Obj. 25 Properties of Polygons

Now, again, a word that we're familiar with now vertex is the common endpoint of two sides, pluralistic vertices. Now, a word that you've heard before and it means really kind of what you think it means is diagonal. Although we're going to kind of define it specifically as a segment that connects any two nonconsecutive vertices. So if I'm connecting this vertex with this vertex, that's a diagonal. Now if it has to be nonconsecutive because if I can't get these two together, it would just be the side of a triangle. So that wouldn't work. So that's a diagonal. 

Now, a regular polygon, because again, if you remember back when we talked about triangles, we had equilateral triangles where the sides were all the same, equal angular triangles where the angles were all the same. Well, polygons can work the same way. And so we get kind of tired of having to say equilateral angular. And so we just say regular. If a polygon is both equilateral and equal angular, we say it's regular. So generally, we name polygons by the number of their sides. And again, some of these names should be really familiar to you. Some of them you may not have heard before. Specifically, for some reason, people always have a trouble with 7 is heptagon, not septic, 9 should be, it should be fairly simple, but again, some people have a little stumble with this 9 is the only one that starts with an N so a not a gun is a 9 sided. 

And then there can be sometimes confusion between the ten and the 12 decay and dodecagon. You might go back to maybe Spanish two is dose, so a dodecagon is two plus ten. So you have dose plus decagon, you might try if that works for you. Now, if we're just talking about a polygon in general, like we don't really care how many sides it has, we'll call it an N gun. That means it could have any number of sides. We don't care. It's just a general polygon. So we want to identify the general name for each polygon. So you go, you get here, you count your sides, you've got one, two, three, four, 5. So this is a pentagon. Here, again, it's tempting to say what's got four, but you notice you've got all the little ones here. So you've got one, two, three, four, 5, 6, 7, 8, 9, ten, 11, 12, 12 is going to be a dodecagon. And then here we've got one, two, three, four. So a quadrilateral. 

Two other words that you're going to be used to classify triangles are concave and convex. Concave, if you want to think about what a something is caved in, it has kind of a dunk in it, so that's what this is we've got here is we have a kind of a dunk in the side of the polygon. All right. Technically, we can draw a diagonal that contains points outside the polygon. But it looks like it has a dip in it. It's side is caved in. It's concave. Convex is basically not concave. So here we have a concave Pentagon because I've got 5 sides. Here I have a convex quadrilateral. Now, we all should remember the angles of a triangle add up to one 80°. And so we'll see what we can do about the other polygons. So let's say we have a convex polygon that has at least four sides, so here we've got a Pentagon. And I'm going to go ahead and draw all my possible diagonals from one vertex. And I want to see how many triangles I can make. 

Well, when I do that, notice here I've got one, two, three triangles. All right? And so I want to know what's the sum of the angles, because if I find that some of the angles, and that will tell me, because again, it's this plus this plus this plus this plus that plus that is that is all the sum of the angles together. So if I find so I've got three triangles, three triangles is each one is a 180°. Three times 180 would be 540. Right, yes. Okay, so some of the angles of my Pentagon here is 540°. Which kind of gives us our angle sum theorem for polygon. Essentially, if I've got insides, then the sum of the interior angles is going to be N minus two times one 80. 

So we had we had 5 sides, 5 minus two is three, three times one 80, was 5 40. So that's kind of how that works. Now, if my polygon is equal angular, so if all of the angles are the same, then I can find the measure of any one angle by find taking the sum and dividing it by the number of sides that I have. Okay? So we've got this chart here. We've got the chart with the names. And so go ahead and I'll start just kind of filling in the number of triangles. Notice here you've got a gap here between ten and 12. We don't really have 11 listed because it doesn't really have an interesting name. And then you've got the sum of each interior angle, some of the interior angles, and then the sum of what each interior angle is for a regular one. Okay, so we already depending on you can do hexagon, hip to God, all those. 

So it's going to work on that. And then here's what it should look like when you get done. To me, the big deal here is you need to know this basic formula right here. Because that will, that way, it doesn't matter how many sides we have. This, I mean, if you happen to remember off top of your head that, okay, the sum of the interior angles of the hexagon is 7 20. Yay, that's fine. More importantly to me is that you know this formula. So it doesn't matter how many sides you can find what the sum of the interior angles is. And then just keeps kind of logic to say, well, if they're all the same, then I take that sum divided by how many I've got. Now that was interior angles now, let's look at exterior angles. Again, reminding you from triangles, if we extend the side of one of the sides, we form an exterior angle on the outside of the polygon. And the deal with that is that this exterior angle and the angle that is next to have to add up to one 80. So let's take the exterior angles of a regular pentagon. 

Well, I know that each interior angle is a 108. I can either find it from my table, I can take, you know, I can use the formula, whatever I want to do with it. So if I know that one of my angles is a 108, that means that the exterior angle that's next to that would then be one 80 minus one O 8, which is going to be 72. If that was 72, all the others are 72, and so if I want to know the sum of the exterior angles, that would be 5 times 72, which is going to be 360. Now they're really cool, weird, creepy part about all this is that some of the exterior angles is always 360. Whether it's any convex polygon, whether it's regular, not regular, no matter how many sides it has, the exterior angles are always going to add up to 360. Interior angles could add up to several 1000° if not more. 

But exterior is always going to be 360, which I find kind of weird and bizarre. So that is our exterior angle some theorem, which is the sum of the X to your angles of a convex polygon, is 360. And if I wanted to find out what any individual one was, if I had a regular polygon, I take 360 divided by the number of sides. Just like we did with the you can see how it kind of works the other way going with The Pentagon. So finishing our table here. Oops. Come back. Anyway. Finish up the table. You can see that what each of the individual exterior angles are. And that's it for polygons.