 Hi, I'm Zor. Welcome to InDesert Education. I would like to talk about prisms today. Well, first let's just very briefly recall what prism actually is. The prism starts with a so-called cylindrical surface, and cylindrical surface in turn is the result of the following construction. If you have some kind of a curve in space, which is called directories, and a line, a straight line, which is called generatrix, and then through each point on this curve you draw a line parallel to generatrix. So all these lines form some kind of a surface, which is called cylindrical. Now, when we want to talk about prisms, we are talking about a specific directories. So as a directories we are using not just any curve, but first of all we have to have a plane, and on this plane we have to have some kind of a polygon. Now, this polygon is supposed to be used as a directories. Now let's do the cylindrical surface construction. So from each point I draw a line parallel to my generatrix, something like this. Now, I draw a line through each point on this polygon. Now, right now I have constructed only the lines which are going through vertices. Now, the next step is, let's call this plane alpha, the next step is to have another plane which is parallel to this one, and let's call it beta. Now, the cylindrical surface which is formed by all these lines intersects both planes, this and that. And what I am stating right now that this particular intersection makes another base, so to speak. And now the definition of the prism is. Prism is a solid object which is bounded by the cylindrical surface on the sides and two bases which are formed by intersection of two parallel planes with this cylindrical surface. So one parallel plane, one of these two parallel planes is given because that's where our directories is located. It's supposed to be flat polygon, flat means lying in one plane, and another is just anywhere in space parallel to alpha. So everything between these two bases and the sides is called a prism. Right, now, there are a couple of theorems which I would like to prove today about this prism. Now, the first thing is we usually draw the prism by connecting only vertices, the corresponding vertices of these two polygons. Now, this is the polygon, I understand that, and obviously the cylindrical surface is intersecting the alpha base along this polygon because basically each line is supposed to go through one of these points on the polygon. So the intersection of the cylindrical surface and lower, in this drawing, lower base alpha, is this particular polygon. Question is, will it be a polygon on any other plane? So that's actually the subject of my first theorem. If I have a cylindrical surface with polygon, flat polygon lying in the plane alpha as a directories and we have any plane beta which is parallel to alpha, then intersection between this particular plane and cylindrical surface is also a polygon and, well, not just any polygon, it would be the polygon which is exactly congruent to this one which means all sides supposed to be the same, all angles are supposed to be the same. Now, this visually obvious fact needs to be proven and again, since the purpose of this whole course is to develop your creativity and analytical abilities, proving some even trivial fact like this one is a very good exercise. So how can we prove that this particular intersection of our cylindrical surface and the plane beta which is parallel to alpha is a polygon congruent to this one? Okay, here is how. And by all means, I do encourage you to basically, instead of watching this lecture right now, try to prove it yourself and then watch. Okay, so first of all, let's just consider two points A and B which are vertices of our base polygon directors on the plane alpha. Now, these are corresponding A prime and B prime points where these lines, I'm talking about only these two lines where they intersect the plane beta. Alright, now, since all these lines are supposed to be parallel to this one, that's how we construct the cylindrical surface, right? From each point on the directors, we draw a line which is parallel to this one. That's how cylindrical surface is built. So A prime and B prime are parallel to each other. Since they are parallel to each other, they belong to the same plane. So these parallel lines always have the plane, they're not skewed, right? So that's basically the definition of the parallelism, lying in the same plane and not intersecting. So these two lines are parallel to each other which is good. And there is a plane and I will use the same letters A, B, B prime, A prime. This particular quadrilateral is a quadrilateral line in some plane. So this is a plane which intersects our parallel planes alpha and beta, right? Now, we know from the previous theorems that if you have two planes parallel to each other and another plane which intersects both of them, the intersections are straight lines and they are parallel to each other. So A prime, B prime is a straight line and it's parallel to AB. So what do we have now? The A, B, B prime, A prime is a quadrilateral which is a flat one. It's all four points belong to some plane and opposite side are parallel to each other. This is parallel to this because that's the definition of cylindrical surface and this is parallel to this because these are intersections of one plane with two parallel planes. Which means it's parallelogram. Within this plane A prime, B prime, B, A, this is parallelogram. Which means that opposite sides are not only parallel but also equal in lengths. And I can say exactly the same thing with every other pair Bc and B prime, C prime, Cd and C prime, D prime, etc. So that actually means that all edges of base polygon exactly equal to corresponding edges of the polygon and now I'm talking about really polygon because I know that these are straight lines, right? So these are all straight lines. So the corresponding sizes of these edges are equal. How about angles? Well obviously angles are also equal because these are angles between correspondingly parallel lines. And again we know this from previous theorems that if you have two angles in space with correspondingly parallel sides the angles are equal to each other. So all equal sides and all equal angles. So correspondingly the result is these are two congruent polygons. So we have let's say on this drawing this is the lower base polygon, this is the upper base polygon. They are completely congruent. Now the second theorem is that these are parallelograms. So in prisms every side face is parallelogram. That's basically a very, very easy consequence of whatever we were just talking about. So every side is parallelogram and opposite bases are congruent. So these are very simple properties. Now there is one more thing which probably needs to be addressed. You see I'm assuming that a, b, b prime, a prime basically this is not only a plane but also part of the cylindrical surface. So let me just for a second pay some attention to this particular fact. Obviously by all means but probably need it to be addressed in some way. So if you have a straight line from a to b in this case as a directress and you are building a cylindrical surface using this straight line as a directress then the cylindrical surface is actually a plane. So as long as a, b is not parallel to generatrix then every line you can draw here would lie exactly in the same plane produced by let's just take two points. So two points, two parallel lines and let's have basically a plane which is going through these two lines a, a prime and b, b prime. And what I would like to actually say that any other c point on this line and if you draw the parallel to generatrix c, c prime it must lie exactly in the same plane. But again it's kind of an obvious thing but to prove the obvious thing is probably more difficult usually. But in this case let's just think about this way. Let's take any point c between these a and b or outside doesn't really matter. But what matter is on the directress. On this directress we have this point c. And now let's assume that the line from c which is parallel to generatrix does not lie in the plane which is defined by these two lines a, a prime and b, b prime. What happens? Well, but let's still consider this plane a, a prime, b, b prime and c belongs to this plane because it belongs to the straight line which has two points belonging to this plane, right? And we know it's one of the axioms actually if two points a and b belong to some kind of a plane then the whole line belongs. Actually the whole directress belongs to this plane defined by these parallel lines a, a prime and b, b prime. So within that plane let's draw the line parallel to a, a prime. Now it's parallel to a, a prime. a, a prime is parallel to generatrix which means that this line is supposed to be which lies within this plane. It's supposed to be parallel to this one. What does it mean? It means that if I draw a line from c parallel to this line and I have assumed that it's not belonging to this plane. It looks like I have two different lines one which belongs to the plane in the plane and parallel to this one and one which does not belong to the plane and also parallel. So we have two different parallel lines to this one drawn from one and the same point which is a contradiction. So this is just a very kind of a plane proof that the cylindrical surface which is produced by a straight line as a directress is a plane. And that's why every side of our prism is actually a plane. And then from this I can talk about every side being a parallelogram and stuff like this. So if you have something like, something like this and it's no longer working. So if you have a prism which looks like this and you draw certain lines parallel to some kind of a generatrix and this is another base, right? Something like this. Okay, so this is visible and this is not visible. This is not visible and these are also not visible. Something like this. So this is in this case a prism with one, two, three, four, five polygon which has five vertices. So these two polygons on the upper base and the lower base are congruent to each other. Now the planes where these two polygons belong to are parallel and each side phase of this particular prism is a parallelogram. So that's the result of all these today's theorems. Well, that's it. Again, I wanted you to understand that certain obvious and intuitively trivial facts still must be proven and to be able to prove it is just one more exercise of your ability to logically make your points. That's basically it for today. Thank you very much and good luck.