 Welcome back to our lecture series, Math 3130, Modern Geometry for Students at Southern Utah University. As usual, we'll be a professor today, Dr. Andrew Miselnine. In lecture 24, we then start the incredible journey, that is the search for a rectangle, the quest. A noble band of heroes will come together to search for the elusive rectangle. Well, what's so elusive about rectangles, one might ask. This is sort of a fundamental shape that many kindergartners draw pictures of and they can identify rectangles amongst other two-dimensional shapes. Why do we need to search for a rectangle? Well, okay, let me give you a little bit of backstory here. Geometers for centuries believed that Euclid's fifth postulate, which of course is equivalent to what we are now referring to as the Euclidean Parallel Postulate, although our version of the Euclidean Parallel Postulate was actually due to Playfair. But the Playfair version of EPP was the one that says that given any line, any point off the line, there exists a unique line passing through that point that's parallel to the original line. So this uniqueness of parallel statements. Euclid's fifth postulate, this is the postulate that Euclid had assumed in his text, the elements here. And the idea of Euclid's fifth postulate was sort of the following. You have a line and you have two lines that cross the line and on the side where you have two acute angles. Let's make sure we spell the word acute correctly here. When you have these two lines, that is to say, you have two lines that are transversed by some other line, call it L right here. On one side of the line, if you get two acute angles, then eventually these lines have to intersect each other. Thus you form a triangle, as Euclid's fifth postulate then gives us. Now, Playfair's version of EPP is logically equivalent to Euclid's fifth postulate in the realm of neutral geometry. So while this is our version of the Euclid and parallel postulate, for centuries, geometers were trying to prove Euclid's fifth as a theorem of neutral geometry, because if you get this one, you get this one over here, but if you get this one, then you get this one, it doesn't matter because they are logically equivalent to each other, okay? Now, many attempts were made to prove Euclid's fifth was a theorem, but all have failed, right? You can think of Indiana Jones right here and how many people tried to drink the Holy Grail? Well, we know at least one dude in the movie and it didn't end well for him, right? He picked the wrong one, right? You never were able to finish the quest. They couldn't find a proof for Euclid's fifth postulate. Now, in the process of doing so, a lot of geometers discovered equivalent forms to Euclid's fifth postulate. Playfairs version of EPP was one such thing. One equivalence to the Euclidean parallel postulate would be that the sum of angles in every triangle is always 180 degrees. Another statement equivalent to that surprisingly is if one triangle sum was 180 degrees, then all triangle sums were 180 degrees and that has to be Euclidean geometry. The Pythagorean theorem is equivalence to the Euclidean parallel postulate. We have that the converse to the alternator angle theorem is equivalence to the Euclidean parallel postulate and many of these we'll prove later on in our lecture series when we talk about Euclidean geometry. So there's all these different theorems that were equivalent to the Euclidean parallel postulate that geometers tried to prove. And in this lecture, I mostly wanna talk about attempts made by Sicari to prove the existence of rectangles because Sicari knew that if a rectangle existed, then we have Euclidean geometry because the existence of rectangles in neutral geometry actually is equivalent to the Euclidean parallel postulate, believe it or not. Now, of course, before we get into that, I do have to tell you that the story didn't end so well and maybe that's not the right perspective. Now, if you were Sicari, you're trying to prove the existence of rectangles, you'll be disappointed to find out that rectangles didn't exist because in fact, we have that Beltrami prove the opposite. In fact, Beltrami prove the independence of the hyperbolic, excuse me, I should say that Beltrami prove the independence of the Euclidean parallel postulate to neutral geometry. In particular, Beltrami constructed a consistent model of what we now call hyperbolic geometry, proving that you can't prove Euclid's fifth postulate as a theorem of the neutral axioms because you could also get hyperbolic, which those two statements are mutually exclusive. You can't have Euclidean and hyperbolic parallel postulates. So if there is a neutral geometry that could be hyperbolic, then that means your neutral geometry, neutral geometries don't have to be Euclidean. And so we have to stop proving Euclid's fifth as a theorem of neutral geometry because it can happen. But nonetheless, a lot of important things were discovered along this journey. So of course, the discovery of hyperbolic geometry is one of the most important mathematical discoveries of all time, particularly in the realm of geometry. But nonetheless, I don't want us to feel bad about Sicari or Lambert. We will be who be two people in lecture 24 we're gonna talk about their efforts to prove the existence of rectangles. But for now, we'll prove why this is the case later on. But for this, for the moment being, we wanna search for a rectangle because if a rectangle exists, then we have Euclidean parallel postulate. And so in order to find a rectangle, Sicari started studying quadrilaterals, which we nowadays refer to as a Sicari quadrilateral. So let me define this. Now, the definition of a Sicari quadrilateral makes sense in congruence geometry. We don't need it in notion of measure or continuity to do this. A Sicari quadrilateral would be denoted as ABCD. It's a quadrilateral such that, well, let me first draw my picture here and label sides, then I'm gonna give you some other things to under... Let me first label the sides here. So we have A, we have B, we have C, we have D here. Whoops, I forgot the fourth side here. When we draw a Sicari quadrilateral, we generally draw something like this. Basically what Sicari was trying to do is come up with a rectangle without assuming it was a rectangle, basically. And so what he did here is they assumed that the angle A and B in this quadrilateral were right angles. And he assumed that the segment AD and the segment BC are congruent to each other. Now, a little bit more vocabulary here. The segment AB is commonly referred to as the base of the Sicari quadrilateral. The segment CD is often referred to as the summit of the quadrilateral. And the sides AD and BC, which are by definition congruent to each other, these are referred to as the legs of the Sicari quadrilateral, okay? So the segment that connects the two right angles together is the base and the segment that connects the other ones are called the summit. These angles right here, C and D here, these are referred to as the summit angles of the Sicari quadrilateral, like so. And so what I wanna do in this video is prove some congruence properties about Sicari quadrilaterals. Now, if you're curious, why did I draw this thing sort of curved, sort of sagging? Well, honestly, in a hyperbolic geometry, a Sicari quadrilateral is not a rectangle, believe it or not, because the angle sum of the quadrilateral is strictly less than 360 degrees. Atop of we mentioned beforehand, when we talked about the Sicari Legendre theorem, same Sicari in that situation there. In hyperbolic geometry, quadrilaterals have angle sums strictly less than 360 degrees. And we're gonna see in a second that, well, we don't need to see that yet, that if the angle sum is less than 360 degrees, you can't have four right angles because that would be an angle sum of exactly 360 degrees. That doesn't happen in hyperbolic geometry, okay? So what can we say about some Sicari quadrilaterals? Well, this first proof, this first proposition will be a proposition in congruent geometry, like I said. We just need the axioms of congruence, betweenness and incidence to make this precise. So in congruent geometry, the diagonals of a Sicari quadrilateral are congruent to each other. What are the diagonals? So the diagonals are the lines BD and CA, like so. We claim that these are congruent to each other. Likewise, we're going to prove that the summit angles of a Sicari quadrilateral are congruent to each other as well. That's the main thing that we're after here. So let's look at the diagonals for a moment. So when you look at the Sicari quadrilateral, we have by assumption that segment AD and BC are congruent to each other, like so. And if you look at the triangle, look at the triangle ABD and BCA. So if you take ABD, that's the wrong one, excuse me, ABD like so, look at that triangle. It has the base, it has a leg of the triangle and it has a right angle, like so. Then look at the triangle, whoops. Look at the triangle ABC, like so. This triangle has the base of the quadrilateral, which the base is obviously congruent to itself. It has a leg of the quadrilateral, which by assumption is the two legs are congruent. And we do have a right angle here. And by Euclid's fourth postulate, all right angles are congruent to each other. So these two triangles here, ABD and BAC are congruent to each other by a side angle, side situation. So this that implies that the segment BD is congruent to the segment AC, like so. And I'm of course taking the whole diagonal right here. So I guess it's the first statement. These segments are congruent to each other. Now I want us to look at the triangle. This time, ACD, like so, ACD, notice that this triangle has this diagonal, which we've now proven that diagonals are congruent. It has a leg of the triangle, which by assumption legs are congruent and involves the summit angle, like so, not the summit angle, it's just the summit itself. And so this triangle has that course, summits congruent to itself. If you look at the other triangle, DBC like so, like I said, it has a diagonal, the diagonals are congruent, it has a leg, the legs are congruent, and they both, both of these triangles have the summit, which is congruent to itself. So by side, side, side, we get that these two upper triangles are congruent to each other. And so just like we did before, right, we knew that the diagonals were congruent because corresponding parts of congruent triangles are congruent, we can do the same thing now with these triangles here. The summit angles are corresponding angles, but in this triangle congruence, and so therefore angle D and angle C are congruent to each other. Pretty nice, right? So there's a lot of symmetry going on with regard to this security quadrilateral. Another proposition that I'm actually gonna leave I'm gonna leave this proposition as a proof to them. So the viewer here, but in congruence geometry, we have that the segment joining the midpoints of both the summit and the base of the security quadrilateral, this is called the altitude of the quadrilateral. So I first should draw this picture here because this is a theorem with a definition in it. So I keep on drawing my security quadrilatals to look hyperbolic. And this is to help us remember that they're not necessarily rectangles because we could be in the hyperbolic geometry here. Okay, so we have the security quadrilateral, AD is congruent to BC, the legs in that situation. Okay, so the first thing we wanna first do is have the definition here. And actually, even before I say the definition, I didn't wanna point out here that since the legs of security quadrilateral have a common perpendicular, right? The base is perpendicular to both legs of the, because the base is congruent to both, not congruent, it's perpendicular to both legs of the security quadrilateral, that actually gives us that the legs of the security quadrilateral are parallel to each other. So it's noteworthy to mention that the line AD, which is the leg on the left, is gonna be parallel to the leg BC on the left because they have this common perpendicular to the base. That's a consequence of the alternate interior angle theorem. Now, what we're gonna do here is we're gonna take the midpoint of the base. We're gonna, and let's call it maybe M or something like that. Do I give it a label here? I don't, but yeah, we'll introduce it here. We'll call it M, and then we'll take the midpoint of the summit, we'll call it N. And we're gonna take the line segment that connects these two points together. So again, these are midpoints. So M, it cuts in half the base. A, M is congruent to BM. And then the point N is gonna be the midpoint of the summit so that DN is congruent to NC. So that's what we buy the altitude in the situation. So then what this proposition claim here is that the altitude is perpendicular to both the base and the summit, like so. So that's what I'm gonna leave it up to you to prove in this video right here. Prove that this altitude is a common perpendicular to the base and summit here. And just by the same reasoning we said a moment ago, the altitude will be a common perpendicular to the base and summit. So then it becomes an immediate consequence of the alternate-tier angle theorem that the base AB is parallel to the summit DC because they have a common perpendicular. And so what this tells us about security quadrilaterals is that security quadrilaterals are an example of a parallelogram. What is a parallelogram? Well, a parallelogram is gonna be a quadrilateral, a quadrilateral, I have first had the right parallelogram on the screen, no tie. A parallelogram is gonna be a quadrilateral that opposite sides of the quadrilateral are parallel to each other. So this proposition gives us this altitude and establishes that security quadrilaterals are gonna be parallelograms to each other. And I wanna mention here that the proof is basically done. Really, the proof is basically one line. And that's because we're gonna use the side angle, side angle, side congruence criteria for quadrilaterals for which in this lecture video, we have not in this video, but in this lecture series, we've, I shouldn't say we proved it, but we took side angle side as an axiom of triangle congruence. In the homework, I've left it for my students to establish the notion of quadrilateral congruence for which you can, since you can dissect, you can dissect every quadrilateral into two triangles. You can then kind of put the congruence of the two triangles together. And so you can get this so-called side angle, side angle, side quadrilateral congruence. And so this proposition basically to immediate consequence of side angle, side angle, side, but I'll allow the viewer to provide the remaining details here. A few other things I wanna mention about security quadrilaterals before we sign off in this video here. We have a theorem, and this is in fact gonna be a theorem in neutral geometry. The notion of measure is gonna come into play here. All right, in particular, we're gonna use the Sicari-Lijonar theorem. Not surprisingly, Sicari worked through what we now call the Sicari-Lijonar theorem as he was studying quadrilaterals. So in neutral geometry, the summit angles of a Sicari quadrilateral cannot be obtuse. Well, he wanted them to be right angles because if they were right angles, they would be rectangles. But because of hyperbolic geometry, you can have a Sicari quadrilateral with acute summit angles. But at least by the Sicari-Lijonar theorem, Sicari was able to show that the summit angles cannot be obtuse. And how is that? Well, we've basically talked about it already. We already have seen as a consequence of the Sicari-Lijonar theorem that a quadrilateral's angle sum cannot exceed 360 degrees. Maybe it does equal 360 degrees, but it could potentially be smaller than that, all right? Let X be the measure of a summit angle inside of a Sicari quadrilateral. Then we get the following. Well, a Sicari quadrilateral has two right angles, A and B, what we were calling them before. So 90 degrees plus 90 degrees gives you 180 degrees. We also know that the summit angles of a Sicari quadrilateral are congruent to each other. So we have two summit angles, so we get two Xs. So two X plus 180 equals, well, it has to be less than or equal to 360. If you subtract 180 from both sides, you're gonna get that two X is less than, is less than 180 degrees. Divide both sides by two, then gives the implication that Xs less than or equal to 90 degrees. So it could equal 90 degrees, perhaps. And then in that situation, your Sicari quadrilateral would be a rectangle. That's what Sicari wanted. And so you can then see why was Sicari interested in the Sicari Legendre theorem, because this was a necessary tool to help him eliminate the possibility of a Sicari quadrilateral that's not a rectangle. He said that he proved that the angles can't be obtuse, but he was never able to prove that the angles couldn't be strictly acute because that's what this theorem allows for us, right? The summit angles of a Sicari quadrilateral, they could be right, given as a rectangle, or they could be acute. And as we will see, the summit angles being right, that is the existence of a rectangle, is equivalent to the Euclidean parallel postulate. And the angles being acute is equivalent to the hyperbolic parallel postulate. And so these different possibilities give us the so-called acute, the acute angle hypothesis. So the acute angle hypothesis is the assumption that the angles of a Sicari quadrilateral are, the summit angles are always acute. That's equivalent to the hyperbolic parallel postulate. We, of course, also have the right angle hypothesis, which is going to be equivalent to the Euclidean parallel postulate. Sicari couldn't distinguish between those possibilities. The last possibility, of course, is the obtuse angle hypothesis. Are the angles, the summit angles of a, of a Sicari quadrilateral, are they obtuse? Well, a Sicari quadrilateral, we could define just using notions of congruence. We don't even need to have a congruence geometry. That is, the, between these axes were not necessary to define a Sicari quadrilateral. And so using, so basically if we have a geometry which has side angle side, we can then prove the summit angles are going to be congruent to each other. So could they be obtuse? Well, it turns out that if you have this setting of where Sicari quadrilaterals are going to be obtuse to each other, this is going to be congruent to the elliptic parallel postulate. Now what Sicari was able to do is prove that the obtuse angle hypothesis is inconsistent with the neutral geometry axioms, but he couldn't separate the acute angle from the right angle because of hyperbolic geometry, the acute angle hypothesis is possible in neutral geometry. Okay, one last result I wanna prove about Sicari quadrilaterals in this video is the following. In neutral geometry, the base in a Sicari quadrilateral is less than or equal to the summit. And this is actually why we've been drawing the Sicari quadrilateral is the way we've been doing it. Let's draw our picture right here. Now we honestly don't need the picture for this proof, but I just did wanna draw it on the screen here so we can see it. The way that I'm drawing my summit on the Sicari quadrilateral seems to suggest that it's hanging, right? You know, sort of like a wire, there's some type of sag going on there. If you look at that, it would appear that the summit is longer than the base, for which they could be in fact equal to each other, like if this was a rectangle opposite sides are congruent to each other, but it could be that the summit is actually longer than the base. And this is something we can prove in neutral geometry. Now, if the summit and base are congruent to each other, that actually implies the right angle hypothesis, which implies the Euclidean parallel postulate. If the summit is strictly longer than the base, then actually would apply the acute angle hypothesis, which implies the hyperbolic parallel postulate. If perchance, the summit was shorter than the base, that actually would apply the obtuse angle hypothesis, which gives us the elliptic parallel postulate, but a neutral geometry, we can't get the elliptic parallel postulate. That's inconsistent. So consider the following situation. We have our security quadrilateral ABCD and consider the diagonal BD, like so. Well, there's basically three possibilities that we have to consider. Consider the angles ADB and BC, DBC, excuse me. So ADB is this angle right here, and then the angle DBC, DBC would be this angle right here. It could be that these angles are congruent to each other. If these angles were congruent to each other, then these two triangles, ABD and CBD would be congruent to each other by side angle side, which then would give us that their corresponding sides, which are the, in this case, the base and the summit, they would be congruent to each other. So in that situation, we would have our statement. They are actually equal to each other. Be aware that this situation we've actually assumed, believe it or not, that this assumption is going to give us the Euclidean parallel postulate, because the assumption there is going to get that the summit angles are both right angles. This is the right angle hypothesis. Okay, not a problem. Another possibility is that what if angle ABD, this one right here, what if it was less than angle DBC right here? Well, then when we look at these two triangles, because these two triangles do have congruent sides, the diagonal was congruent to itself and the legs are congruent by assumption. In this situation, the Heng's theorem then applies, which then would tell us that the segment AB, which is the base, is less than the segment DC in that situation. And so that also is compatible with what we have written here. We have a less than in that situation. So if you're assuming that ADB is less than DBC, you're actually assuming, believe it or not, the acute angle hypothesis, this situation you're going to get the hyperbolic parallel postulate, okay? So lastly, consider the situation where angle ADB is greater than angle DBC. So again, the same two angles are in consideration. Now we're just changing our mind who's bigger than the other one. Without going any further here, you might be able to suppose that this hypothesis here is actually assuming the two-sangle hypothesis, AKA you're assuming the elliptic parallel postulate. That seems like it should be inconsistent with neutral geometry. Aha, we're going to get a contradiction in this third case with the secular genre theorem. So consider the sum of the angles ADB plus angle A plus DBA. So what are our angles in play here? So we have ADB, which is this angle right here. We have angle A, which is this angle right here. And we have DBA, which is this one right here. So we're just looking at this lower triangle and we know that it's angle sum needs to be less than 180 degrees. Now angle A we know is a right angle, like so. And so we can make that substitution here. We also know that angle DAB, this one right here is complementary to angle DBC because their union forms angle B, which is a right angle. So we get that DBA is the complement to the angle DBC. So some things to mention here. Of course, if you take 90 plus 90 add that together, you're going to get 180 degrees. But also our assumption angle ADB is less than angle DBC. So I can replace ADB with DBC and that'll make things get smaller. That's how our assumption comes into play. Like I said, 90 plus 90 is equal to 180. So this sum right here is greater than this sum, DBC plus 180 degrees minus DBC. But the DBCs then cancel each other and we end up with 180 degrees. So the sum of the angles is greater than 180 degrees. That doesn't happen. The Secure Legendre theorem forbids that. And this is something we mentioned earlier, right? That if you have the elliptic parallel posh that is, if you have the obtuse angle hypothesis, that implies the angle sum of triangles will be greater than 180 degrees. That's inconsistent with the Secure Legendre theorem. So that third possibility is not possible here. And so therefore by the Secure Legendre theorem, the base of a Secure Quadrilateral is always less than or equal to its summit.