 I'm Zor. Welcome to Unisor Education. I would like to present problem number four in the area of mathematical induction. As the previous problem number three, this will be about geometry. Okay. Let's consider you have a polygon. The shape of this polygon, it should be like this. I don't want to have something like this. No such thing, right? I think it's called convex. The question is, if you will summarize these angles, internal angles of this convex polygon, what will be the sum of these angles? Now, in case of a triangle, everybody knows that the sum of internal angles of triangle is 180 degree. I'm not going to prove it by the way. I'm just taking this for granted. This is a school math. Okay. So question is, what's the sum of internal angles of this polygon? Well, again, let's start with some kind of an analysis of this situation and do it step by step. Let's say you have a four-sided polygon. Well, obviously, I can divide it in two triangles. Sum of these angles is 180, and sum of these angles is 180. So it looks like we double 180. So for triangle, we have 180. For a polygon with four sides, it looks like it's 360. Okay. Fine. Let's do five-sided polygon. Well, it's very easy. Again, cut this triangle out. This is 180, sum of these triangles, and this is a polygon with four sides, which we can already know. All right. So it looks like we have added another 180. So for n is equal to five, it will be 540. So it looks like every time we are adding a triangle, we are adding another 180 degrees. Right? So basically, this is the basis of the proof using the full mathematical induction, which I'm going to introduce right now. So the formula seems to be 180 degree times n minus 2, where n is number of sides of the polygon. For 3, it's 183 minus 2 is 1. For 4, it will be 4 minus 2, 2 times 180 360, etc. All right. So this is the formula, and how do I prove it? Well, again, number one step for n equals to 3, we have already proved we know it as a school mess. It's 180. The formula is holding fine. Number two, let's assume that if n equals to k, if we have a k-sided polygon, the formula is 180 degree times k minus 2. And let's think about what happens when we have a k plus one-sided polygon. Well, as I said, if this is a convex polygon, it means I can always take three vertices and cut a triangle out of it completely. So this will be a polygon with k sides, right? Because I just cut out one vertices and replaced two sides with one. So I cut down the number of sides by one. So now I have a inside of this side, this is a k-sided polygon. And I know that the formula for all these angles is 180 times k minus 2 by our assumption, which is number two. And what do I add? I add these three angles of a triangle, which is always 180. So I have to add 180. And what will be? Well, obviously, it will be 180, k minus 1, which is 180 k plus 1 minus 2, because that would correspond this number k plus 1 and this number k plus 1. So the formula is exactly the same. I substitute it with k plus 1 here, with k there. Formula is exactly the same, so formula holds. And that basically proves this fact that for any n-sided convex polygon, the sum of all internal angles is 180 times m minus 2. That concludes it. Thank you very much.