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Triangle angle challenge problem

Interesting problem finding the sums of particular exterior angles of an irregular pentagon. Created by Sal Khan.

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Video transcript

Now this looks like an interesting problem. We have this polygon. It looks like a pentagon right over here, has five sides. It's an irregular pentagon. Not all the sides look to be the same length. And the sides are kind of continued on. And we have these particular exterior angles of this pentagon. And what we're asked is, what is the sum of all of these exterior angles. And it's kind of daunting, because they don't give us any other information. They don't even give us any particular angles. They don't start us off anywhere. And so what we can do, let's just think about the step by step, just based on what we do know. Well, we have these exterior angles. And these exterior angles, they're each supplementary to some interior angle. So maybe if we can express them as a function of the interior angles, we can maybe write this problem in a way that seems a little bit more doable. So let's write the interior angles over here. We already got to e. So let's call this f, this interior angle f. Let's call this interior angle g. Let's call this interior angle h. Let's call this one i. And let's call this one j. And so this sum of these particular exterior angles, a is now the same thing as 180 minus g, because a and g are supplementary. So a is 180 minus g. And then we have plus b. But b we can write in terms of this interior angle. It's going to be 180 minus h, because these two angles once again, are supplementary. We do that in a new color. So this is going to be 180 minus h. And we could do the same thing for each of them. c, we can write as 180 minus i, so plus 180 minus i. And then d, we can write as 180 minus j, so plus 180 minus j. And then finally, e, I'm running out of colors, e, we can write as 180 minus f, so plus 180 minus f. And so, what we're left with, if we add up all the 180s, we have 180 5 times. So this is going to be equal to 5 times 180 which is what, 900. And then you have minus g, minus h, minus i, minus j, minus f. Or we could write that as minus-- I'll try to do the same colors-- g plus h-- I'm kind of factoring out this negative sign-- g plus h-- I'll do the same color as g, that's not the same color-- g plus h, plus i, plus j, plus f. And the whole reason why did this and why this is interesting now, is that we've expressed this first thing that we need to figure out. We've expressed it in terms of sum of the interior angles. So it's going to be 900 minus all of this business. So this is 900 minus all of this business, which is the sum of the interior angles. So this is the sum of the interior angles. So it seems like we've made a little progress, at least if we can figure out the sum of the interior angles. And to do that part, I'll show you a little trick. What you want to do is divide this polygon, the inside of the polygon, into three non-overlapping triangles. And so we could do that from any side. Let's say that they're all coming out of that side right over there. So there. I have divide it-- let me do this in a neutral color, I'm doing it in white-- so that's one triangle right over here. And then let me make another triangle just like that. So there you go. I've divided into three non-overlapping triangles. And the reason why I did that, the reason why this is valuable, is we know what the sum of the angles of a triangle add up to. And so to make that useful, we have to express these angles in terms of angles that we can figure out based on the fact that the sums of the angles, or the measures of the angles in a triangle add up to 180. So g is kind of already one of the angles in the triangle. F is made up of two angles in the triangle. So remember, f is this entire angle right over here. So let's divide f into two other angles, or two other measures of angles, I should say. So let's say that f is equal to-- we've already gone as high as, let's see, a, b, c, d, e, f, g, h, i, j-- we haven't used k yet. So let's say that f is equal to k plus l. It's equal to the sum of the measures of these two adjacent angles right over here. So f is equal to k plus l. So that way we've split it up into angles of these other triangles. And then we can do that with j as well, because j, once again, is that whole thing. So we could that j is equal to-- let's see, we already used l. So let's say j is equal to m plus n. And then finally, we can split up h. h is up here. Remember, it's this whole thing. Let's say that h is the same thing as o plus p plus q. This is o, this is p, this is q. And once again, I wanted to split up these interior angles if they're not already an angle of a triangle. I want to split them up into angles that are parts of these triangles. So we have h is equal to o, plus p, plus q. And the reason why that's interesting is now we can write the sum of these interior angles as the sum of a bunch of angles that are part of these triangles. And then we can use the fact that, for any one triangle, they add up to 180 degrees. So let's do that. So this expression right over her is going to be g. g is that angle right over here. We didn't make any substitutions. So it's going to be g-- actually, let me write the whole thing. So we have 900 minus, and instead of a g, well, actually I'm not making a substitution, so I can write g plus, and instead of an h I can write that h is o plus p plus q. And then plus i. i is sitting right over there. Plus i. And then plus j. And I kind of messed up the colors. The magenta will go with i. And then j is this expression right over here. So j is equal to m plus n instead of writing a j right there. And then finally, we have our f. And f, we've already seen, is equal to k plus l. So plus k plus l. So once again, I just rewrote this part right over here, in terms of these component angles. And now something very interesting is going to happen, because we know what these sums are going to be. Because we know that g plus k plus o is 180 degrees. They are the measures of the angle for this first triangle over here, for this triangle right over here. So g plus o plus k is 180 degrees. So g-- let me do this in a new color. So for this triangle right over here, we know that g plus o plus k are going to be equal to 180 degrees. So if we cross those out, we can write 180 instead. And then we also know-- let me see, I'm definitely running out of colors-- we know that p, for this middle triangle right over here, we know that p plus l plus m is 180 degrees. So you take those out and you know that sum is going to be equal to 180 degrees. And then finally, this is the home stretch here. We know that q plus n plus i is 180 degrees in this last triangle. Those three are also going to be 180 degrees. And so now we know that the sum of the interior angles for this irregular pentagon-- it's actually going to be true for any pentagon-- is 180 plus 180 plus 180, which is 540 degrees. So that whole thing is 540 degrees. And if we want to get the sum of those extra angles, we just subtract it from 900. So 900 minus 540 is going to be 360 degrees. And we are done. This is equal to 360 degrees.