 to our lecture series, Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Angela Missildine. In lecture 14, we're going to start talking about the congruence axioms. So Hilbert, as he axiomized Euclidean geometry, put his many axioms into one of several groups, two of which we've already talked about. The incidence axioms we've talked about previously and the between the axioms we also talked about already in this lecture series. So the third family of axioms that we're going to discuss are the so-called congruence axioms. Now, the incidence and between this axioms both came into groups of four. The congruence axioms are actually, there's six axioms total. And so actually our discussion is going to break it up over two lectures. In this lecture, number 14, we're going to talk about the first three congruence axioms that have to do with the notion of congruence of line segments. And then the other three congruence axioms we'll discuss in the next lecture, which those deal with the, well, three axioms of congruence related to the idea of congruence of angles. And so that gives us a natural dichotomy between congruence of segments and congruence of angles. The two are related to each other, of course, but we will define those a little bit later. So with Hilbert's axioms of congruence, we introduced two new undefined terms. The first one, well, actually both of the words are the same, but they do have a slightly different meaning. So in this video, we'll give us one of the undefined terms. The other one will come next time, of course. So we introduce an undefined term called congruent. And this is a relationship between line segments. So we say that a line segment AB and a line segment CD can be congruent. And we would denote that using the following notation right there. AB is congruent to CD. So just like any undefined terms, without any axioms, it means nothing. It's open to complete interpretation. All that we have here is that congruent is a binary relation on line segments. Okay. So we'll introduce the first three axioms of congruence, which say something about what it means for two segments to be congruent to each other. Like always, when we talk about these axioms, these Hilbert's axioms for including geometry, we're going to attach some names to them so that it'll be easier for us to describe in this lecture series. But this naming system by no means is universal. All right. So with that, let's start with the first three axioms of congruence. So Hilbert's axiom C1, which we will refer to as segment translation reads as the following. Given two distinct points A and B and a ray emanating from some other point A prime, there exists a unique point B prime that's on the ray R such that AB is congruent to A prime, B prime as segments. In particular, on the ray AB, that's a unique point. That is B prime is equal to B. All right. So before we talk about the other two axioms, I want to try to unpack this one a little bit. So we start off with two distinct points. And with those points, we're thinking of it, we have a line segment, right? So we have these points A and B, they're distinct. Any two pair of points determine a line. So there is a line, there is a line between the two. And even when I say where it's between, we're assuming as we define these axioms, notions of betweenness, notions of incidence, they're very much in play right here. And so then suppose we have some ray, some ray R, which doesn't necessarily have anything to do with the line segment. It could, they could be co-linear, but they don't necessarily have to be. We have some ray R, but we do know the vertex of the ray is the point A prime, like so. So this is our setup. We have a segment AB, we have a ray whose vertex is given as A prime. So then what the what the segment translation acts and guarantees is that on this ray, there exists a unique point. This uniqueness is important here. We'll use this all the time. There's a unique point B prime that lives on the ray, R here, such that the line segment AB is congruent to the line segment A prime B prime. And we'll often use this little naming convention that if we have a correspondence, a congruence of segments or angles or whatever, we'll often label the same vertices, the same points using the same Roman letter, but we might add something like a little mark like a prime or whatever to make that correspondence more apparent, of course, to the viewer here. All right. So the segment translation then gives us that if we have a segment and we have a ray, we can then find a segment on that ray that's congruent to the other one. And so this is where it gets the name segment translation. The idea is, oh, it's as if we could move this segment onto the ray. And so in essence, we can move congruent segments anywhere we want throughout the geometry. But like I said, this point B is unique. There isn't more than one point that can accomplish the same thing. If we had some other points, B double prime, B triple prime, or maybe B prime is between them or something, right? This is the only, this is the only segment on R that'll be congruent to AB. If you pick some other point, not going to happen. So there's a very important uniqueness statement put inside of this. In particular, of course, if you were to take the ray that emanates from A in the direction of B, right? So if you're looking at this ray right here, then the uniqueness statement tells us that if you translate the segment AB onto the ray AB, this ray right here, then of course, this unique point that we were guaranteed must have been the point we started off with. So this uniqueness statement is built into the cake here. We'll actually use this a lot. In particular, the uniqueness statement of segment translation is very, very, very important. Okay. So let's now talk about the second congruence axiom of Hilbert. This is axiom C2, for which we will call this transitivity of segment congruence, which as the name seems to suggest, we do have, since congruence is a binary relation, we assume that this binary relation is transitive. So if the segment AB is congruent to A prime B prime, and if the segment A prime B prime is congruent to the segment A double prime B double prime, then we can conclude that the segment AB is congruent to A double prime B double prime. So with our usual intuitive notion of congruence, this should be satisfied, right? In fact, we might think that congruence is actually an equivalence relation. We'll say some more about this in just a moment. But the second axiom of congruence assumes that congruence of segments is a transitive relation. We will see that it's an equivalence relation. We'll prove that's reflexive and symmetric in just a moment. Okay. But before doing that, let's also talk about axiom C3, which we will call segment addition, which tells us if we have three points AB and C, these are all three points on a line. And so such that there's a between this relationship, B is between A and C. So I want you to be aware here that we're trying to make a connection between the congruence notion of segments with betweenness, because these are some undefined terms we put into play earlier, right? I mentioned as we talked about segment translation, how there's these notions of incidence playing around there, right? That you have these line segments, you have these rays, these are all parts of lines. There's of course a unique line determined by A and B, there's unique line determined by A prime and B prime. So segment translation does make a connection between congruence of segments to lines. A segment addition is primarily going to allow us to connect congruence of segments to the notion of betweenness that we studied previously with order to geometries. So we have three points AB and C, such that B is between A and C. And then we have three other points, A prime, B prime, C prime, that live on a different line, L prime, right? Such that B prime is between A prime and C prime. So notice what's of course happening here, right? This naming convention I said before, the point A coincides with A prime, B coincides with B prime, C coincides with C prime. This is exactly what I mentioned earlier, L coincides with L prime. There's this natural correspondence between the betweenness there, all right? So those are just the assumptions. And so then segment addition tells us that if A prime, excuse me, if AB is congruent to A prime, B prime as segments, and BC is congruent to B prime, C prime as segments, then the segment AC is congruent to the segment A prime, C prime. So let's illustrate exactly what we have in mind here. So we have some line L for which there are three points on this line, A, B, and C, for which B is between A and C as illustrated. We have some other line that we can see right here. This is called L prime. And it contains three points as well. Now of course with our naming convention, we'll have A prime, B prime, and C prime. So what segment addition then tells us is that if the segment AB is congruent to the segment A prime, B prime, and if the segment BC is congruent to B prime, C prime, you'll see that we'll have to put these little tick marks on a segment to suggest it's congruent to another one. This is a very common geometric convention. So if this segment is congruent to that one and this segment is congruent to that one, notice that the union of these two segments is AC and the union of these two segments is A prime, C prime. Well, if the two parts are congruent to each other, then it turns out that these ones are going to be congruent to each other as well. And so that's why we call it segment addition. So the idea is if we have a congruence between the pieces, then when we put them together, when we take their union, we will retain congruence. And this is a foreshadowing to the idea of addition that we will talk about later on, because after all, if these segments have a length to them, because in Euclidean Geometry, for example, we say that two line segments are congruent if the distance between their endpoints are the same. So we can use the metric, the distance function of the geometry defined segment congruence. The way that we've talked about congruence here, we don't need distance, because congruence is just an undefined term, we can interpret it in many different ways. Distance is one possible interpretation, that's how you do it in Euclidean Geometry. But foreshadowing that, right? If these two distances were the same, and if these two distances were the same, then the sum of the distances must be the same as well. And so that's if you think about it in terms of distance, that's why we call it segment addition. So now summarizing what we have here, like I said earlier, axioms C4, C5, and C6 all relate to congruence of angles, and we're going to list them next time. But despite not being explicitly listing the axioms, all of the axioms of congruence, we are ready to define what a congruence geometry is. We say that in ordered geometry, remember what an ordered geometry is. An ordered geometry, it's an incidence geometry that satisfies Hilbert's four axioms of betweenness. An incidence geometry of course is a geometry that satisfies Hilbert's four axioms of incidence. So an ordered geometry has the four incidence axioms, it satisfies the four between this axioms, and then we say that a congruence geometry will be an ordered geometry that satisfies the six congruence axioms, the three we just discussed, plus the three that we haven't yet been explicit about as well. So congruence geometry means that we have the four incidence axioms between the axioms and the six congruence axioms. We call this a congruence geometry. Now, I want to relate this to the very, well, relatable topic of what's called a neutral geometry, which we will define later on. Neutral geometry is very similar to congruence geometry. A lot of people who study geometry, when they study neutral geometry, a lot of what they're studying is actually congruence geometry. The main difference is that the neutral geometry will also have a notion of continuity for which we haven't introduced yet. We'll do that, of course, later on. And then, so a lot of what people call neutral geometry is really just congruence geometry, because if you don't use continuity, it's a congruence geometry, not just a neutral geometry. So we're going to delay our introduction of the continuity axiom until we really need it so that we can develop this theory of a congruence geometry. All right. So I mentioned earlier when we talked about the transitivity axiom for segment congruence that it turns out that congruence of segments is, in fact, an equivalence relation. Well, why is that? Well, we have transitivity. We're taking that as an axiom. But it turns out that the symmetric property and the reflexive property can be proven using the axioms we already have in play here. So our proposition we're going to prove here is that, in fact, the congruence of segments is a symmetric relationship. So imagine we have four points, a, b, c, d, in a congruence geometry. If the segment a, b is congruent to c, d, then that means that the segment c, d is congruent to a, b. So we want to prove that it's symmetric. And so this is going to follow by using segment translation and transitivity of congruence. So let's first start off with segment translation. I guess I should first start off with our segments here. So we have our segment a, b. We have our segment c, d. So maybe something like this. Let me label it here, of course. So we have a, b right here. We have b, d right here. And then we're going to have c, d right here. And we're going to have, there you go. And so we know by assumption that a, b is congruent to c, d. That's an important observation in this consideration here. All right. So by segment translation, there exists a unique point. We could call it b prime that's on the ray a, b, such that c, d is congruent to a, b. So what we're going to do is think about this line segment. It could be extended, of course, the extension axiom from between us. We can extend this into some ray, right? Think of the whole ray. Well, by segment translation, there's a unique point, call it b prime, that lives on this, that lives on this ray, such that the segment a, b prime is then congruent to c, d. I actually should write the other way around because we haven't yet proven that this thing is symmetric. So that c, d is congruent to a prime, a, b prime, like so. Okay. And so by, so that's what we get. We get this point b prime with that property. But by transitivity of congruence, right, this is what we assumed. We know that a, b is congruent to c, d. We know that c, d is congruent to a, b prime. So then we have to conclude that a, b is congruent to a, b prime. Okay. But a, b and a, b prime live on the same ray. They emanate from the same point. And so by the uniqueness statement that's part of segment translation, it's got to be that a prime and b prime are actually one in the same point. And therefore we get that c prime is actually congruent to a, b prime. Thus proving, proving this statement right here that a, c, d is congruent to a, b. So we can see very quickly here using transitivity and translation, of course, that a, that this relationship of congruence is a symmetric relationship between them. The last property we need to be in equivalence relation, so we need to be reflexive. But that actually goes back to the very last sentence in segment translation where it said when you in particular on the ray a, b, on the ray a, b, if you were to translate the segment a, b back onto itself, yeah, you're going to get some point a, b prime, but you were told very explicitly that this point is in fact, this new point b prime was actually the point you already had here. So you translate a segment onto itself, you get back the original segment. And since a, b is congruent to a, b prime, and since b prime equals b, it's reflexive. So the reflexive property is built into a segment translation, it's not a completely explicit, but it's there. Transitivity we take as a maximum. And then thirdly, some symmetry we can improve using this translation action, which has reflexivity plus a little bit extra and then transitivity. And so we do get that congruence of segments is a equivalence relation.