 In the previous video for lecture 24, we introduced the notion of a secondary quadrilateral and then prove some congruent statements and some measure statements about secondary quadrilaterals and then referenced how they have to do with the right angle hypothesis, AKA the Euclidean parallel postulate, how it relates to the acute angle hypothesis, AKA the hyperbolic parallel postulate and how it relates to the obtuse angle hypothesis, AKA the elliptic parallel postulate. Now, we're gonna take a different approach to the search for rectangles, because after all, Secary was talking about secondary quadrilaterals because he wanted to prove that the summit angles of a secondary quadrilateral are necessarily right. Therefore, secondary quadrilaterals are just rectangles, which then would prove the right angle hypothesis, which then would prove the Euclidean parallel postulate, excuse me. In this video, we're gonna take a slightly different approach, the approach due to Lambert here. And so we're gonna define the notion of a Lambert quadrilateral. This definition, of course, is valid in a congruence geometry. In fact, we don't even need notions of betweenness here. We just need to know notions of congruence of angles and particularly what a right angle is. And with that, we can define a Lambert quadrilateral. A Lambert quadrilateral ABCD is a quadrilateral with three right angles, ABC. We make no assumptions about the fourth angle, angle D here. And so I'm gonna draw a picture of what a typical Lambert quadrilateral might look like, maybe not makes it so long here. I think it's something like this, for which this is vertex A, vertex B, vertex, actually the way I'm labeled, I'm gonna call this one C here. And then D is our fourth one. We usually label things a little bit differently here, but that's okay. We're gonna get a picture of something like the following, like so, is vertex D. So by assumption, angle A, angle B, angle C are all right angles, but angle D, the so-called fourth angle. We don't actually know. I mean, that's actually what we wanna talk about in this video here. So can you have a quadrilateral with three right angles where the fourth angle is not necessarily right? Hmm, hmm, that's interesting here. It doesn't seem possible, but that's only if you're thinking Euclidean. Lambert, of course, knew that the fourth angle of a Lambert quadrilateral was right, if and only if you have the Euclidean parallel postulate. This is another example of the so-called right angle hypothesis. The right angle hypothesis assumes that the fourth angle of a secondary quadrilateral is right. And that's equivalent to Euclidean parallel postulate. It could be that this other angle is acute. The way that I've drawn the diagram of a Lambert quadrilateral on the screen actually does make this look like an acute angle. And that's because I'm drawing a hyperbolic version of a Lambert quadrilateral. Maybe this fourth angle is obtuse, right? Is that a possibility? If the fourth angle of a Lambert quadrilateral was obtuse, this actually would satisfy the obtuse angle hypothesis, which is equivalent to the elliptic parallel postulate, which is not possible in neutral geometry because of the Sicari-Legender theorem. But if your fourth angle is acute, that actually will give us the acute angle hypothesis and that's equivalent to the hyperbolic parallel postulate. And so yeah, like what I have drawn on the screen right now, this is an example of a hyperbolic Lambert quadrilateral. But I keep on talking about the right acute and obtuse angle hypotheses. I thought those had to do with some angles of Sicari quadrilaterals. Why does it also have something to do with a fourth angle of a Lambert quadrilateral? Well, we're gonna see this in just a second, but it turns out that these Lambert quadrilatals are closely related to Sicari quadrilatals. The two notions are actually equivalent to each other when you take the right change of perspective. So let me mention some propositions about Lambert quadrilaterals here. So the first proposition here in neutral geometry, the fourth angle of a Lambert quadrilateral cannot be obtuse. I mentioned that in just a moment ago. This is a consequence of the Sicari-Legender theorem. I'm gonna leave it as an exercise to the viewer here to prove that with a Lambert quadrilateral, the fourth angle cannot be obtuse. And I promise you in just a second, I'm gonna show you a nice little trick that will show you that we've actually basically already proven it because we proved the equivalent statement for Sicari quadrilaterals. And so if you can transition to a Lambert quadrilateral, the proof will be immediate. Another proposition that's worth mentioning here is that in neutral geometry, the side of a Sicari quadrilateral between two right angles is less than or equal to the side opposite to it. So my diagram is now erased right now, but the sides that are between right angles are less than or equal to the sides opposite them. So remember with that really distorted side over here versus the other side over here, it looks shorter than the really long one. That's a consequence of the theory here, which I'm gonna leave this also as a proof to the viewer here. And so let me switch this one up here. This final theorem I am gonna prove here and this theorem is actually gonna establish the connection between a Sicari and Lambert quadrilateral. This is why I wanna prove this one. It's not that this one is easier than the others or anything like that, but the connection is very, very much important. I want that to be very available to the student right now. So what's the theorem? The altitude of a Sicari quadrilateral is less than or equal to either of its leg. What does that have to do with Lambert quadrilatals? Well, let me show you. So imagine we have a Sicari quadrilateral, which remember by definition, a Sicari quadrilateral is gonna be a quadrilateral with two right angles. So we have angle A right here, we have angle B, those are right angles. And then we're gonna have our summit, which, whoops. Let me try it one more time, something like this. So we have angle C, angle D. These are the summit angles, they're congruent to each other. The altitude, remember, is going to be, you take the midpoints of the base and the summit, we'll call those M and M, M and N, connect those together. We've proven previously that these are right angles. So the altitude does cut it into right angles like so. These will be congruent, these will be congruent. And of course by assumption, the legs of the Sicari quadrilateral are congruent to each other like so. All right, so these are properties we already know about Sicari quadrilaterals. What I want us to look at now is, look at the quadrilateral AM and D. Look at that thing right there. So if you take this quadrilateral right here, you'll notice that this quadrilateral, which is basically half of the Sicari quadrilateral, this is an example of a Lambert quadrilateral. Cause after all, we have a right angle here, we have a right angle here, so we have three right angles and then the summit angle, bum, bum, bum. It could be right, it might not be, but this is the summit angle of a quadril, of a Sicari quadrilateral is the fourth angle of a Lambert quadrilateral. And so this is the important connection between them. This is why I can talk about the acute angle hypothesis, the right angle hypothesis, the obtuse angle hypothesis, when you're talking about a Sicari quadrilateral or a Lambert quadrilateral, cause it's the same thing. If the summit angle of a Sicari quadrilateral is acute, that means there is a Lambert quadrilateral with an acute fourth angle. And it also goes the other way around, what if you started off with your Lambert quadrilateral in this situation? Then you can copy it over here so that this side right here is shared and then you can use copy it, right? We have like, we have a triangle translation property. You can do the same thing for quadrilaterals because translating a quadrilateral just means translating a triangle twice because quadrilaterals are just two triangles glued together. And so if you start off with a Lambert quadrilateral, you can move it into a Sicari quadrilateral. So look at some of these statements we have above here, right? In neutral geometry, the fourth angle of a Lambert quadrilateral is not obtuse. Well, if you have a Lambert quadrilateral, double it. And then your fourth angle is the angle, it's the summit angle of a Sicari quadrilateral. We've already proven in the first half of this lecture that those things can't be obtuse. Then boom, we have the theorem. We took a result from Sicari quadrilaterals and pushed it on to Lambert quadrilaterals. It goes the other way around that we're doing right now. So we have this Lambert quadrilateral, AM and D, okay? And so look at the proposition we have right here, which I'm leaving it up to you to prove here, 368. 368 says that if you take the side that's between two right angles, that's less than or equal to the side opposite of it, okay? And so notice, what are we trying to prove here? We're trying to prove that the altitude of a Sicari quadrilateral is less than or equal to either of its legs. Well, the altitude, the altitude is going to be a side between two right angles on a Lambert quadrilateral. The leg is gonna be the side opposite of it. So then by proposition 368, we get the statement we want right here. So proposition 367, you can push it from Sicari's onto Lambert quadrilaterals, but in this theorem 369 here, we took a Sicari quadrilateral, we cut it into two Lambert quadrilaterals, we took the corresponding result for Lambert's and then push it onto Sicari. So the reason I'm illustrating this is that the theory of Lambert quadrilaterals is basically equivalent to the theory of Sicari quadrilaterals. So why Sicari and Lambert took two different approaches to try to prove the existence of rectangles. In essence, it really is the same approach and neither one was successful because in a hyperbolic geometry, you can't have rectangles and neutral geometry, excuse me, hyperbolic geometry is a neutral geometry. So you cannot prove the existence of rectangles because that would imply the Euclidean parallel posture, which is not valid in hyperbolic geometry. And so that's where we're gonna leave our search for the rectangles for now. In the next lecture, we're gonna talk about the so-called angle of parallelism and then make some connections to these notions of quadrilaterals that we're talking about in this lecture. Thanks for watching. If you learned anything about Sicari or Lambert quadrilaterals, please like these videos, subscribe to the channel. If you wanna see more videos like this in the future and always, if you have any questions, please post them in the comments below and I'll be glad to answer them as soon as I can.