 Hi, I'm Zor. Welcome to Unizor Education. Today we will continue talking about basic elements of geometry, of planimetry actually, and today's topic will be polygons. Well, everybody knows more or less that polygon is something like this, right? So let's try to define it in more rigorous terms. First of all, as we see, polygon contains segments. So segments, we know what it is. These are just pieces of a straight line between two points, endpoints. Well, let's call the segment endpoints beginning and end. It doesn't really matter. Left and right, number one, number two, doesn't matter. So let's consider we have certain number of segments which are arranged on the plane in such a way that the beginning of every segment corresponds to the end of the previous segment. So it's an ordered set of segments. This is segment number one. This is segment number two, et cetera. Every one of them has a beginning and an end. And the end of the segment number one coincides with the beginning of the segment number two. So if this is number one, this is number two, this is number three, number four, and number five in this case. So the endpoint of every segment number n, end of number n, coincides with the beginning of number n plus one. At the very end, when we have exhausted all the segments which we have, the ending point of the last segment should coincide with the beginning of the segment number one. And that actually closes the whole loop and that's how we have the polygon. Well, we can have it drawn in this way, but there are different ways to draw a polygon. So let's try to differentiate different polygons. Well, number one, polygons can be differentiated by the number of segments or the number of vertices which is exactly the same thing. So every endpoint of every segment is called vertex and the segment itself usually called a side or a leg sometimes. That's as far as the terminology is concerned. Now, what's interesting is that there are polygons which can be drawn this way. Let's say this is segment number one, this is number two, this is number three, number four I will draw this way and then number five, this way and number six, this way. So this will be number four, this will be number five, this will be number six. Is this a polygon? Well, the answer is yes, but it's not the polygon which we will ever talk about in our course of geometry. The polygons which have this property of crossing segments are not traditionally considered as basically a subject of studying in the geometry. So we will concentrate on polygons that do not have any two segments crossing each other. And these polygons which we will consider are called simple. So this is not a simple polygon although it is a polygon but we will never consider anything like that. So that's one classification. Well, another classification can be also related to the shape of the polygon. For instance, we can have this type of polygon. Now, what's interesting about this particular polygon and what's the difference between this one and let's say this one? For some reason today I'm drawing comedy polygons with five vertices. Anyway, so what's the difference? Well, consider this one and consider all the points which belong to all the segments comprising the polygon. If we take any two points on any two segments and draw a segment between them, this segment always lies inside the polygon itself, inside the area which is bordered, which is bounded by the polygon. Now, in this case, this is not true because if you will take for instance this and this two points which belong to two different segments and draw a segment, then part of the segment, this one, is inside the polygon and another part is outside. So these polygons are called convex and these one can keep. It's more customary to talk about convex polygons. It's just a tradition basically because whenever we are talking about triangles which are the simplest polygons which contain only three vertices and three sides, well, all triangles are convex just because there are only three sides and they cannot be outside of each other. It's always in between something. With quadrangles, polygons with four sides, it is already possible to have a concave polygon like this. But it's quite rarely actually that we will consider them. Most likely, all the polygons which we will be talking about will be concave ones, sorry, convex ones. Concave is something which is rarely occurring in practice and in problems. So simple and convex. Actually, there is another interesting property of convex polygons. If you would take any side, so let's forget about this property, that the line which connects two points and any two segments lies within the polygon. So let's consider a different property. What if you would take any side and make a straight line out of it so any segment can be extended to the straight line. Then the polygon itself always lies on one side of this line. So no matter which side we will take, the polygon which is convex always lies on one side of the line. And again, as an example of concave polygon, if you will take this side for instance, as you see, piece of the polygon is on one side of the line and piece of it is on the other side. So that's just another property of convex polygons and actuates an equivalent to the one which we talked about before, like two points and two different segments always have the segment which completely inside the polygon. These are two completely equivalent properties of the convex polygons and one can be used as a definition and another can be proved as a theory. So we will always consider simple and in most cases convex polygons. Now as far as terminology related to number of vertices and number of sides, so we all know that this is triangle. Now this is quadrangle and this is and this is five vertices. It's pentagon and the next one is hexagon etc. is all from legend enumeration. Now as far as the shape is concerned, polygons can be regular or not regular. Now regular polygons are those polygons which have all their sides and all their internal angles the same. Well the same means congruent to each other. Now in case of triangle all sides are equal or all angles are equal, well I should say congruent to each other. These three angles and these three sides. Now this is equilateral triangle. Again it's just the terminology. In case of quadrangle the regular quadrangle is as we all know a square. Again all angles are equal to each other 90 degrees and all sides are equal as well. In case of regular pentagon it looks exactly like the building of the Department of Defense in Washington and hexagon is more like whatever bees like to put their honey in. So basically the whole lecture is very short one and it's only about definitions. We are talking about basic elements of geometry not exactly the properties or theorems about it. That would be further lecture. So today it's just the definition of certain kinds of polygons and basically some terminology related to this. Thank you very much.