 Hi, I'm Zor. Welcome to Unisorification, the Kingdom of Knowledge. Today we will talk about triangles and the corresponding sizes between sides and angles. Basically, it's a very short lecture and it's dedicated to comparison between two different sides and two different angles in a triangle. Two different theorems we will prove. Well, actually four considering we will have it back and forth. The first theorem is about equilateral triangles. Namely, if two sides of a triangle are congruent, then the corresponding two angles will also be congruent. We actually touched this in one of the earlier lectures. It's a very simple theorem and I will prove it in a second just once more for clarity. And another theorem is that if one side is bigger than another side, then the corresponding angle is also bigger. Okay, let's start in the first one. If sides are equal. Well, if sides are equal, then it's quite easy to prove that the corresponding angles are equal as well. Just draw a bisector and consider two triangles, ABG and GBC. One side is shared by them. Now, these two sides are equal as a condition of this particular theorem and the angles are equal because it's bisector, angle bisector. So by an axiom of side-angle side, these two triangles are equal. Corresponding elements are also equal, congruent I should really say, and that's why these two angles are congruent as well. Okay, so this is a simple case. Now, a little bit more complicated is the case when I would like to prove the inequality between sides. So let's say this side AB is longer than BC. Then the opposite angle to this side, which is angle C, should be bigger than the angle opposite to side BC. Once again, I will use the proof which goes back to Euclid and we will just have a couple of words about why this might not be exactly Ruder's proof in any case. Let's take the point D here in such a way that the segment BD is congruent to segment BC. Since, as I said before, AB segment is longer than BC, this point D will be somewhere between A and B. Well, just as I'm talking about this point D being in between A and B. Again, the word in between in the Euclid's time wasn't really precisely defined. It's Hilbert in the end of the 19th century actually formulated much more rigorously all these concepts like in between part of being inside, etc. But anyway, intuitively it's quite obvious that the point D is somewhere between A and B if AB is longer than BC. Now, consider the triangle BDC. Well, as I was saying these two segments are of equal size and that's why they are congruent. And that's why this angle, by a theorem which has just proved a second to go, if two sides in a triangle are equal then the angles corresponding to these sides are equal as well. Okay, so that's easy. The angle BDC is greater than BCD. Now, in one of the prior lectures I was talking about exterior angles of a triangle, basically proving that exterior angle is greater than any interior not supplemental to it. Well, consider now this triangle, triangle ADC. Angle BDC, this angle is exterior and that's why it's bigger than angle A because it's an exterior angle and it's bigger than any interior not supplemental to it. Right, so angle BDC measures greater than BAC. Well, when I'm using the symbol greater, I mean measure of these angles. I could have put more traditional letter N in front of it. But it's quite well understood without. Okay, now at the same time angle BDC, this angle, is equal to BCD. That's why angle BCD is also, I'm sorry, I should have put greater than, not less, exterior angle is greater than interior. BDC is exterior, BAC is interior. Okay, so this angle is also greater than angle BAC. Why? Because they are congruent to each other. Right? BDC and BCD, BDC and BCD congruent to each other. This one is greater, that's why this one, for example. But now, here is again, this not exactly rigorous logical consideration. Angle BCD is inside the angle BCA and that's why it is a smaller angle since it's a part of the whole. So, this angle BCD, BCD is greater than BAC but at the same time it's smaller than angle BCA. So, angle BCA, BCA is greater than BCD, this one, which can turn greater than angle BAC. And as you know, this relationship between different things, greater than, being greater than, is transitive. Which means if x greater than y and y greater than z, then x greater than z. This is called transitive. What follows from here is x is greater than z. That's the property of being greater, transitive property. That's why angle BCA, which is this one, is greater than angle BAC, which is this one. Which is supposed to be fruit, actually, that's exactly the theorem is all about. Since this side AB is greater than the BC, then the corresponding angle, which is opposite to this side, which is this one, is bigger than this one. So, in a triangle, against equal sides, we can find opposite to equal sides, we can find equal lengths. And opposite to a bigger side, you will find a bigger angle. Okay. These are two direct theorems, which I wanted to prove. Now there are two even easier converse theorems. Number one, if angles are equal in a triangle, then the corresponding sides are equal. Well, it's actually very easy to prove from the contrary. Well, if you consider that these sides are not equal, then one would be greater than another. And that's why the corresponding opposite angle would be greater than another, because of the direct theorem, which I just proved. Which contradicts our initial preposition that the angles are equal. So again, if they are proving from the contrary, if we assume that one side is greater than another, it means that the corresponding angle would be bigger than another. And that contradicts the preposition of the theory. Easy, right? And the next theory is, if one angle is greater than another, let me just draw it slightly differently. So it will be obvious that an angle is greater. So, if this angle is greater than this, then the corresponding side would be longer than the corresponding side. Again, we prove it using the same logic. Assume the contrary. If this side is not greater than this, what happens? Well, according to a direct theorem, there are only two cases when the sides are either equal or this one is smaller, right? Not bigger means either equal or smaller. Well, if these are equal sides, then the angles will be equal, which contradicts our preposition. Or if this side is smaller than this, then again, the corresponding angle would be smaller than this, according to the direct theorem, which also contradicts the preposition when this angle is bigger. So in both cases, using the direct theorem, we can prove what we wanted to do as a converse theorem. But we do have to go and consider all the cases. Like, if it's not bigger than what is it? Well, it's either equal or less. And again, from the logical standpoint, we have to be a little bit more rigorous with the definition of what is bigger. We base our logic on the comparison of numbers that the numbers can be either bigger or smaller or equal to each other, which goes to a completely different algebraic numerical theory and the definition of what is bigger and what is the number, etc. But again, we assume that all this is already defined and intuitively we basically understand that the bigger means not smaller and not equal. Alright, that basically concludes this particular relatively small and relatively simple lecture. And I would like to refer you once again to the website Unisor.com where parents can actually very actively participate in the education of their children by enrolling them in certain programs and checking the progress by basically seeing what's the results of the exams which students can take on that particular site using materials like this. So welcome to this site and that's it for today. Thank you very much.