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# 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 in particular exterior angles of the hexagon so that this angle equaled A, this angle B, C, D and E. 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 angle. So it was a bit of an involved process. But I want to show you in this video that 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 polygon (if you're picking these particular exterior angles I should say) and so the way to think about it is you can just redraw the angles. So lets just draw each of them, so let me draw this angle right over here, we'll call it angle A or the measure of this angle's A, either way let me draw right over here. So this going to be a convex angle right over here it's going to have a measure of A, now let me draw angle B, angle B, and i going to draw adjacent to angle A, and what you could do is just to think about it maybe if we draw a line over here, if we draw a line over here that is parallel to this line then the measure over here would also be B,because this is obviously a straight line, it would be like transversal, this of course a responding angles, so if u want to draw adjacent angle, the adjacent to A, do it like that, or whatever angle this is the measure of B and now it is adjacent to A, now let's draw the same thing to C We can draw a parallel line to that right over here. And this angle would also be C and if we want it to be adjacent to that, we could draw it there, so that angle is C C would look something like this, like that then we can move on to D, once again we do it in different color, you could do D, right over here or you could shift it over here it'll look like that, or shift over here, it'll look like that If we just kept thinking of parallel, if all of this line were parallel to each other So, let's just draw D like this, so this line is going to parallel to that line Finally, you have E, and again u can draw a line that is parallel to this right over here and this right over here would be angle E or you could draw right over here, 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 going all the way around the circle, either way it could be going clockwise or it could be counter clockwise but it will going all the way around the circle. And some of this angle, A+B+C+D+E is just going to be 360 degree And this is work for any convex polygon, and when I say convex polygon I mean it is not that dented words Just to be clear what I'm talking about, it would work for any convex polygon that is kind of I don't want to say regular, regular means it has the same size and angle, but it is not dented, this is a convex polygon. This right here is a concave polygon Let me draw this, right this way, so this would be a concave polygon Let me draw as it having the same number of side, So i just going to dent this two sides right very here. Is it right? Let me do the same number sides, So i do that, that, that, that and then that's the same side over there, Let me do that and then like that. This has 1,2,3,4,5,6, sides and this has 1,2,3,4,5,6 sides. This is concave, sorry this is a convex polygon, this is concave polygon, All you have to remember is kind of cave in words And so, what we just did is applied to any exterior angle of any convex polygon. I Am a bit confused. This applied to any convex polygon and once again if you take this angle and added to this angle and added to this angle, this angle, that angle and that angle and I'm not applying that all It's going to be the same and I just drew it in that way I could show u that they are different angles, i could say that one green, and that one some other colour they can all be different but if you shift the angle like this you can see that they just go round the circle. So, once again, I'll just add up to 360 degrees