 Hi, I'm Zor. Welcome to Unisor Education. I would like to devote some attention to congruence of triangles. Well, basically traditionally in all the schools, etc., they started three basic statements about congruent triangles. These three statements are specifying certain conditions which are sufficient for triangles to be congruent. Now congruent means everything is the same, basically, right? So the conditions which we are talking about are not everything, just some elements of triangles are the same. And from this it's implied that everything else will be the same and the triangles will be congruent. Now, so these three different statements about congruent triangles sound like this. First statement is, it's called side-angle-side or SAS. It means that if you have two triangles and you have two sides congruent to each other and an angle between them as well, then triangles are congruent, which means everything else will be the same, which means this side will be equal to this side, this angle to this angle, and this to this. Well, you know, statements can be either axioms which we accept without the proof or theorems which we must prove, otherwise we have no right to state it. All right, so what about this particular statement? Well, the interesting thing is that contemporary axiomatic foundation of geometry accepts this particular statement as an axiom. Actually, it's axiom number 3.5. I have this little thing written for me, so I don't forget it. It's Hilbert's axiom 3.5, which he has included into his very, very comprehensive set of statements as an axiom, and we just take it without any kind of a proof. Why should we actually take it without the proof? Because we cannot really derive it from something more elementary. So this is an axiom and we don't really talk about any kind of a proof. So side angle side is Hilbert's axiom and in contemporary geometry, it's not really supposed to be proofed. It's an axiom. But to others, to other statements which I wanted to talk about, they are supposed to be proofed and they will be proofed right now. So the second is angle side angle ASA in abbreviation. Now, this is actually a theorem which can be proven. So basically we are dealing with two different triangles, ABC and GPF. And we know that one side is equal to another side, congruent, and an angle congruent to an angle and another angle congruent to another angle. And it is important that the side is between these two angles. So these two angles share one particular side. Then these two triangles are congruent. All right. How we will prove it? Here is the proof. Let's take this side AB and find a point here, call it G, in such a way that segment AB is congruent to segment DG. Now, eventually we would like to prove that G and E coincide with each other. So the whole segment DE is actually congruent to AB. But right now we don't know that. So we are assuming that there is some point G on this side DE, on the side or its continuation, doesn't really matter, which is congruent to our segment AB. Now, let's connect this to this and consider two triangles, original ABC. And instead of this DEF we will consider DGF. Now, what can we say about ABC triangle and triangle DGF? This side is congruent to this side by the condition of our theorem. Now, this side DG is congruent to AB because that's how we have constructed, we have chosen the point G, and the angle is congruent. So these two triangles are congruent because of the side angle side axiom, which you remember we don't have to prove it's an axiom. So we have side, angle, and side, side, angle, and side. This side is congruent to this by construction, and this congruent to this by the condition of the theorem. Now, if that's true, then this particular angle in this triangle should be congruent to this particular angle of GFD. So angle ACB is congruent to angle DFG. Now we have a very interesting situation that the angle DFG is congruent to this one. But in the original triangle, DFG, this angle also by condition DFG is also congruent. DFG is also congruent to ACB. Now, is it possible? So we have one segment, two rays from the end point of this segment, and therefore we have two angles. One is DFG and another is DFD. Is it possible that both these angles would be congruent to one angle ACB? Well, I mean, from some obvious, if you wish, standpoint, it is not possible. But we can't use the word obvious. We have to really either prove it or take it as an axiom. Going back to Hilbert and his system of axioms. Okay, I go back to my original thing, and it's in, again, in the third section of Hilbert, there is a very interesting axiom which we can use. Here it is. He is taking as an axiom the following statement. If you have an angle, you have a segment. Then there is only one ray which can be constructed on one side of this segment and another ray on another segment with these angles congruent to the given angle. So let's say this angle is given and this segment is given. And there is only one ray on one side of the segment and only one ray on another side of the segment with these angles congruent. Here we have the situation when we have on one side of the segment DF. You have two different rays and both angles, angle GFG and EFG, suppose we congruent to the same angle C here. Well, because of this axiom, it's not possible. There is only one angle, which means GF and EF should really be one and only one ray, which means G and E are supposed to coincide. And that's basically the end of the proof. G and E are coinciding and since we have built the point G in such a way that GDs congruent to AB, G and E coincide, which means that the whole side GE coincides with AB. And that's sufficient because now we have two sides, two sides, and an angle in between them by the first axiom, the triangles are congruent. All right, so that's the second statement about congruent triangles. And the third one is side, side, side. SSS for equation. All right, now this is also a theorem and I'm going to prove it. Here is the proof. Now, let's consider you have one triangle, ABC, and another triangle, which has exactly the same sides. G, E, F. Okay, how can we prove that if all three sides pair by pair congruent to each other, then the triangles are congruent? Okay, here's how. First what we do, we reflect this triangle ABC relative to this axis AC as an axis of reflection, and we will build AB prime C triangle. So everything is the same. By the way, I will use this to mark equal sides here. Okay? So AB prime C is a reflection of ABC, and as a reflection, this is a completely congruent triangle because reflection is one of those non-deforming transformations. Now, we take this AB prime C and move it to DG, such that the point A will correspond to G, coincide with G, and C will coincide to G. Now B prime will go to something else. Let's call it, you know what? Let me just do it alphabetically. I don't know why I use G. Let's put F here, and I will use G here. Now, whenever I move AC B prime to this location, I move the point A to G, and since the segment AC is congruent to segment GF, my point C will coincide with F, and B prime will move somewhere into location G. Now, since I moved this triangle AB prime C into this location, it's again non-deforming transformation, and these are equal elements. This triangle GFG is congruent to AC B prime, which in turn congruent to ABC. Since this is congruent to this and this congruent to this, it's sufficient to prove this congruent to this. Instead of proving the congruence between this triangle and this, knowing that this is congruent to this and this congruent to this, I can prove that this is congruent to this. Now, how can we prove that? All right, what do I know right now about this interesting geometric figure? You see, this side equals to this, and this equals to this, and this is the common side, and these are two triangles. All right, fine. To do what I want to do, I need a small theorem, if you wish. Small theorem about equilateral triangles. Let's say this is an equilateral triangle. No, I don't even need a new equilateral. I need a socialist triangle. You have a socialist triangle M and P, and I will use bisector here. Angle bisector Q. So these angles are the same. It's quite obvious that triangles P, M, Q and Q, M, N are congruent to each other because this side is common for them. They share it, and these two sides are congruent by definition of a socialist triangle. And since it's an angle bisector M, Q, the angles are the same. So again, by the first axiom, we have side, angle, side of one triangle, and side, angle, side of another triangle, which means triangles are congruent, which means that everything, all elements are the same. In particular, it means that this particular piece of the base is equal to this one, which means that M, Q in the isosceles triangle is not only the bisector of the angle on the top, but also a median. And also it means that this angle equals to this angle, and this angle equals to this angle. So the angles on the sides of the base are equal to each other, which are against equal legs of the isosceles triangle. And also these two angles are equal, but some of them is equal to 180 degree, which means that they are 90 degrees at each one. So it's the altitude as well. So by the way, we have proven that in the isosceles triangle, the bisector of the angle on the top is also a median and an altitude. But what we really need is equality or congruence, rather, between these two angles at the base. So if you have a isosceles triangle with two equal sides, then you have two equal angles which lie across these sides. That's a very important property. This is the mini theorem which we are going to use right now. Here's how. We connect these two points and consider a triangle EGG, which happened to be, as you see, isosceles. Since it's isosceles, the two angles at the base are equal to each other. Similarly, in the triangle EFG, which is also isosceles, since these sides are equal, also angles at the base are equal to each other. So this angle equals to this, this angle equals to this. And if you remember from angle arithmetic, we can add angles, right? And if these two angles are congruent to these two angles, then some of them will be congruent to some. So the whole angle will be equal to the whole angle. So the whole angle will be equal to the whole angle. And now this is sufficient for congruence of these two triangles because we have a side, an angle, and a side. And here we have a side, angle, and a side. So in this triangle, DFG, you have a side equals to this side, angle equals to this angle, and the side equals to this side. So these two triangles, just to see it better, let me just wipe this particular thing so it doesn't really disturb us. So we have equal angles and sides on both sides of the sample. So we have side, angle, side, side, angle, side. Again, using the first axiom, these two triangles are congruent. And since this is congruent to this and this congruent to that, we have congruence between these two original triangles. The theorem is proven. Okay, that's basically all for congruent triangles, which I wanted to talk about. As a summary, let me just wipe out everything which we don't really need, and I will repeat the summary which we can use in the future. There are three different statements about congruence between two triangles. The first statement is side, angle, side, which is taken as an axiom. Now, two other statements, angle, side, angle, and side, side, side are theorems which we have just proven. By the way, which is very important in case angle, side, angle, we have to have exactly this sequence, angle, side, angle, angle, side, angle. Because if you have one of the angles in one of these triangles not immediately adjacent to the side, but the side does not really make one of the angles, then actually the theorem is not correct. Because let me just give you an example of when it's not correct if you have angle, side, angle, and something else in another case. Let's say you have the same side, you have the same angle, and then instead of this angle, I will use this angle. So these angles are the same, this side is the same, but the angle is not the one near the side, it's an opposite side. So this angle is equal to this. Now, as you see, in this case you have angle, side, angle, in this case you have angle, angle, side, and these are not congruent to each other, as you obviously see. So it's very important for this second theorem when you have angles and sides together in one statement you have to understand that it's very important that the side is really between two angles, otherwise the theorem is not correct. Okay, and the third one is when all three sides of one triangle are congruent to three sides of another triangle, and the whole triangles are congruent to each other. So these are three very important statements about triangles which we will use in many, many different cases. And also, just by the way, we have proven a little mini theorem about isosceles triangles that in the isosceles triangles, a bisector of the angle on the top and the height, altitude and the median are exactly the same, they coincide and two angles at the base are equal to each other. Okay, that's it for today. Don't forget to look at Unisor.com website where this particular lecture has notes and many other notes actually, there are some exercises and what's important for parents and supervisors they can actually use the site to control the educational process using exams and scores and enrollment feature which this website allows you to do. So you can actually control the whole educational process just sign in and follow some guidelines. All right, that's it for today. Thank you very much.