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## Geometry (all content)

### Unit 3: Lesson 5

Angles with polygons# Sum of the exterior angles of a polygon

Sal demonstrates how the the sum of the exterior angles of a convex polygon is 360 degrees. Created by Sal Khan.

## Video transcript

Several videos ago, I had a figure that looked something like this. I believe it was a pentagon or a hexagon. And what we had to do is figure out the sum of the particular exterior angles of the hexagon. So it would've been this angle, we should call A, this angle B, C, D, and E. And the way that we did it the last time, we said, "Well, A is going to be 180 degrees "minus the interior angle that is supplementary to A." And then we did that for each of the angles. And then we figured out we were able to algebraically manipulate it. We were able to figure out what the sum of the interior angles were using dividing it up into triangles, and then use that to figure out the exterior angles. And it was a bit of an involved process. What I want to show you in this video is there's actually a pretty simple and elegant way to figure out the sum of these particular external angles, exterior angles I should say, of this polygon. And it actually works for any convex polygons when you're picking these particular external, these particular exterior angles, I should say. And so the way to think about it is you can just redraw the angles. So let's just draw each of them. So let me draw this angle right over here. We could call it angle A or maybe the measure of this angle is A, either way. Let me draw it right over here. So it's going to be, this is going to be a congruent angle, right over here. It's going to have a measure of A. Now let me draw angle B, angle B. And I'm going to draw adjacent to angle A. And what you could do is think about it. Maybe if we drew a line right over here, if we drew a line right over here that was parallel to this line, then the measure of this angle right over here would also be B, because this obviously is a straight line. It would be like a transversal. These are corresponding angles. So if we wanted to draw the adjacent angle be adjacent to A, you could do it like that or the whatever angle this is, its measure is B. Then now it's adjacent to A, and now let's draw the same thing for C. We could draw a parallel line to that right over here. And then this angle would also be C. And if we want it to be adjacent to that, we could draw it right over here. So that angle is C. So C would look something like this. C would look something like that. Then we can move on to D. Once again, let me do that in a different color. You could do D. D could be right over here, or you could shift it down over here to look like that. Or you could shift it over here to look like that. If we just kept thinking about parallel... If all of these lines were parallel to each other, so let's just draw D like this. Let's just draw D like this. D like this. So this line once again's gonna be parallel to that line. And then finally, you have E. Finally, you have angle E. And once again, you could draw a line. You could draw a line that is parallel to this right over here. Right over here, and this right over here would be angle E, or you can draw it right over here. You would draw it right over here. And when you see it drawn this way, it's clear that when you add up the measure, this angle A, B, C, D, and E, you're going all the way around the circle. Either way, you could be going... You could be going clockwise, or you could be going counter-clockwise, but you're going all the way around the circle. And so the sum of these angles are just going to be... So A plus B, plus C, plus D, plus E is just going to be 360 degrees. And this will actually work as I said, for any convex polygon. When I say convex polygon, I mean one that's not dented inwards, one that's kind of... So just to be clear, what I'm talking about... It would work for any polygon that is kind of... I don't want to say regular. Regular means it has the same sides and same angles, but it's not dented. So this is a convex polygon. This right here is a concave polygon. So let me draw it this way. So this right over here would be a concave, would be a concave polygon. I'm gonna draw it as a having the same number of sides. So I just kind of dented these two sides right over there. And did I do that right? Let me see. Let me do it the same number of sides. So I want to do that, that, that, that, and then I know that's the same side over there. Let me draw it like that. And then like that. This has one, two, three, four, five, six sides. This has one, two, three, four, five, six sides. This is concave. Sorry, this is convex. This is a convex polygon. This is a concave polygon. And the way I remember it is kind of caved inwards. And so what we just did would apply to any. If we're trying to find these particular external, exterior angles of any convex polygon, I afraid, I apologize ahead of time if I've confused them all, because I have a feeling that I might've. This applies to any convex polygon. And so once again, if you take this angle and add it to this angle, and add it to this angle, add it to this angle, add it to that angle, and add it to that angle. And I'm not implying that they're all going to be the same. I just drew it that way. I could show you that they are different angles. So I could say that one in green and that one in some other color, I think you get the idea. They can all be different, but when you if you shift the angles like this you'll see that they just go around the circle. So once again, they'll just add up to 360 degrees.