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Triangle angle challenge problem

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 like they don't give us any other information they don't even give us any particular angles they don't give it you know they don't start itself 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 are all there each there 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 so let's say we have this we already got to letter 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 into your angle H let's call this one eye and let's call this one J and so this some of these particular exterior angles a is now the same thing 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 are 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 see we can write as 180 minus I so plus 180 minus I and then D we can write 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 plus 180 minus F and so what we're left with if we add up all the 180s we have a 180 five times so this is going to be equal to five times 100 in 85 times 180 which is what like 900 and then you have minus G minus H minus I minus J minus F or we could write that as minus minus I'll try to do the same colors G Plus H I'm kind of factoring out this negative sign G Plus H do the same colors G Plus that's not the same color G Plus H plus I plus I plus J plus J plus F plus F and the whole reason why I did this and why this is interesting now is that we've expressed this first thing that we need to figure out we have expressed it in terms of the sum of the interior angles so this is going to be 900 minus all of this business so this is 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 if we can 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 can do that from any side let me just say let's let's say they're all coming out of that side right over there so there I have divided it let me do this in a neutral color do it in white so 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 till you make that useful we have to express these angles in terms of the sums or in terms of angles that we can figure out based on the fact that the sums of the angles or the measures of an 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 let's divide F into two other angles or two other measures of angles I should say so let's call it call f is equal to let's say that f is equal to c we've already gone as high as let's see ABCD EFG 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 that way we've split it up into two part in two angles of these other triangles and then we can do that with J as well we can do that with J as well we can say because J once again is that whole thing so we could say that J is equal to let's see we already used L so let's say J is equal to M plus n so 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 parts of already an angle 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 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 could use the fact that those that for any one triangle they add up to 180 degrees so let's do that so this this expression right over here 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 900 900 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 instead of an H I can write that H is Oh plus P plus Q plus o plus P plus Q and then plus I plus I I sitting right over there plus I and then plus J and I kind of messed up the colors or the I with the magenta will go with I and then J is this expression right over here so J is equal to M plus n so plus M plus n instead of writing a right there and then finally we have our F and F 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 + k + 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 + o plus K is 180 degrees so G let me do this in a new color so for this triangle right over here this triangle right over here we know that G + o plus K is 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 P plus L plus M is 180 degrees so you take those out and just you say you know that sum is going to be equal to 180 degrees and then finally this is the homestretch here we know that Q plus n plus I is 180 degrees in this last triangle Q plus n plus I Q plus n plus I is 180 degrees that those are 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 is sex you're gonna 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