## anonymous one year ago 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?

1. 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

2. ganeshie8

like for example |dw:1436929826981:dw| ?

3. anonymous

Yeah, exactly.

4. ganeshie8

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

5. ganeshie8

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

6. ganeshie8

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

7. ganeshie8

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

8. 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.

9. anonymous

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

10. ganeshie8

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

11. ganeshie8

|dw:1436930692495:dw|

12. anonymous

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

13. anonymous

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

14. ganeshie8

sry forgot angle $$f$$ : |dw:1436931109510:dw|

15. ganeshie8

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

16. anonymous

Ah, okay :)

17. anonymous

No worries

18. ganeshie8

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

19. 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}

20. anonymous

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

21. ganeshie8

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

22. anonymous

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

23. 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|

24. 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

25. 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

26. 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.

27. ganeshie8

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

28. 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 :)

29. ganeshie8

Are you doing axiomatic geometry ?

30. ganeshie8

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

31. 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.