 Hi, I'm Zor. Welcome to UniZor education, the Kingdom of Knowledge. We will talk about right triangles today. Well, obviously we start with definition and I'm sure everybody knows that the right triangle is the triangle which has one right angle, 90 degrees. And from the terminological standpoint, the side which is opposite to this 90 degrees angle is called hypotenuse. And two other sides are called legs. That's a short name. There is a more scientific name if I can say so. It's catatous. Catatous. And in plural when you're talking about two sides, it's catatous. That's plural. Alright, that's terminology. That's it for a while for this terminology, not more than that. Now, first of all, you remember this little theorem which we have proven in some other lecture about exterior angle of a triangle that it is greater than any other angle, interior angle, not supplemental with it. So, exterior angle obviously will be 90 degrees as well here. And since it's greater than any interior, not supplemental with it, it means that two other angles of the right triangle are acute. So, one right angle, 90 degrees, and two others are less than 90 degrees, called acute. Okay, now let's talk about something which seems to be unrelated to right triangles, but in COH it is really very much part of the COH. Here it is. Let's consider you have a line and point outside of this line. Now, first of all, from axioms of the cleat, you know that there is one and only one perpendicular. If it's more than one perpendicular, there is an axiom about parallel lines which will be broken. So, one and only one perpendicular. Now, so the first property of this perpendicular which I would like to discuss is the fact that it is shorter than any other segment which connects the same line with the same point with this line. So, if this is 90 degrees, then any other segment would be longer than the perpendicular. Okay, the second theory which I would like to talk about is that if we have two different segments on different distance from the point where perpendicular actually is projecting, then if the point is further from this base, then the segment connecting this point would be longer as well. So, the further we go, the longer this segment actually becomes. Intuitively, it's quite obvious. But let's do it in a more rigorous way. But first of all, let's talk about PA and PV. Another theorem which was proven in one of the prior lectures where I was discussing sides and angles with the triangle and their relative dimensions, we were discussing the fact and actually proved that in the triangle against a bigger angle lies a bigger side. So, in any triangle, if this angle, this one, is bigger than this, then this line, this side is bigger than this side. Now, we have proven that and we are going to use this particular theorem as well. So, two theorems we are going to use. One about exterior angles. The exterior angle is greater than any interior not supplemental with it. Like in this case, this exterior angle is greater than this and greater than this. And the second theorem is that in any triangle opposite to bigger angle lies a bigger side. Like in this particular case. Now, that's basically all we need to discuss these particular properties. Well, first of all, PAB, triangle PAB. Now, obviously it is the right triangle since PAB was a perpendicular. So, it's a 90 degrees angle. And as I pointed before, all other angles in this triangle are acute, less than 90 degrees. Which means that the angle PAB, the right angle, is bigger than angle PBA, acute angle. And since against bigger angle, we have the side which is bigger. Now, this is bigger than this. Because PAB lies opposite to right angle, 90 degrees. And PAB lies opposite to acute angle PBA. So, the bigger angle, the bigger side opposite it. The smaller angle, the smaller side. So, no matter what kind of point B is chosen, if it's not coinciding with A, if it's a real triangle, right triangle, then this line, PB, which is not a perpendicular, will always be longer than the perpendicular line, I should say segment PA. So, there is some other formulation, so to speak, of the same property. We used to talk about the distance between two points, which is the length of the segment, which connects these two points. So, the length is basically a distance. Now, if we do not have two points, but something a little bit more complex, let's say we have a point and a line. Now, what is the distance from point to a line? Well, it's not obvious from the first glimpse, but there is a definition for this. The distance is basically the length of the perpendicular to this line by definition. Now, but why didn't we choose this particular distance? Well, the answer is because this is the shortest distance from this point to any other point on the line. And in general, if we are talking about distance between a point and any other object, then the definition is choose the point which belongs to this particular object and which is closest to our point. And the length of this particular segment would be, by definition, there is nothing to prove about, will be the distance between a point and whatever the object is. So, if this object is a straight line, then it's perpendicular because the perpendicular is the shortest distance. Now, if you have two objects, there is also a concept of the distance between these two objects. Again, we have to choose two points and we have to choose them in such a way that this segment has the shortest length among all other segments which connect any other pair of points belonging to these two objects. So, the shortest distance is the distance between the two closest points which belong, each one belongs to its own, its own object. All right, so we have proven that the perpendicular is shorter than any non-perpendicular line. Now, if you have two non-perpendicular lines, then the one which falls further from the perpendicular would be longer. How to prove it? Well, actually, it's also same simple logic which we used before. Let's consider triangle PVC. Now, we know that exterior angle, again exterior angle, is always greater than any interior which is not supplemental with it. So, if you consider this interior angle or this interior angle in the triangle PVC, they both are smaller than this PVA. But this angle, as we know, is acute because it belongs to a right triangle which means that the supplemental angle, this one, is of two. Acute is less than 90 degrees, some is supposed to be a hundred and 18 since these two angles are supplemental to each other. Since this is less than 90, then this is greater than 90. So, it looks like this acute angle is greater than this one which means this is also acute less than 90. But this angle is greater than 90. What's following from this? That this angle PVC is greater than this angle PVC. And opposite to the bigger angle, this one lies the side which is bigger than the one opposite to the smaller angle which is this one. So, that's basically the proof that the further point where this particular segment falls on a line, the bigger the length of this segment will be. Well, there is one little thing which we actually missed in this particular proof. What if the point B lies on the opposite side? B prime. So, C is on one side, B is on another side. We cannot really make the same logic if we will consider B prime instead of B. Well, the answer is very simple actually. Let's just reflect back to the side where this point C is located. So, if A B prime has certain lengths, we just put exactly the same lengths on this side, it will be A B. And it's very simple to prove that these two sides P B prime and P B are equal to each other. Why? Because these two triangles are equal. You have a side which is the one which they share. This is by construction exactly the same as this. A B prime is equal to A B and both angles are right, 90 degrees. So, it's a side angle side. Triangles are congruent and therefore the lengths of the P B prime is equal to P B. And then we continue basically the logic which we have already went through that the P B is shorter than A C. And what's very important obviously is that if A B prime is shorter than A C, then the B will also be closer to the A. That basically proves it. All right. So, now that's all I wanted to talk about perpendicular and non perpendicular lines between a point and a line. Now, we are still talking about right triangles and what's more important about triangles than theorems about their congruence. All right, so let's talk about congruence between two different right triangles. Certain things are basically following from the general theorems about congruence triangles or axioms actually. Now, you remember we have three major statements about congruence triangles, general triangles. Side angle side, which is by the way an axiom in a more rigorous Hilbert system. Then we have angle side angle and we have side by side, side side side. These are theorems. Now, how is it applicable to right triangles and what do we know about right triangles? Obviously we know that one angle is 90 degree. So, immediately from this what follows is that if you have two legs of one triangle congruent to two legs another triangle, then triangles are congruent. Why? Because the angle between them is 90 degree. So, immediately have to basically use the side angle side theorem and the angle is 90 degree. Sides are two legs, two cavity, if I may say so. So, this is a leg, this is a leg and this is a scapontaneous, right? So, leg and leg and angle between them, that's basically exactly what side angle side theorem is. Now, fine. So, that's the obvious and very, very trivial statement about right triangles. Another very trivial statement is that if a leg plus an acute angle adjacent to this leg, which is this one. So, we are considering this leg and adjacent angle, which makes, which is between this particular leg and the scapontaneous. So, these two elements are congruent. Then obviously using this theorem, angle, we know about 90 degrees side and angle. If they are congruent, then again using this theorem we have this particular statement about right triangles. Again, in these general cases, we needed three elements to be congruent to each other. In case of right triangles, since we already know that one element and angle is 90 degree, is therefore it's congruent in all right triangles, that's why we need only two elements, like leg plus leg or leg plus acute angle, which it makes with the hypotenuse. Now, there are less trivial statements, theorems actually, about right triangles, which are also true and we're going to prove them. Here they are. Let's say you have a hypotenuse and one leg equal to each other. So, congruent triangles, to prove the congruence between these two triangles, we basically have to prove that either an angle is equal to an angle between them or something else and then we can just refer to a general case. All right, so let's think about how can we prove that if two particular elements, hypotenuse and hypotenuse, are equal to each other, then hypotenuse and the leg are congruent to each other, then the whole thing is basically true. Well, very simple. For convenience, I would like actually just to have my drawing a little bit more looking like the previous one, let's use not this leg, but this leg. All right, so let's consider we have this leg congruent to this one and hypotenuse congruent to hypotenuse. How can we prove that this particular leg, the second leg is also congruent to this one? Very easy. Let's go back to our line and point outside it. What we can do is we can have this particular triangle positioned here. Now, this is a perpendicular, right? This is a non perpendicular line. So this is our leg and this is our hypotenuse. Now, so what we have right now here is that hypotenuse is equal to hypotenuse. Now, how can it be that if we will put this triangle into exactly the same position, which means this point and this point will coincide and let's say that this point will not coincide. So this is a, b, c. This is x, y, z. Then this is a and x. This is b and y. Now this is c and let's say z is projecting something further. Now, how can b, z not be equal to b, c? Well, it can't because as we know from the explanation which we went through before, the further the point is from the point where the perpendicular falls, the bigger this particular segment is, which means that if z is further from b or y from c, then the x, z would be longer than the a, c. They cannot be equal. So if they are equal, it means that z should really coincide with c and that's why this particular leg is exactly the same for both triangles. So that brings us into a statement that basically the three sides of one triangle are congruent to three sides of another triangle and then we can use the general theory. So everything is basically going back to the original to the beginning of this particular lecture when I was considering a point and the line, a perpendicular and a couple of other segments connecting this point with points and the line. Now, so this is something which we have just proven which is less obvious, less trivial than two prior statements. So this is what this is leg plus hypotenuse. All right. Now, are there any other interesting statements about congruent right triangles? Well, there is one more. As I was saying before, for right triangles, we are using only two elements because we know about the angle 90 degree, which is always the same in all right triangles. So this is just another example of the theorem where only two elements of the right triangle are needed to be congruent to have the whole triangle, triangle's congruent. Now another is if you have a hypotenuse plus one of the acute angles. All right. So let's do exactly the same thing. Let's say this is our triangle, this is hypotenuse and this is an acute angle. Now, that's all we know about these two triangles. So the hypotenuses are the same and acute angles are the same. Is that sufficient for two right triangles to be congruent to each other? Well, the answer is yes. Now, how can we do it? Well, very easily. Let's do it this way. Let's say again this is ABC and this is XYZ and we will try to bring them together using the following technique. We will have this angle somewhere and first we will bring into this angle this particular triangle. So this is X somewhere and this is Y and this is the right angle, right? Now let's bring this particular triangle starting again from the hypotenuse. Now, since hypotenuse AC is congruent to XZ, so A and X would coincide each other, right? Now, what we will do is we will try to find location of the point B, right? Where can B be located? Well, let's assume that B is not coinciding with the point Y. It's somewhere else. So the BC, let's consider BC is longer than YZ in this particular case. And then we will connect them this way. So the triangle A, now this is Z and C, right? So the triangle ACB on this particular picture is congruent to this triangle because that's how we basically construct it. We have a side, side, we have an angle, angle and we have another side, another side. So the triangle ABC on this picture is exactly the same, exactly congruent to triangle ABC here. Now, if that's true, then this angle is supposed to be also 90°, right? This is 90°, so this is 90°. And what do we have right now? We have two different perpendicular from the same point A outside of this line which are falling into two different points. We cannot have two different perpendicular from the same point to the same line as you remember it contradicts some Euclidean axons without parallels. So basically this is the proof that B and Y should coincide since there is only one perpendicular and that's why the BC is congruent to YZ and then the whole triangles are congruent to each other because of the side, angle, side, side, angle, side. So we have proven two different theorems about right triangles which are slightly different obviously than three original congruence theorems, axon and theorems about the general triangles and in both cases we are using only two elements from the right triangle, not three as in this case. So in this case it's black and hypotenuse or hypotenuse and an acute angle. Now, two others which I have already specified before are actually trivial. I mean I can put it here as well. So these are non-trivial. Trivial ones are when two legs congruence to each other because the angle between them is 90° so we have side, angle, side and leg plus acute angle which it makes with the hypotenuse because this is actually angle, side, angle because on this side of this leg you have a straight, you have a right angle so you have right angle, leg and acute angle. So these are non-trivial and trivial relative to these theorems about congruent right triangles. Well, that's probably about it as part of the theory of right triangles is I might actually have some more information which I will include in the notes for this lecture and obviously you are invited to visitunisor.com where this lecture is recorded and there are notes on it, exercises, exams and I would like parents to pay special attention to this website as a source of not only educational materials for your children students but also for you to be able to control the educational process because you the parents are basically the customers of the system and you can enroll your children in one of the programs. You can control how their progress actually is going because there are exams which they should take and score based on these exams so you can see the score, what exactly your child has accomplished on any particular topic and again it's up to you to basically consider this particular topic finished or should be repeated and the exam should be retaken etc etc so it's all up to you. Okay, that's it for today. Thank you very much.