Challenging triangle angle problem Interesting problem finding the sums of particular exterior angles of an irregular pentagon
Challenging triangle angle problem
- Now this looks like an interesting problem.
- We have his polygon. It looks like a pentagon right over here,
- has five sides it's an irregular pentagon; not all
- the sides looked to be the same length, and the sides are kind of
- continued on and we have this particular exterior angles of
- this pentagon and what we're asked is what is the sum of all
- these exterior angles, and it's kind of daunting 'cause like
- they don't give us any other information,
- they don't even give us any particular angles,
- they don't even give us -- you know 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 all, they're each supplementary to some interior angle,
- so maybe if can express them as the function of the interior angles
- we can maybe write this problem in a way maybe 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 G,
- let's call this H, let's call this one I, and F let's call this one J
- And so this some of this 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 we can write in terms of
- this interior angle It's gonna 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 can 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 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 right 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
- We have a 180 five times
- So this is going to be equal to five times 180
- which is what like 900
- and then we 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 we 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 have expressed it
- The first thing we need to figure out we have expressed it
- In terms of the sum of the interior angles
- So this is gonna be 900 minus all of this business
- So this is this is 900 all of this business
- Which is the sum of all interior angles
- So this is the sum of all interior angles
- So it seems like we've made a little progress if we get
- 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 over lapping triangles
- and so we can do that from any side let me just say
- let's say they are all coming out of that side right over there
- so there I have divided it let me do this is a in a neutral color
- I'm doing 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 it in three non overlapping triangles
- And the reason why I did that, the reason why this is valuable
- Is that we know what the sum of the angles of the triangles add up to
- And so to 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 let's call F is equal to let's say
- that F is equal to so we've already gone as high as let's see
- ABCDEFGHIJ we haven't used K yet
- So let's say that F is equals 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
- Into part into angles of these other triangles
- And then we can do that with J as well with J as well
- We can say coz J once again is that whole thing
- So we could say that K is equal to let's see we already used L
- So let's say J is equals to M plus N so J is equals to M plus N
- And then finally we could split up H
- H is up here remember this is 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 or already an angle of a triangle
- I wanna 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 gonna be G
- G is that angle right over here we didn't make any substitutions
- So this is going to be G
- And you want me to 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 instead of an H instead of an H I can write
- that H is O plus P plus Q plus O plus P plus Q and then plus I
- plus I, I is sitting right over there, plus I
- plus J...
- I kinda messed up the colors, the magenta will go with I
- and J is the expression right over here,
- so J is equal to M plus N
- so plus M plus N instead of righting J right there
- and then finally we have our F
- and F, we've already seen,
- is equal to K plus L, plus K plus L
- so once again I just rewrote
- this part right over here
- in terms of these component agles.
- And now something very interesting
- is going to happen
- because we now
- what these sums are going to be
- because we know that G plus K plus O
- is a 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 a 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 plus O plus K
- are going to be equal to 180 degrees.
- If we cross those out
- we can write 180 instead
- and then we also know
- let me see, I'm definiftely out of colors here...
- 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 you say,
- You know that the sum
- is going to be equal 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.
- Q plus N plus I
- Q plus N plus I is 180 degrees.
- THose three are also gonna be 180 degrees
- and so now we know
- the sum of the interior angles
- for this irregular pentagon,
- which is actually 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 the 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.
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