 This video is sum of interior angles and we're asked to find the sum of the interior angles for each polygon. Make sure you're understanding exactly what they're asking. The sum and the interior as we're going to be learning some different formulas going forward. We did just learn this formula though, that we're just going to find n for each of these and plug it into n minus 2 times 180. For a nonagon, we know that that is a nine-sided polygon. So if n equals 9, we're going to plug that into our formula and you will want a calculator for these again. And remember we're going to subtract 2 from the number of sides and multiply times 180. When we do that, we get 7 times 180, which is 1260 degrees. So a nine-sided figure, the sum of the interior angles, 1260, that's a degree sign. An 18-sided figure, once we give it above 10, we usually just call it an 18 gone, a 23 gone, etc. We know n equals 18. Again, we're going to plug it into the formula and we're going to get 18 minus 2 times 180, which if you don't have a calculator, will be a little bit more difficult. 16 times 180 is 2,880 degrees. So all we need to do is find the number of sides. And so for the third one here, we have to count the number of sides because it's not given to us. Make sure you count correctly. It's easy to miscount when the number is not given to you. If I mark them off, I see that n equals 6. Six-sided figure is a hexagon and then when I plug that into my formula, I'm going to get 6 minus 2 times 180, or 4 times 180, which is 720 degrees. So that is just finding n and plugging in the number into the formula we have. So now we can use what we just learned to solve more complex problems. We're asked to solve for x in this figure and these two angles here are each labeled x. So we're going to set up an equation, but first we need to know the the sum of the interior angles of this polygon. So this is a two-part problem. First we're going to use this formula to find out what all of these angles add up to and then we'll use that to set up our equation. First we're going to have to find out what n equals. Well, I don't know that it equals 9. Sorry about that. n equals 1, 2, 3, 4, 5. I know that this is a 5-sided figure. So if n equals 5, I need to first plug that in to find the sum of the interior angles of this pentagon. And we did this before, but we're going to just write out the work again. 3 times 180, which is 540 degrees. We know a pentagon, the sum of the interior angles is 540 degrees. Now we can set up our equation because we know the sum of all these angles is going to equal 540. So we can actually write out and it doesn't matter the order. I'm going to start with the x's. I'm going to do x plus x plus 150. I'm going to go all the way around that polygon. Get each of those values of the angles and set that equal to 540 degrees. I have five angles, so I should have five numbers that equal the sum of the interior angles of this polygon. And now that I have my equation, I can go ahead and solve for x. I'm going to combine like terms first. I have x plus x is 2x and then I have three numbers here. 150, 52, and 86. Again, you'll want your calculator for that. When you add that up, we're going to get 288 equals 540. And now it's just about solving for x. Subtract 288 from both sides, and I'm going to get 2x equals 252. Final step is to get x by itself. I get 252 divided by 2, which is 126. That 126 represents each of these angles, and I want to just make sure that I did answer the question. It says solve for the variables. That's what I did. I don't have to plug it into anything else, and I can move on to my next problem. We have a similar setup in the next problem here. Just some more complex terms or expressions for the angles. The first thing we want to do again is figure out the sum of the interior angles for this polygon. We know that this is a six-sided figure. This is a hexagon. So we will start off by solving for the sum of the interior angles of that hexagon. And again, we have done that before, but we're just going to plug that in again. 6 minus 2 times 180, which is 4 times 180, or 720 degrees. We know a hexagon, the sum of the interior angles is 720 degrees, and we're going to do the same thing we did last time, and set up an equation where we're going to add up each of those angles and set that equal to 720. Now this can be a big long equation. I know some people will try to take shortcuts on this, but I would recommend that you write out each of those if you are prone to making mistakes when doing these kind of problems. I'm going to add up each of those six angles, set it equal to 720 degrees. And then when you combine like terms, I have three x terms here. Four plus two plus three is nine x. And then I'm going to add up all of the numbers. Keep in mind there's some negatives in there. And when I add those all up, I'm going to get 441 equals 720. If you don't want to write out this big long equation, you could just kind of cross off as you go and say 3x plus 2x plus 4x. That gives me my 9x there, and do the same thing with the numbers, but cross them off as you go or make sure you're not missing any because these are really easy to make mistakes on. So when I go ahead and finish solving my algebraic equation, I'm going to get 9x equals 279. Dividing both sides by 9 gives me x equals 31. I want to go back and make sure that I answered the question that was asked. Solve for the variables. That's what I did when it says solve for the variables. That means solve for x in this equation. If it asked you to find an angle measure, you would take this x equals 31 and plug it back into one of the angles. But all I had to do was solve for x, so I'm done with this problem, x equals 31.