How would you prove that the sum of the interior angles of a n-gon is 180(n-2) degrees in the case that at least one of the interior angles is greater than 180 degrees?

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- anonymous

Basically how would you prove the case where its a concave polygon? You wouldn't always be able to create a line segment that can connect any two vertices

- ganeshie8

like for example
|dw:1436929826981:dw|
?

- anonymous

Yeah, exactly.

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## More answers

- ganeshie8

It will be easy to prove if we consider "signed" angle for sum of exterior angles of a polygon

- ganeshie8

I'm thinking of using the fact that the sum of "signed" exterior angles of any polygon add up to 360

- ganeshie8

count counterclockwise as positive and clockwise as negative
|dw:1436930131588:dw|

- ganeshie8

\[\color{blue}{a-b+c-d+e = 360}\]

- anonymous

How can we then extend that to what the interior angles would be? If we had triangle constructed, I know the exterior angle is equal to the sum of the other 2 interior angles. But I don't think we can make a triangle construction, if that would even help.

- anonymous

Well, at least make a triangle construction that would work in any arbitrary case, but Im not positive on that.

- ganeshie8

We may use this :
\[\text{interior angle} + \text{exterior angle} = 180\]

- ganeshie8

|dw:1436930692495:dw|

- anonymous

Alrighty, Ill mess with that idea for a minute or two.

- anonymous

And then we would need some angle f at the starting point, right?

- ganeshie8

sry forgot angle \(f\) :
|dw:1436931109510:dw|

- ganeshie8

I also don't have the proof yet, still trying..

- anonymous

Ah, okay :)

- anonymous

No worries

- ganeshie8

If we agree that above theorem works, looks we're done!

- ganeshie8

\[\text{interior angle} + \text{exterior angle} = 180\]
\[\sum\limits_{1}^{n}\text{interior angle} + \text{exterior angle} = \sum\limits_{1}^n180\]
\[\begin{align}\sum\limits_{1}^{n}\text{interior angle} &= \sum\limits_{1}^n180 - \sum\limits_{1}^n\text{exterior angle} \\~\\
&=180n - 360\\~\\
&=180(n-2)
\end{align}\]

- anonymous

Well that just dumbed everything down exponentially, haha. Yeah, that's very basic, concise, makes sense :) Awesome, thanks :3

- ganeshie8

XD that signed angles thingy is a bit confusing though, it always trips me and makes me doubt the exterior angle sum thm...

- anonymous

Actually yeah, what exactly did you mean when you said "signed" angles?

- ganeshie8

for convex polygon, it is easy to see that the external angles must add up to 360 for one complete revolution :
|dw:1436932101810:dw|

- ganeshie8

for definiteness, lets agree that we always measure angle with respect to that blue extended line,
|dw:1436932485018:dw|
if the polygon is convex, notice that all the angles will be measured counter clockwise. so we don't have to worry about "sense" here

- ganeshie8

But if we allow the polygons to be concave, some angles will be measured counterclockwise and some clockwise
|dw:1436932611913:dw|
we need to distinguish between these two

- anonymous

Oh, duh. Signed as in positive or negative with respect to the direction of rotation. I have no idea why I was thinking of signed as in labels or something else. Brain fart, lol.

- ganeshie8

Yes haha thats it I guess. For "concave polygons", most of the usual theorems work if we replace "angle" by "signed angle"

- anonymous

Gotcha. This course so far has basically been a bunch of random proofs and then homework that is not aided at all by the proofs done in class or in the textbook, haha. In the end, though, I think being forced to investigate it so much will help me out (gotta find some positive). Thanks again :)

- ganeshie8

Are you doing axiomatic geometry ?

- ganeshie8

I remember ikram doing this course sometime ago and it was very interesting

- anonymous

It doesnt really have a specific name. The course itself is called "College Geometry". I guess right now we're just dealing with Euclidean stuff, but I know we will be touching on spherical and hyperbolic geometry at some point as well. I don't believe it ever becomes anything beyond calc 1 integration. Although I think if we get far enough (doubt it), there is a chapter that requires line integrals.

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