Common Core Geometry Unit 3 Lesson 2 The Axioms of Equality

Many of the axioms of equality, you've actually seen in prior courses. Remember, an axiom is something that we assume to be true just based on common experience, but we can't actually prove it to be true. Um, that seems a little dubious. So let's jump right into it. All right, so let's take a look at one of the first and most important of all axioms of equality or axioms of congruence. Let's take a look at exercise number one. In each diagram, make a statement about the whole measurement about how the whole measurement relates to individual parts of the segment. All right, so in other words, here we've got a segment. A T all right? And what we're going to look at first is how you can kind of take this segment and write something down in terms of how it can be broken apart. All right? So let me do this for the segment and then maybe in the second part of the problem you can do it for the angle. What we're getting at here is given that we know that these three points are collinear AFT. 

We can say that AF plus FT is equal to at now literally what we're saying here is when we take the length of AF and we add to it the length of FT, we get the length of a T it would also also be correct to say something like this. Segment AF plus segment FT is congruent to segment 80. All right? Either way, they're saying slightly different things, but in essence they're the same. When I take these two parts and I add them together, I get the hole. All right? In letter B, we're talking about doing exactly the same thing, except with angles. So pause the video now and take a few minutes to do that. All right, let's go through it. Again, the idea is that we can add angles together to get an entire angle. In this case, what I might say, if I'm going with equality, is I might say that the measure of angle N R, K, which is this angle. Plus the measure of angle K R S, which is this angle equals the measure of angle N R, S this angle. Now, if that seems obvious to you, excellent. The whole idea of axioms is there are things that are supposed to be obvious, all right? But still, obvious things need names. And specifically, what we're invoking in both of these is one of the most important things in all of physical space. And that's the fact that the whole is the sum of its parts. 

The whole is the sum of its parts. In other words, AF and FT are the parts that make up the whole, which is a T likewise over here, angle NRK and angle KRS are the parts that make up this whole angle. And we're going to need that little phrase. The whole is the sum of its parts a ton, especially in this lesson, but in future lessons as well. So let's do some more in terms of the axioms of equality. Okay. Now, all of these that we look at that remain are ones that you saw actually in algebra one when you were justifying how to solve equations. So the next one we're going to look at is what's called the addition axiom or the addition postulate. Sometimes you'll just call it audition. All right? And it's a very simple idea. If equals are added to equals the sums are equal. And again, this is just so blatantly obvious, right? If I've got something like 5 equals 5 and three equals three, then when I add them together, 8 equals 8. Of course they do. Right? But in geometry, in geometry, we might use it in some other way than just kind of these number statements. Let me kind of get rid of this. And let's take a look at exercise number two. 

The addition axiom or just addition. Given CE equals FE. And EB equals ED. Prove that CB is equal to FD. And again, make sure you understand what all these statements mean. CE equals FE just means that these two line segments here and here are the same length. That EB is equal to ED. These two line segments are the same length. Prove that CB is equal to FD. And again, if I replaced all of these things with congruent symbols and then had little line segments above them all, basically the entire line of reasoning would work out the same. And we'll be using that in the course. So let array first make a test case to make sure you understand what the givens illustrate. So in other words, right? CE is the length of CE. So let's say that that was a length of three. And FE was a length of three as well. That's our first given. All right, likewise, E B is equal to ED, maybe this would be a length of 6. And maybe this is a length of 6. Right? Then, of course, what that means would be CB would just be equal to 9. And of course, FD would be equal to 9 as well. Honestly, what we're invoking there, then, is that the whole is the sum of its parts. Obvious, it's an axiom. 

So let's go ahead and do the proof. Now, this is the first time we're really kind of seeing a proof in this form. But we're going to see in a lot. I'm going to erase those numbers. Okay? And what do I mean by this form? A lot of times when we do proof and you're going to do a ton of proof and geometry, we have one column where we lay out statements. And we have another column where we lay out the reasons that we know the statements are true. Now sometimes we will have those statements fill band and you'll only have to fill in the reasons. Other times, you'll have to do it all on your own. Both sides of this table, okay? Sometimes we'll also do proof in what's called a paragraph form, but not for today. All right? So let's take a look at this particular proof. Our first statement was that CE is equal to FE and EB is equal to ED. Well, what reason would we give for why we can make that statement? That's a very simple reason. 

Because it was given. All right? So any time that we're given something in a problem, eventually we will put that in a proof. All right? It may not go as the first line. We don't want to do a given dump, which means we throw them all in in the first line and move on. But in this case, there we have it. Now, the second part of this proof says that CE plus EB is equal to FE plus ED. All right? So in other words, what I've done is I've taken CE and I've added it to EB and I've taken FE and I've added it to ED. Why can I do that? Well, that is the addition property of equality. In other words, if I know that these two things are equal, and these two things are equal, then when I add them together, I still have things that are equal. Finally, I write down CB is equal to FD. Now why do I know that CB is equal to FD? Well, what have I done? I've really replaced CE plus EB with CB. And FE plus ED with FD, why can I do that? I can do that. Because the whole is the sum of its parts. Very, very simple. And the line of reasoning is easy enough. 

Overall, I wanted to prove that these two longer line segments were the same length. And what I knew was that these pieces of the longer line segment were equal. And these pieces of the longer line segment were equal. So I used the property of equality, the addition property of equality to add the two sections together. And then I used the fact that the hole is the sum of its parts to replace those sums of parts with the hole. Therefore, giving me what I wanted, which was that this whole segment is the same length as that whole segment. All right? Let me step back for a second so that you can write down any of these reasons that you need to. And then we'll move on. All right, let's do it. Okay. Now, if there's an addition property of equality, there's a subtraction property, or axiom of equality. All right? And it says exactly what you would think. If equals are subtracted from equals the differences are equal. Now we oftentimes use this in situations very similar to the last one, except situations where perhaps we know that two wholes are equal to each other. And we need to take parts away to establish that other parts are the same size. 

I don't know how easy that was to follow. Let's take a look at this exercise. Okay. So exercise number three, given the measure of angle BAD is equal to the measure of angle CAE. Prove that the measure of angle BAC is equal to the measure of angle DAE. Letter a, why would this have to be true? Come up with some example measures of angles to show why this would work. Well, let's just make sure we really understand the given again. This is where right from the beginning of the course I mentioned that it's a very critical. To be able to look at these three letter names for angles and be able to know what angles they're talking about. So we know from the outset that angle BAD is congruent to angle CAE. Let's say that they're both, let's say that the measure of angle BAD equals the measure of angle CAE and I don't know. Let's say that they're both 40°. All right? Then what we're asked is we're asked to prove, let me change it to a different color. 

We're asked to prove that the measure of angle BAC is equal to the measure of angle DAE. Well, do you notice something? Of the larger angles BAD and CAE angle CAD, one more color. Angle CAD, this thing. Is part of them both. So if CAD let's say, let's say the measure of angle CAD was 25°, let's say that, well, that's a measure of angle CAD was 25°. Then the measure of angle BAC, which would have to be the measure of angle, DAE, well, they would both have to be 15°, right? Because they overall angle was 40, and if I took out that 25, they'd both be 15. And this really gives you some sense for how we're even going to think about the proof here. All right? Very similar to before. Let's take a look at it. All right. Step one, measure of BAD is measure of CAE. Why do we know that BAD and CAE are equal? Because we were given that. We just know that's true. All right, now the second part is I'm saying BAC plus CAD is equal to CAD plus DAE. So what am I really doing? I'm really, let me go back to blue. I'm really replacing the measure of BAD with this. And I'm really replacing the measure of CAE with this. 

Now why can I do that? Why can I take this larger angle and break it down into these two smaller angles and each case? Well, that's because the hole is the sum of its parts. Right? I can take a hole and I can break it into the sum of its parts. That's just what I'm doing. And notice now. That CAD is part of both of those sums, CAD is part of both of those sums. Now, I state something that's very, very curious then. In line three, so let's pause for a minute right now, not you pause, but me pause. Measure of angle CAD is equal to the measure of angle CAD. There's going to be many times in this course where we're going to have to state that something is equal or congruent to itself. And that could be the most obvious axiom of them all. Something is equal to itself, or congruent to itself. That's what I call the axiom. It's got a more complicated name. It's called the reflexive property of equality or the reflexive property. It kind of makes sense. It's like the reflection in a mirror. The reflexive property of equality or of congruence just says that any geometric object is congruent to itself. Or the measurement of any geometric object is equal to the measurement of the same geometric object, which is really what we're saying right here. 

Now why do I even state this line? Why do I state CAD is equal to CAD? Well, I state it so that I can subtract this equation from this one. And when I subtract the two, I get what I want. The measure of BAC is equal to the measure of DAE. And again, what I'm doing is I'm just subtracting CAD from both sides. This is like solving an equation, right? It's like subtracting 5 from both sides when you're solving some equation to get a final answer, a value for X and again, look at the overall structure of this problem, right? We've got two larger angles equal to each other. We can break those larger angles into their component parts. One of those component parts is common to both of the angles. So I state what that component part is, angle CAD. I subtract it from both sides, and I'm left with these two smaller angles. BAC and DAE being equal in measure. Or being congruent. All right. Let me step out of the way. Bring this up. So you can take a look at it, copy down anything you need to. Are you ready? Let's move on. All right, the substitution axiom. All right, the substitution axiom. 

Quantity is equal to each other, or to the same quantity, may be substituted for each other. Again, this is this is so, so very basic, right? I know. That 6 divided by two is equal to three. And therefore, any time I see something like the number three, I could replace it by 6 divided by two, or with the fraction 6 halves. Equals can be substituted for equals. This is actually the general principle that you used when you did things like solving systems of equations by substitution. You would simply substitute one equation into the other because equals can be substituted for equals. Now one of the great things is we can now use this along with other properties to prove something that we saw in the last unit using transformations, we're now going to prove that vertical angles have the same measure. So let's take a look at that. 

Exercise number four, given lines M and N intersect to form angles one, two, and three prove that the measure of angle one is equal to the measure of angle three. So what we're really doing now is we're proving that these two vertical angles have the same measurement. And it's all blank in here now, right? It's all blank. So how do we even go about doing this? I mean, we have so little in terms of our givens. We just have two lines intersecting. Where do we go with this? Well, think for a moment about what you can say based on this diagram and on what you know. All right, now make sure be careful. You already know that vertical angles are equal, but that's what we're trying to prove here. Using the axioms of equality. So one thing I know basically the only thing I know because I've got straight lines going on here is that this must be a 180° angle. And this must be a 180° angle. So I'll start there. Measure of angle one plus measure of angle two is a 180° and the measure of angle two plus the measure of angle three is a 180°. It's as simple as that. 

Now, what would I write down for that? Look, the plain fact is, what you write down in the reason column has to be reasonable. All right? It has to follow from logic. It has to explain why you know something. All right? So why do I know this? Well, I know it. Because when two angles sum to form a straight angle, they sum to be a 190°. In other words, right? This is a straight line, so it must be a straight angle. This is a straight line, so it must be a straight angle. So when two angles sum to form a straight angle, they sum to be a 180°. All right? So what can I now do with that? Ah, well, this is where the substitution property of the equality comes in. Because both of these two sums are equal to 180. They must be equal to each other. All right? The idea of the substitution property of equality when two things are equal to a common third thing then they're equal to each other. 

The measure of angle one plus the measure of angle two equals the measure of angle two plus the measure of angle three, that is the substitution property of equality. All right? Ah. But what am I trying to prove? I'm trying to prove that the measure of angle one is equal to the measure of angle three. Do you notice what's coming up on both sides? And the measure of angle two. So really, I can get to my final statement if I can get rid of the measure of angle two from both sides. How do I do that? I have to subtract it away. Now, before I can subtract it away, it'd be great if I could just be like, yeah, one is equal to three by the subtraction property of equality. One of the silly things is I literally have to write down the measure of angle two is equal to the measure of angle two. Right? Do you remember what the reason was for that? The great reveal is coming. Ah, the reason for that. Is the reflexive property of equality. 

Any time we say that two things that the same thing, sorry, is equal to itself for congruent to itself, it's the reflexive property of equality or congruence. And again, it's completely okay if you just say the reflexive property. That's completely all right. All right? And now what do I want to do? I want to subtract that from both sides. When I do subtract it from both sides, I get the measure of angle one is equal to the measure of angle three. And literally, I say the subtraction property of equality. Again, you can just say subtraction property, and most teachers will simply accept subtraction. All right? Because you're saying what you did. I subtracted, right? And I was left with the result. Now isn't that kind of cool? Back in the last unit, we proved that when two lines cross each other. That these two vertical angles have to be equal. And we did that by doing a 180° rotation about their intersection point, and then we talked about rigid body motions and things like that. They're not going there. 

This time it's purely algebraic. Well, I know that these two angles have to add up to be one 80. I know that these two angles have to be add up to be one 80, and therefore the sum of those two must be equal to the sum of those two. Because angle two is common to both of those two sums, I can subtract it away, and that leaves me with the measure of angle one equal to the measure of angle three. There again, proving that vertical angles are equal or congruent. All right. Today's lesson will make a lot more, it will make a lot more sense. Once we do longer proofs in which we need to either add geometric objects together to get a larger object. Or take a larger object and subtract something from it to get a smaller geometric object. All right? Right now it's just kind of like, all right, well, sure, when I add two things together, I get a larger object. That's the hole as the sum of its parts. Or when I take a large object and I break it down into its component parts, same line of reasoning. But then my ability to add things to parts or subtract things away from them are just given the most obvious names. The addition property and the subtraction property. 

Finally, we also saw the substitution property of equality, which is when we know two things are equal to each other, we can substitute them for one another. All right. We're going to get a lot more practice on these axioms in future lessons. For now, I want 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.