 Hi, I'm Zor. Welcome to Unizor Education. That's what you can find on this website, Unizor.com. I would like to continue talking about similarity of geometrical figures and the first thing which is addressed is the angle. Now, before actually talking about angles, I would like to say a few philosophical things about similarity. Let's consider a similarity with some center, and some factor, whatever the factor it is. It doesn't really matter. Consider you have a straight line. My first question is, what would be the image of this straight line after the transformation of scaling by this factor relative to this center? Well, let's just think about it. You connect the point on the line with the center, and let's just for the finitiveness, put it to. Then you basically stretch it by the factor of two, getting this point. Now, from this, you get this, from this, you get this. It looks like the image will be another straight line. Actually, it looks like it will be parallel, but that's a different question. Let's talk about the straightness of this line. It's not easy to prove that the image of a straight line is a straight line. And I would like to be completely open about this. I'm not going to prove it. I'll just consider this as a given, and I will use this property that the transformation of straight line by scaling relative to any center and any factor would result in a straight line. So let's just forget about proving of that thing. But the consequence of this, if I take it as a given, then I can prove that the image would be not just any straight line, but the straight line, which is the parallel line to the one given. And here's how we can do it. Okay, this is the center. This is the line. And let's consider that image is a non-parallel straight line. And we will come to some kind of contradiction, right? So let's consider that the image of this line is this line. Well, if you will connect these two lines with the segment from the center, by definition of scaling, this point goes into this point. Now, if you do exactly the same thing, you would find out that this point on this line is transformed into the same point on that line. Which means in this case, it's obvious that the factor is equal to one, since the length of this segment is exactly the same. But in this case, factor is not equal to one. So for any factor which is not equal to one, this is impossible, because the factor is supposed to be exactly the same for any pair of points wherever you take it. So the factor between this length and this length and this length should be exactly the same as between this and this. And in this case, it's definitely one, so it cannot be true statement. So they are not intersecting each other, which means they are parallel. Okay, so assuming that scaling or homotopy, as sometimes people call it, the property that scaling transforms straight line into straight line, if you take it as a given, then we can prove that the image would be parallel to the source. Okay, how can we use this property? Well, if we are talking about angles, what is angle? Angle is basically two different rays connected at one point. So let's just have any center of scaling. Now, we know that this line would be transformed, let's say our scaling is one half. So this line would be actually this, closer to the center. And this line would be something like this, I guess, I'll use another color. So we used to have this angle, and now we have this angle. Now, what's the most important property of these two angles? They have parallel sides. And you remember from one of the prior lectures that if two angles have usually parallel sides, then they are either congruent as in this case, or they add it together and they make up the hundredth age of degree. So basically, this angle becomes equal to this one, this one equals to this one. And all other, obviously, these are vertical and these are vertical. So angles are preserved by the transformation of scaling, by commodity, and this is a very, very important factor. Because if you remember in the prior lecture, whenever I was talking about two triangles being similar, I was actually drawing triangles which have similar angles, the same angles. And that's actually a very important property. So any scaling preserves the angles. So whenever you have any geometric object which has an angular form, like a triangle or quadrangle or something like this, the result would be a very similar, in some sense, geometrical object in what sense? Because it will have exactly the same number of sides. Because straight line goes into straight line. So every segment which is a side would be transformed into some, again, segment which will be a side of a new geometric figure. And the number of vertices will be exactly the same, obviously. Number of angles will be the same and angles themselves would be preserved. So that's a very important property. Now, is it really a characteristic property of scaling? The fact that angles are preserved. And anything which has all angles exactly the same, like two triangles which will have the same angles, are they always similar? Well, with triangles actually it's true, but I will address it in different lectures. But I would like actually to offer you a counter example in case of quadrangles. If you have a square and a rectangle which is definitely not square-ish in shape, you wouldn't call these two figures similar, right? Even the angles are all 90 degrees, so number of vertices is the same, number of sides is the same, all angles are exactly equal to each other, and still you cannot really call it similar. So my point is that from similarity you can derive the congruence of angles and the number of sides and vertices should be the same. The reverse theorem, like if two different figures have the same number of vertices, the same number of sides, the same number of angles, and all angles are equal to each other, then it's similar. This is not right, because these two have exactly this property which I was just saying, but they're not similar to each other, so for similarity we need something else. And this would be a subject to the next lecture. I would like to finish this lecture on this. So from similarity we have congruence of angles, and this is a very important property. So congruence of angles is a necessary but not sufficient condition for similarity. You remember, necessary and sufficient, from sufficient you can derive something, but necessary is something you derive to from a sufficient condition. So similarity is sufficient to equality of the angles, and equality of the angles is necessary for similarity, but not the vice versa. There's no other logic reverse in the reverse order. Okay, that's it for this lecture. Next would be about segments and their proportionality. That would be another property which would be required for similarity, at least of angular figures. And I would like to refer you to Unizord.com, where parents and group leaders or teachers can actually, they can use this particular website to supervise the education of their students and basically enroll them in different courses, check their exams, et cetera, et cetera. So please do examine the website. And this lecture will be also there, obviously. The website is free. Thank you very much for your attention, and the next one will be, next lecture will be about similarity of segments and their proportionality. All right, thank you very much.