 Let's return to discussing properties about elliptic triangles, that is triangles and elliptic geometry. We saw in the last video that the obtuse angle hypothesis holds in elliptic geometry, and essentially it's actually equivalent to the elliptic parallel posh. I mean, we'll have to be careful because we're not in the context of neutral geometry anymore. But I mean, Euclidean geometry has the right angle hypothesis. Hyperbolic geometry has the acute angle hypothesis. And the obtuse angle hypothesis, that is, the sum of angles of the secondary quadrilateral are obtuse, characterizes elliptic geometry. And we're going to see that the obtuse angle hypothesis actually implies that the sum of angles of any elliptic triangle is going to be larger than 180 degrees. Now, in this proof, we're going to actually prove the special case where the triangle is right. I'll leave it as an exercise to the viewer here, the reader, so to speak, to show that this is true for general triangles. But it's sort of the type of argument we've seen before where you cut the triangle into two pieces using an altitude and then use the property of both right triangles to bring it back together to show that the original triangle exceeds 180 degrees. So let's look at right triangles here. So imagine we have a right triangle, that's elliptic, and we'll say the vertices are A, B, C, and we'll assume that angle C is our right angle right here. And so what we're going to do is we're going to do the following construction. So we're going to take angle B right here and we're going to translate it to the other side, the other air quote side, right? The other side of the line A, B, C. That's we want on the opposite side of C. So there's some point X over here so that when we look at the ray, we add this ray A, X here, X. And the angle X, A, B, we want to be congruent to the angle A, B, C. So clearly, when you're a projective geometry, there is no other side of the line. So you have to do this in a relative between this setting. For a spherical geometry, of course, there is plane separation, so there's no confusion right here. But we can perform this construction in elliptic geometry. So we have that point right there, X. We're going to introduce a new point Q and choose Q to be the midpoint between A and B, like so. And then we're going to drop a perpendicular from Q onto the line B, C. So we get this perpendicular right here, call R the foot of this perpendicular. And be aware, I'm not necessarily trying to make a statement about betweenness. Again, in terms of we can make some statements about relative betweenness, but I'm not trying to say that R is between B and C in any relative sense. Just for the sake of diagram, we have something like this. And then what I also want to do is I want the intersection of the line Q, R to be the point P. So we get something like this, P. Now one might be concerned, does the line Q, R even intersect the line AX? What if they're parallel? Oh, let me remind you, everyone, we're in elliptic geometry. There is no such thing as parallel lines. Guarantee the two lines intersect is characterizing the geometry in right now. Many of those arguments we had before about whether or not we have intersections is a non-issue in elliptic geometry. The concern, of course, is with betweenness, right? Am I being too loose in my argument here about betweenness, relevance betweenness? I'm going to let slide right here. So I want you to consider the triangle B, Q and R with respect to A, Q, P. So take B, Q, R and A, Q, P. I want you to notice that by this choice of the point Q, segment BQ is congruent to AQ. Angles RQB is congruent to AQP because they're vertical angles. And then by construction, angle B was congruent to angle A. So we get that these two triangles are congruent to each other. And so then if we look at corresponding parts, which are congruent, we see that angle R, so that is BRQ will correspond to APQ. And so we get there's a right angle right there so that this triangle we constructed is a right triangle right here. And so the reason we care about that is if you look at the quadrilateral illustrated right here in red, that is the quadrilateral CRPA. This we now have constructed a Lambert quadrilateral. We have four right angles. Let's count them. One, two, and three. We did it. So we have a Lambert quadrilateral. So we're concerned of course with this fourth angle right here. We know by the obtuse angle hypothesis this fourth angle is going to be obtuse. But notice this fourth angle is just angle CAB plus angle PAB. But PAB by construction was formed to be congruent to angle ABC. So when you put these together, these form together to be a right, to be obtuse angles, this is going to be greater than 90 degrees. If we turn this the statements of measure, right, and this gives us two of the angles inside of that right triangle. If you slap on the measure of angle C, which angle C is itself a right angle, then this will show you that the right triangle exceeds 180 degrees. And so like I said, it is left as an exercise for the reader here to show that all elliptic triangles have an angle sum greater than 180 degrees. But if you take your typical triangle, try to dissect it using an altitude and go from there. It's very similar to what we did with hyperbolic geometry as well. As an immediate consequence, the angle sum of every elliptic quadrilateral will exceed 306 degrees as it's just two triangles put together. And similar statements can be said for higher vertexed polygons in elliptic geometry. Also, some other parallels I want to make mention between hyperbolic geometry and elliptic geometry is because the triangle sum of a elliptic triangle always exceeds 180 degrees, we could measure how large of a surplus do we have in terms of angles or in this are the word we'll use is what's the excess of the elliptic triangle that is not the defect the excess here. Sorry about that typo. The excess measures how much this thing exceeds 180 degrees to the excess avail of an elliptic triangle is going to be the sum of the angles here. So I'll actually write those specifically out so we have the measure of angle a plus the measure of angle B plus the measure of angle C. And the sum of these things will always be larger than 300 or 180 degrees. So we track from 180 from and see how much beyond 180 degrees you get. Now this function we define here sounds a lot like the defect, which is probably why I had the typo here in the first place saying defect of the elliptic triangle. It's very similar to the defect. You're trying to measure how far away from a Euclidean measure are you. And without going through all the details here, I want to mention that the excess function in elliptic geometry forms an area function. It satisfies the three axioms of area. Remember, the axioms of area is positive for elliptic triangles that excess is always going to be positive because their angle sum goes beyond 180 degrees. It is congruence preserving if two triangles congruent will have the same area of the same excess. And that's because it comes from the angle sums and the angle sums will be preserved for congruent triangles because the angles are congruent. And then finally, it's additive that one takes a little bit more of an argument. But if you go back and look how we showed that hyperbolic defect is additive that same argument is going to apply and show that elliptic excess is additive as well. So area in elliptic geometry really just comes down to calculate the excess of the triangle. So area of a triangle ABC will just equal K times the excess of that triangle ABC. Larger triangles have larger excess, have larger area, and conversely as well, smaller triangles will have smaller excess. There of course is this constant K that affects things. You want to think of this as like a unit conversion type of thing or another way I like to think of it is in terms of the radius of the so-called elliptic geometry. If you want to think of it in terms of the spherical model, we have a sphere, we can take the unit sphere, but we can also take a much smaller sphere. Oh no, we're teeny tiny, right? Doing geometry on Jupiter might not feel the same as geometry on Earth or something like that. And since the diameters, I'll just draw the radius, the radius of these spheres is different, that affects the area, right? Because if you take a triangle on the surface of Jupiter, like take a triple-right triangle on Jupiter, that's going to be a bigger triangle than a triple-right triangle, say on Earth. And so this K right here has a lot to do, I think, in my opinion, with the radius of the sphere. This of course in spherical geometry, if you think of like the projective geometry, a similar notion does exist. But an immediate consequence of the excess function being additive is that in elliptic geometry, we get the angle, angle, angle triangle condition for triangles. That is, if the three angles are congruent, then the two elliptic triangles will be congruent as well. And so this might not seem very normal, right? Because this also shows the notion of similar triangles doesn't exist in elliptic geometry. Elliptic geometry is like hyperbolic geometry in this regard. I'll point you to that proof we did in hyperbolic geometry of AAA. It's the same proof in elliptic geometry where you switch from defect to excess, and it's really just using the additivity axiom there. I should mention on the other hand, though, that in elliptic geometry, the angle, angle, side condition fails. Angle, angle, side worked in neutral geometry, so it worked in Euclidean and hyperbolic geometry. But angle, angle, side does not work. And the issue really comes down to double right triangles, right? We know that in a right triangle, an elliptic right triangle, the sides opposite of a right angle are going to be the polar length. So if you have a double right triangle, the sides, the two hypotenuses are going to be polar lengths. But that third side associated to the other angle could be potentially any side, any size you want, right? I mean, you could have a really skinny double right triangle, but then you have, like, say a triple right triangle over here. You know, it could, that third side could be anything, so it's not determined. So basically, if you have a right angle, right angle, and a polar length, that doesn't determine the triangle. So angle, angle, side doesn't work in elliptic geometry, even though it did work in neutral geometry. The proof of angle, angle, side essentially followed from the exterior angle theorem, which the exterior angle theorem we get from the alternative angle theorem. So maybe it comes as no consequence that we can't capture this in elliptic geometry. We can't, we can't recover things like the exterior angle theorem. So if we keep track of what we do have, right? Whoops, let me fix up my mistakes there. If we keep track of what we do have in elliptic geometry, we are going to have side angle, side, which is just taking their action. We get angle, side, angle, which we can show is logically equivalent to even in elliptic geometry. We got angle, angle, angle. Let's see. And we showed here that angle, angle, side doesn't work. Side, side, angle, that didn't work in neutral geometry. It doesn't work here as well. Now in neutral geometry, we are able to prove the special case of hypotenuse leg. But that also doesn't work in hyper, or in elliptic geometry for sort of the same reasons we had before, that when you have a double right triangle, things act really funky in this setting. That if you have a right angle and a hypotenuse, that's going to have to necessarily be the polar length. But if the other sides are congruent, which could also be this double right triangle, it's essentially the same problem that's happening here. So hypotenuse leg doesn't work in, hypotenuse leg doesn't work in elliptic geometry. What about side, side, side? Does it work in the elliptic geometry? Now if you were to go back to the original proof we did in lecture 18 for side, side, side, what are things we used to prove side, side, side? Well, it took triangle translation, which is valid still in elliptic geometry. Let's see it used transitivity congruence, which is still valid. It used the Asosceles triangle theorem, which is still valid. Let's see it used the between cross lima, which in a relative sense that still is applicable angle addition still works. I mean, there is some issue about trichotomy going on there. Do we actually use trichotomy in that proof? I'm trying to actually remember it here. Maybe I should have prepared this a little bit better ahead of times. No, no, I think it's okay. Trichotomy is not exactly used, angle subtraction, side, side, side, angle, side. So looking at the proof really quickly, I think the answer is yes. Maybe I should verify this. If it's not, I'll put a comment below. But it should work here. So we get these properties for elliptic geometry. In comparison, if you look at hyperbolic geometry, in hyperbolic geometry, we get everything from neutral geometry, side, angle, side, angle. We get side, side, side. That's a definite in hyperbolic geometry. We get angle, angle, side. We get hypotenuse leg. We also get angle, angle, angle. But be aware in hyperbolic geometry, we do not get side, side, side, angle. That doesn't work in hyperbolic still. I mean, you know the properties for Euclidean geometry, so I won't say much more about this right here. And just wanted to say a little bit about how triangles work in these three geometries. None of them are exactly the same. There are some subtleties for each and every one of these things. But try to pay attention to these things. Why does it work in one geometry and not the other? I will see you next time. Bye.