 Welcome back to our lecture series Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misledine. So we finally come to it, Lecture 25, what's so significant about Lecture 25? Lecture 25 is when we're really going to open up the cage and let calculus run free in our geometric system. So this lecture, if you've been following along with our lecture series here, is going to be the most calculus-based, the most analytic lecture of the entire lecture series, where we're really going to use the full strength of our continuity axiom for neutral geometry. Beforehand, we've talked about neutral geometry, we've talked about continuity, but we've never really used it, right? We use the Archimedean principle here or there, maybe a circular continuity, but those axioms do not require the full-blown deacon axiom. It doesn't require the full-blown least upper bound property of the real numbers. For example, the rational plane, which doesn't satisfy deacon's axioms, it has the Archimedean principle. The constructible plane, that is all the points and lines you can create in a plane using a straight-edge encompass, that'll have the Archimedean principle, that'll have circular continuity, and thus it'll have elementary continuity as well, but it doesn't have the full-blown deacon axiom. What do we need it for? Well, now we've arrived upon that issue. While we were studying the topics of Zachary and Lambert quadrilaterals, we have seen that the summit and fourth angles respectively of those quadrilaterals must be less than or equal to 90 degrees. I made some comments about, well, if the fourth angles or summit angles were equal to 90 degrees, that was equivalent to the Euclidean parallel postulate, but if they were properly less than 90 degrees, they were authentically at the acute angles, then we were in hyperbolic geometry. It begs the question, how small can they get? How much smaller than 90 degrees can these angles get? It turns out that even in hyperbolic geometry, we can get the summit angles of the secure quadrilateral to get really, really close to 90 degrees, in particular if it's a very small secure quadrilateral, then the measure of the summit angles will be very, very close to 90 degrees. Therefore, if we as creatures living in a hyperbolic universe, if we don't know we're hyperbolic, but we suspect we are, we could try to measure the summit angles of a secure quadrilateral. But if the quadrilateral is small relative to the size of the universe, then it turns out their angle measures will be so close to a right angle that we might not be able to measure them. That is, our instruments might not be able to detect the difference because the difference between 90 degrees and the actual angle measure might fall within the margin of error of our instruments. But on the other extreme, if our quadrilateral is large compared to the size of the geometry, then it turns out the summit angles can get arbitrarily small. These summit angles can in fact, well, we know by the Sicari-Legioner theorem that they're bounded above by 90 degrees. But we actually want to argue in this situation that the smallest they can get is zero. And zero is never actually going to be obtained. 90 degrees is only obtained in Euclidean geometry. But in hyperbolic geometry, we can get arbitrarily close to 90 degrees and zero degrees if we want to. And it depends on basically how big the quadrilateral is. And so in this lecture, I want to make explicit what we mean by how big it is. And so we're going to talk about this notion of an angle of parallelism. So let me explain this, this definition is a lot more cumbersome than other ones we've seen before. So let's explain what's going on here. So we have some line in our geometry for which we'll place two points on that line A and B. And so let's call this line L. L, of course, is the line determined by A and B. Then we are going to take a line T, which is the unique perpendicular line to L that passes out of the point A like so. So this is our point T. And we're going to select a point D on the perpendicular line and so I should indicate that this is in fact a perpendicular line like so. So we're going to select some point D that it's on the perpendicular line T, but it's not itself A. And then we're going to pick some point C that's on the same side of the line T that B is on. And so we get some point over here, call it C. And so if we take the line determined by D and C, you might get something like this, we're going to call this line M like so. And we ask ourselves the question, are the lines M and L parallel to each other? Now, in a Euclidean sense, if I were to extend this picture on, beyond the edge of the screen right here, you would think that these lines would eventually intersect each other, but that's because I've been drawing Euclidean lines in this diagram. If we're in a hyperbolic setting, the hyperbolic lines could look very, very different. And because of their curvature, they actually might avoid each other. Because remember, parallel doesn't mean anything to do with equidistance, it doesn't have anything to do with slope, because those are notions we have not introduced into our geometric framework yet. Parallel just means do they intersect each other? And so with this framework, we can ask, do the lines L and M intersect each other? Or are they parallel? So in that direction, consider the following set of numbers M, which this is gonna be a subset of the real numbers. And what we're gonna do is we are going to gather together all of the possible angle measures ADC, where we're treating C as a variable. So A, we don't even care about B so much in this conversation. Well, I mean, we do, we do, we do, because we have to talk about the angle ABD. With C is selected to be a point interior to the angle BAD, that's what I meant to say. And so we are going to gather all the points C, all the points in the interior of the angle BAD, such that the lines AB, which is L, and the lines CD, intersect each other, all right? So such an example would be something like this, right? If this was our choice of C, this line, DC would intersect the line L. So the measure of this angle would then be something we throw inside this set, okay? What's another choice? We could take a point C over here, take the line intersects. We could take some point over here, take the line, oh, intersects off the screen, but there's that. And we then measure what are the angle measures of these angles? And we gather all of those together, all of the angle measures into this set M. Now, the first thing to mention is that the set M is, it's a set of real numbers, of course, but it's a non-empty set of real numbers. Why is that? Well, because if you choose C to equal B, which after all, I said interior earlier, but I guess actually, if I correct myself the way I said it, you just have to be in the angle, which includes the boundary as well. But even if you don't like that, that there's an easy workaround of that, I'll say in a second. If you just take C to be B and you take the line that connects them together, well, that line clearly intersects L because the point B is on there. Now again, if you want an interior point, it's not such a big deal. Just take some point between D and B, call that point C, it'll be interior to the angle and so that C will do it. So again, if you want an interior point, like I originally said, not a big deal. But in particular, there do exist lines, there do exist, excuse me, I should say there do exist points so that the line imitating that passes through D, I should say, it will intersect L. So this isn't just an artifact of my diagram right here. No matter what the geometry is, as long as it's a neutral geometry, in every neutral geometry, there will be at least some point that causes the line DC to intersect the line L and therefore if you take the measurement of that angle, whatever it is, there's something there, it'll be non-intersecting. Excuse me, it will be intersecting, it won't be parallel. And so this set is non-empty, but I want us also to consider the case, what if M was perpendicular to T itself? And we're again, I'm treating the point C as a variable in this situation. There does exist, for example, the perpendicular that erects out of the point D that is perpendicular to the line T in that situation. So what if C was such a point where you get this right angle? Well, by the alternate interior angle theorem, this line D determined by this point C right here, that version of M would in fact be parallel to T in that situation. This is what we call the guaranteed parallel line. Such a line always existed in neutral geometry here. And so because there does exist a guaranteed parallel line and the guaranteed parallel line occurs when the angle in question here, ADC, has measure equal to 90 degrees, this gives us an upper bound to the set M. So I want us to consider that for a moment. The set M is not empty, it's a set of real numbers and it has an upper bound, 90 degrees. So the least upper bound property of the real numbers comes into play and describes this set M right here. It has to have a least upper bound, a real number for which anything strictly less than that number will make the lines be intersecting. And so we call such a number D naught. And we say that this value D naught is the so-called angle of parallelism with respect to the line L and with respect to the point D. Now I want to make it very clear in this situation that when we talk about an angle of parallelism, the angle depends on the line L and it just depends on the point D in this situation. Different lines and different points will give you different angles of parallelism potentially, right? In particular, even if you keep the same line L, if you choose different points D, those can give you different angles of parallelism, okay? So in particular, if you start moving D closer and closer and closer to A, that actually will enlarge the angle of parallelism. And if we take the limit as D approaches A, that actually will then obtain the upper bound of 90 degrees. So when A equals D, you would actually get the angle of parallelism there as 90 degrees. That doesn't really say so much because if D equals A, you're no longer a point off the line. But the idea is these angles of parallelism converge towards 90 degrees when D converges towards A. But it also goes the other way around that if we take points farther and farther and farther and farther away from A, in that situation, the angle of parallelism is going to converge towards zero degrees. That is when D diverges away from A, then you could say that D is D approaches infinity. Then in that situation, the angle of parallelism will converge towards zero degrees. And this is why we can get essentially any angle measure, any acute angle we want, because it depends how far apart the point in the line are. So when it comes to this discussion about angle of parallelism, it's important that we keep in mind that the segment AD is paramount in that discussion. It doesn't just depend on the line, it depends on the segment. In fact, in a neutral geometry, all lines are congruent to each other. So really the defining factor for the angle of parallelism is going to be how far off the line L is the point D. So since 90 degrees is an upper bound for any line and point, and this is a consequence of the alternate to your angle theorem, we see that we get D is bounded above by 90 degrees. And I mentioned as we converge, as D converges towards A, we can actually, we can get closer and closer. We can get arbitrary close to that 90 degrees as well. And as, like I said before, as D diverges away from A, we can get that this thing is going to sit between 90 degrees and zero degrees, like so. So the reason we care about the angle of parallelism is because we care about how other angles interact with it. So let's draw our picture again. We have L, we have T, like so. And we pick some, we pick some ray that obtains the angle of parallelism. So I'm going to add my points to the screen here. We have A, we have B, we have D. We're going to have then here in this angle, we'll call this angle the angle of parallelism. Okay. So since D naught is the least upper bound to our set M, what that means is the following. If we take, if we take some ray emanating out of D inside the angle D AB right here, in this case, if our angle is smaller than the angle of parallelism here, so we have some point C in consideration here. So the measure of ADC is less than D naught. Then what this tells us, because it's the angle of parallelism that's eventually this ray will intersect the line and that these lines have to intersect themselves somewhere. So this line L and M are not parallel to intersect because we are less than the angle of parallelism in this situation. Now, conversely, if we choose some point C so that when you look at the ray DC, if you look at the angle in this situation, we actually are getting an angle measure for ADC that is greater than the angle of parallelism and that situation that lines L and M are gonna in fact be parallel. So if you're larger than the angle of parallelism, you're gonna be parallel. If you're smaller than the angle of parallelism, then you're gonna be intersecting. And so this angle of parallelism gives us a boundary between whether you're intersecting or whether you are parallel. But it does beg the question, what happens if your angle is equal to the angle of parallelism? Does it intersect or are they parallel? It's the boundary point. It's the least upper bound but Dedekin's action doesn't tell us which one it belongs to. We have a Dedekin cut here. So the boundary point exists but it could belong to either the parallel side or the intersecting side. So the good news is we can actually prove that if you obtain the angle of parallelism, so you have a line AB and some point D and suppose that the angle of parallelism relative to this line and point is D naught in that situation, then if you have an angle measure ADC that's equal to the angle of parallelism, then the two lines are gonna be parallel in that situation. And so we can do this by contradiction of course. So considering our picture again, here's our lines T and L like so with their usual labels A, B and D. And it's important to note here that T and L are always right angles in this consideration and suppose that we have some line determined by the points D and C and suppose that the angle between these two lines here that is the angle between the angle between line T and line M is in fact the angle of parallelism. I'm gonna label my line L or excuse me, my line M in this situation. So suppose for the sake of contradiction that the line AB and the line M actually intersect each other. And so in such a situation we can actually without the loss of generality assume that of course the point of intersection turns out to be B in that situation. I'm actually gonna extend. No, no, this is fine right here. We'll just put that there. So we can actually take C to equal B and suppose that we have the angle of parallelism right here. So basically what we're saying is yep, we've obtained the angle of parallelism but the line turns out to be intersecting in that situation. What happens there? Well, by extension there's gonna exist some point E over here so that B is between A and E like so. Now because of that you can then consider the angle ADC right? Which of course C I didn't draw the second time but the angle ADC is equal to the angle ADB. And so if we take the line segment that connects D to E here, sorry about the crudeness there. Let me try that one more time. If we take the line segment that goes from D to E right there I should say it's the line determined by D and E there. This angle because B and C are interior points to the angle ADE. This is gonna be an angle that's bigger. It's gonna be bigger than the angle of parallelism. So D naught is less than the angle ADE. And as such that means the lines AB and DE are parallel to each other. But that's not the case, they contradict. That the contradiction because these lines do in fact intersect L contains E so does the line M here. And so we do get a contradiction. So if you obtain the angle of parallelism then that actually means that you are parallel itself. And so this line that obtains the angle of parallelism is essentially the smallest line that would be parallel to L passing through the point D. Now this concept deserves a definition here. So we have our line CD it's parallel to AB. We call this a limiting parallel. Some people like to call it an asymptotic parallel if the angle ADC obtains the angle of parallelism. So the limiting parallel is a parallel line that obtains the minimum boundary. So if you make that angle just a teeny bit smaller by any margin, any level of smaller it actually would intersect. So the idea with the limiting parallel is that these lines intersect each other like at infinity. So if you think of like the hyperbolic disk model here we could have for example a line that looks something like this. I'm actually gonna draw one with some semicircles. You might get semicircles that look something like this. It's possible and despite the crudeness of my drawing here it's possible that you have these semicircles which they share a point on the boundary but they don't share any points in the geometry. And as such, these are what we call limiting parallels. Another example of we use the half plane model for hyperbolic geometry. A vertical line is example of a hyperbolic line. We can also take a semicircle which contains that same point, something like this. These line and circle intersect on the x-axis but the x-axis doesn't belong to the geometry. These are both examples of limiting parallels. They touch each other at infinity but infinity doesn't belong to the geometry. And so they're asymptotic to each other. Notice they converge towards the same location but they never actually touch each other much like a rational function which is converging towards its vertical asymptote. They never touch each other but as you go towards the extreme they are both approaching the same value. So hyperbolic geometry does have these limiting parallels. Euclidean geometry has them too. They just don't look so weird. In Euclidean geometry, your all lines are gonna be limiting parallels. And that situation, your angle of parallelism is actually a right angle. But as you go from Euclidean geometry to elliptic geometry, that is you add points at infinity, these points intersect at infinity. I should say these lines intersect at infinity. It doesn't happen in the affine geometry but it does happen in the covering projective geometry. A topic we'll talk about a little bit later. All right, so the of a parallel line obtains the angle of parallelism we call it a limiting parallel. And yeah, in Euclidean geometry the angle of parallelism is always 90 degrees. We'll talk about that in just a second. Now, I wanna also clarify something before we go much farther here. In this situation, when we talked about parallel lines we really were talking about a line being parallel to a ray. And so if we were to draw our picture again, we have L, we have T. I'm actually gonna move T to put a little bit more in the middle right here. Whoops, one more time. So we have T like so. L, we have A, we have B. We have our point D, say like so. When we talk about the point C, it's always as we're talking about our ray, something like this, for which we do know that the angle of parallelism has to be an acute number or right. It can't be obtuse because after all the angle of parallelism was the least upper bound to the set M that we described earlier where M was the set of all angle measures that make the two lines intersect each other. Once you have 90 degrees, those rays aren't gonna intersect anymore. If you have some of the two-sangle going on like this, those aren't gonna, they're not gonna intersect each other. Now, mentally, if you go on the other side of the ray, then maybe you get some intersection going on there. But I want you to be aware that while we're talking about rays here, it's not such a big deal. The other side isn't going to intersect, but in particular, the concern I have is something like the following. What if we pick, let me distort my picture a little bit. What if we have something like the following? What if we have two rays on opposite sides of the line right here? What if in that situation, we have an angle over here that's smaller? Okay? So what I'm saying is the following, let's take some point C over here on this side of the line. Let's take some point C prime over here, and let's consider the angle D naught, which let's call that the angle of parallelism for D on the right side of T. And let's take this one over here, D naught prime, which we'll say that's the angle of parallelism on the left side of the line T. And again, for the sake, without the loss of generality, let's suppose that D naught is greater than D naught prime. Could you have something like that? Could it be unbalanced? That one side, it's easier to converge on than the other? That other one has maybe a bigger angle of parallelism, so that's actually harder? I don't know. Well, I do know. I'm just trying to speak hypothetically, of course, right here. Could it be possible that such a thing could happen? Well, I claim the answer is no. And why is that? So under the assumption that D naught is greater than D naught prime, then consider the point E that lives inside of the half plane AB that contains C. So we have some point over here. Some point over here, we're gonna call it E, such that the angle ADE, something like that, ADE, then is the angle D naught prime. Now, this is going to imply, since D naught prime is smaller than D naught, this implies that our line, DE actually must intersect the line L, and without the loss of generality, we can assume that point of intersection is gonna be B, no big deal there. In fact, we can say that B and E are the exact same point for all we care about, the same point in this consideration. I want you to then consider the triangle ADE, or ADB, whichever you prefer. It doesn't matter what you call the point there. So you have this triangle over here. Well, by triangle translation, we can translate this onto the other side of the line, for which case we're then gonna get a triangle over here, right, and so there's gonna be some point E prime over here so that ADE prime is congruent to ADE, and as this is a right angle, this is gonna be a right angle. So when we translate it over, we actually do get these supplementary angles at EAD and E prime AD like so. So these are triangles are intersecting, but then this, excuse me, these triangles are congruent, and that then forces an intersection between the ray, C prime D with the line L, but that refutes the fact that D not prime was in fact the angle of parallelism. So this tells us that D not prime cannot be smaller than D not, but by the same argument, D not cannot be smaller than D not prime. And so because of this argument here, we see that the angle of parallelism has nothing to do with the ray itself. It has entirely to do with the point D, right? Because whether you're on whichever side of the line you're on, it doesn't matter. You're gonna get the same angle of parallelism. So our definition of angle of parallelism is in fact a well-defined concept. All right, another thing we need to mention here is what about security quadrilaterals? We've seen previously that if you take a security quadrilateral, let me draw one for us right here. So security quadrilateral, remember the thing we know about security quadrilaterals is that we have these two right angles. The summit here, maybe it's potentially longer than the base. That's the possibility. But we do know that if we call the vertices A, B, C and D, we know that angle A and B are right angles thus congruent to each other. We know that the legs A, D and B, C are congruent to each other. That's what it means to be a security quadrilateral. We've discussed previously the summit angles of a security quadrilateral. They're congruent to each other, but in particular, they cannot be obtuse angles. They're either both acute angles or they're both right angles, all right? What we can then say because of the angle of parallelism is that these angles, these summit angles actually have to be greater than the angle of parallelism because of a security quadrilateral, as we also discussed, if we take the midpoint of the summit and the base and connect them together, you get the so-called altitude of the security quadrilateral. This shows us that the base and summit of the security quadrilateral are in fact parallel to each other because they have a common perpendicular, all right? I want you to look at this picture right here. This kind of looks like the picture we've been drawing over and over and over again, right? So we have the point B over here, the point A. I'm gonna throw these midpoints in there for the altitude as well. But we have the line L, which is determined by A and B. We have the line T, there's some point D right here. There's some point C, right? That determines this thing here. The summit is the line D, C. It is parallel to the base AB. And as such, this angle must be larger than the angle of parallelism. So for every security quadrilateral, the summit angles have to be greater than the angle of parallelism determined by the length of this. Of course, if the legs have different lengths that could change the angle of parallelism. But with whatever this length is given here and that's what determines the angle of parallelism, that length, how far apart are the points A and D? That length determines this. But for a security quadrilateral, this angle has to be, the summit angle has to be larger than the angle of parallelism. Now a nice little trick about security quadrilateral is if you introduce the altitude, this will cut the security quadrilateral into two congruent Lambert quadrilaterals because we actually now have three right angles in this situation. But nonetheless, when you look at these Lambert quadrilateral right here, this angle D, which was a summit angle for the security quadrilateral, this is the fourth angle for a Lambert quadrilateral. The same argument applies right here. The fourth angle of a Lambert quadrilateral has to be greater than or equal to the angle of parallelism because these two sides of the Lambert quadrilateral are parallel and it has to do with this distance here. So whatever this length is, we'll determine what the angle of parallelism is. And for a Lambert quadrilateral, those things have to be bigger than the angle of parallelism. So this is very much related to this topic of security and Lambert quadrilaterals that we had introduced in the previous lecture videos. All right, so let me mention two theorems about the angle of parallelism that are extremely important. The first one, theorem 3611 in our lecture series, I'm gonna leave this as a exercise to the viewer here. It's a very important theorem, don't get me wrong, but it's one that I believe the viewer should be able to prove. If the angle of parallelism with respect to a line L and with respect to a point P is acute, then there exists more than one line through P, which is parallel to L. And as such, this means that the hyperbolic parallel posture has to hold. And there's a couple of ways you could think about this. If your angle of parallelism is acute, one way to construct multiple parallel lines would be to consider a ray that obtains the angle of parallelism on one side and a ray that obtains the angle of parallelism on the other side. Those can't be the same lines because they're acute. That's gonna give us two different parallel lines, right? But we can also use the guaranteed parallel line. Let me use a different color there. The guaranteed parallel line is going to use a right angle, right? By the ultimate here, angle theorem, this gives the guaranteed parallel line. Since this is a right angle and these are acute angles, those lines can't be the same. So I've actually constructed three and then you could even make the argument that any ray that sits between the limiting parallel and the guaranteed parallel is also gonna be a parallel ray for which on the other side, that gives you then parallel lines. So in fact, if your angle of parallelism is acute, then you're gonna have an infinitude of parallel lines that live in sort of a spectrum. But again, I've given you the basic idea here. I'm gonna leave the viewer the opportunity to provide the necessary details there, but a pretty good hint has then been already executed here on the screen. So the next theorem here, this one I'm gonna prove this one in our video right now. In neutral geometry, the Euclidean parallel posh is equivalent to the following statements. The measure of the angle of parallelism is 90 degrees for all lines and all points. Very similar to the theorem 3, 6, 11 that we just talked about a moment ago. So if your angle of parallelism is ever acute, then you actually have the hyperbolic parallel posh in neutral geometry. But if your angle of parallelism is 90 degrees, then you get the Euclidean parallel posh. And so this measurement, this angle of parallelism really determines the parallel posh. And so looking at angles of parallelism, which is why we call it the angle of parallelism gives us a parallel posh for our neutral geometry. So there's two directions we need to go here. It's logically equivalent. So there's an if and only if statement. We can assume the Euclidean parallel posh and then prove that the limiting parallel has to be a right angle. And then we have to go the other direction as well. So we will assume that the angles of parallelism are always 90 degrees and then argue that we have the Euclidean parallel posh. So let's first assume that we have the Euclidean parallel posh. Let L be a line that we have, we have the same picture here and let P be a point not on L. Usually we call it in that D, but to be consistent with my notes here, let's call that point now P and let T be the unique perpendicular line to L through P. So that's this picture we have so far. Let D not be the angle of parallelism with respect to the line L and with respect to the point P. So we wanna argue that this angle of parallelism has to be 90 degrees. Let M be a limiting parallel through the point P with respect to the line L. Such a limiting parallel exists by D to consaxium. So we're gonna get something like this. This is our line M and let's claim that the angle between T and M is in fact this angle of parallelism D. Now by the Euclidean parallel posh, M is the only line parallel to L through P. That's what EPP gives us. On the other hand, by the alternator angle theorem, the unique line perpendicular to T that passes through L that's parallel to L is gonna be that guaranteed perpendicular line. So let's use a different color for that one. Because we do have our guaranteed perpendicular that if we form a right angle between T and this line we just directed out of P, that's gonna be parallel by the alternate interior angle theorem. But because of the Euclidean parallel posh, there can only be one parallel line to L that passes through M. So it turns out that passes through P, excuse me. So it turns out that these lines have to be one and the same thing. And since the angle between M and T was a right angle, this then gives us that the angle of parallelism was 90 degrees. Now given that the line L and the point P were arbitrary, this shows us that the angle of parallelism is always 90 degrees in Euclidean geometry. Now let's go the other direction here. So this time, let's suppose that the angle of parallelism is always going to be 90 degrees. Let's then argue that we have a unique parallel line. So to prove the Euclidean parallel posh, what we're gonna do is we always start off with a line L and we then take some point P that is off of the line. We then wanna show there's only one parallel line that passes through P in that situation. Well, let's take the perpendicular. It's perpendicular to L and goes through the point P. Such a construction exists in neutral geometry. We know that by the alter interior angle theorem that there's at least one parallel line, the so-called guaranteed parallel that will erect out of the point P here. I never gave this line a name. It's T of course. So we have this guaranteed line M. That it's gonna be a right angle in that situation. Now this line of course forms a right angle with T that we see right here. And hence it is a limiting parallel since it obtains the angle of parallelism. We're assuming right now the angle of parallelism is 90 degrees. Therefore no other parallel line can form an angle with T less than 90 degrees. But the two angles forms by such a parallel line either side, right? If we go on the other side as well, this one's also a right angle, right? Right angles are their own complements here. So either side, the angles are gonna be both right angles. You don't have one acute and one right angle either. So if we had some other line that passed through P, whoops, we had some other line that passed through P. Could it be possible to L? Well, one of the angles associated to that line would have to be an acute angle because we're thinking of this as a distinct line. One of those angles would have to be acute and therefore it's smaller than the angle of parallelism which then means that somewhere the lines are gonna intersect each other which means it's not in fact a parallel line. So we get a unique parallel line passing through P that's parallel to L and that then proves the Euclidean parallel postulate. So in neutral geometry, these two notions of the 90 degree angle of parallelism and the Euclidean parallel postulate they are logically equivalent to each other. Let me provide one last result about the angle of parallelism which then is going to connect the notions of hyperbolic geometry to the angle of parallelism. In neutral geometry, the hyperbolic parallel postulate is equivalent to the following statement. The measure of the angle of parallelism is acute for all lines and all points. I'm not saying they're equal to each other but if you have the hyperbolic parallel postulate all angles of parallelism will be acute for any line with any point and the direction is then reversible. If you have ever an angle of parallelism that is acute then actually implies the hyperbolic parallel postulate. Now I want to point out that of course theorem 3611 that we had mentioned on the previous slide but did not prove, it gives us the first direction we had mentioned in that situation that if your angle of parallelism is acute then you're gonna get multiple parallel lines and therefore that gives us the hyperbolic parallel postulate and given that this is true for all lines and all points that construction is arbitrary and therefore you always get multiple parallel lines for any line and any point. What about the other direction? So the direction we're trying to go now is assuming this statement, excuse me, no. The direction we need to go this time is that we're gonna assume the hyperbolic parallel postulate and argue that the angle of parallelism has to be acute. Well, let the geometry be hyperbolic. That is, it satisfies the hyperbolic parallel postulate. So there exists more than one line parallel to L through some point P. Thus the angle of parallelism cannot be 90 degrees. If the angle of parallelism was 90 degrees by the previous theorem we just did then there would only be one parallel line and therefore that's not hyperbolic. So we get that the angle of parallelism can't be 90 degrees. Now, since the angle of parallelism is less than or equal to 90 degrees, we know this in general, that's the only possibility. It can't be 90 degrees. It can't be obtuse and therefore it's gotta be acute and that's the last possibility. And so that brings us to the end of course of lecture 25 about the angle of parallelism. Like I had mentioned at the beginning of this video we definitely let our calculus run wild in this lecture but it was very necessary to grapple these notions of angle of parallelism and what it has to do with the parallel postulates inside of neutral geometry. The fruit of course of this will bear very quickly in the next couple lectures as we talk more about our search for a rectangle in neutral geometry and then ultimately we lead to the schism between Euclidean geometry and hyperbolic geometry. If you learned anything about the angle of parallelism in this video, please like it, subscribe to the channel to see some more mathematical videos like this in the future. And if you have any questions about angles of parallelism or anything you see in any of these lecture videos feel free to post your comments below and I'll be glad to answer them.