. Use measurement to justify your answer. All right, this is going to be a little bit dicey, but what we want to know is does this array does ray AG bisect cut in half angle BAD. Go ahead and try to use your protractor to justify whether or not it does, in fact, or it is in fact an angle bisector. All right, are you ready to find out? It's close. But I believe the answer is that it isn't.
Let's take a look. All right, I'm going to measure these two angles. Angle BAG, bag, so the measure of angle BAG looks to be about 24°. Somewhere between 23 and 24, but I'm going to go with 24. And now. Let's measure the angle of DAG. And the measure of angle DAG. Ah, it's also 24°. So the answer is yes. I thought it was close, but no. But the person who invented this problem says yes. Right? Angle bisectors. And there's only one for any given angle. We'll talk about that more in a future lesson. Any given angle has one in exactly one angle bisector. That kind of makes sense. If I have a 48° angle, there's only one ray I'm going to put in there to form two 24° angles. Okay? Let's see if you can make that in the next problem. Letter B says use your protractor to draw the angle bisector of angle G shown below. Mark all relevant measures. All right, so using your protractor, see if you can bisect that angle.
You'll need a ruler as well to draw a straight line segment. Go ahead, spend a little time on that. All right, are you ready? First things first, I simply need to know the overall measure of the angle. Measure of angle G so what is it? Well, it looks like on my compass. Stretch that up a little bit. Hitting right at that 110° angle. So the measure of angle G is 110°. Of course, what the angle bisector should do is it should take half of that. And half of 110°, or 110 divided by two. Is 55°. So if I want to bisect the angle now, I need to figure out where 55° is right about there. I'm going to move this out of the way. And now I think I'll grab my line tool. Oh, maybe I'll even grab one that'll make it look like a ray. Sorry about that. Give me a sec. There it is. I add it. And there's my angle bisector. Right? Now, you're not likely to have to do this on any kind of a standardized test.
This is something more for maybe like, let's say a middle school activity. But it really is important to understand simply what an angle bisector is, right? It doesn't matter how long this is, right? It doesn't matter whether it's a ray or whether or not it's a segment. The plain fact is this now is a 55° angle, and this is a 55° angle. All right. Let's move on and see if we can work with a little more terminology. Now this, this is starting to be a nice diagram. Let's see what we're working with in this problem. In the following diagram, I think I need to change back to pen. Let me do that really quick. In the following diagram, CE is perpendicular to AB. Important, I don't know why I have the spinning blue circle. And CF is perpendicular to CD.
Now, both of those are marked on the diagram, but it can be a little bit confusing with those right angle markings. Let's keep going. If we are given that AC is 5 and measure of angle BCD is 25°, then they ask us about letter a, find the measure of angle DCE and the measure of angle ECF. And letter B, do we know that AB is ten? Wire, why not? Now this is the last problem of the lesson, and I think you're very, very well prepared to take care of it right now. So what I'd like you to do is again take some time and work on this problem. Yeah, ready? We're going to do both parts of it. Okay? First, we have to be able to properly identify and find things on this diagram. So when we're told this piece of information, the measure of angle BCD is 25, we have to be able to find it on the diagram. BCD is this angle. 25°. Then, when they go on and say, find the measure of angle, DCE, maybe make it in a different color. We'll go for red.
So when we're finding the measure of angle DCE, DCE, we have to know which angle we're looking for, specifically this one. Now, this isn't going to be too hard because of that piece of information they gave us about CE being perpendicular to AB. Because we know that we know that angle, the measure of angle B, is 90°. And due to that, we can now say the measure of angle DCE must be 90° minus that 25°, right? Which is 65°. I'm going to write that right on my diagram. 65°. And again, we know that because this overall angle has to be 90. It's got to be 90. Now they asked us to find another angle. Again, let me change colors just so that we can probably shouldn't use green. We'll go with black. So come out here.
We're looking for the measure of angle ECF. That's going to be this angle. But the great thing about the measure of angle ECF is, again, just like before, we know that this angle has to be 90. Because CF is perpendicular to CD, and therefore the measure of angle ECF will then be 90 minus that 65° angle and strangely enough, wow, something strange is going on here. We're back to that 25° angle. I've got all sorts of weird markings. Don't want you to think that's a decimal point. Let me make that actually hopefully look like an equal sign. There we go. Technology. It's awesome. When it works. Okay, so the last part of this problem, though, asks us, do we know that AB is ten? Do we know that? We were told in the problem very, very clearly that AC was equal to 5, do we know now that AB is ten? What did you write down? Well, I hope I hope that you said no.
Now remember, I'm not saying no, AB doesn't equal ten. I'm not saying that. I'm saying no, we don't know that it is. Right? And in geometry, it's very important to know what we know and as well know what we don't know, right? And what we don't know in this problem is whether C is the midpoint of AB. We've been given no information on that. So I'm going to say we don't know. We don't know if C. Is the midpoint and often you'll see me abbreviate midpoint like that. And it's okay for you to abbreviate midpoint in the same way. We don't know if C is the midpoint of AB. If we did, then we could certainly conclude that AB was ten. But we weren't told that in this problem, there was nothing in the setup that told us that so we shouldn't assume it. All right? Very, very important. Let me kind of just step back for a moment. Let you write down anything you might need to. I went through that pretty quickly. All right, let's wrap this one up.
Now, as I mentioned at the outset of the lesson, in geometry, terminology is going to become important. Later on, we're going to be doing something that's going to be one of the most challenging things that you'll ever probably do in math, which is geometric proof. And in geometric proof, terminology is going to come up a lot. It's kind of like the bare key to open up the door to the proof. You can't even, you can't even walk inside of the great hall of proof. Now I don't even know what I'm talking about. But you can't even get anywhere. If you don't know what the terminology means. It's like not knowing how to use a hammer and trying to construct a house. It's not going to happen. All right? So it's important that you learn the terminology in this lesson and any other lesson.
So when we introduced you to the technical definition of what a circle was, you got to know what that is, right? Midpoint, bisector, angle bisector, perpendicular, all of those things. All right. We'll be using them a lot. So you get lots of practice. But work on trying to memorize them. 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.