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There are a number of claims from elementary school that I just remembered I don't actually mathematically know. Let's start with some specific examples and perhaps the rigorization will inspire me to know how to do it in any questions I remember in the future. For example, it is said that "the sum of the angles of a polygon in 2D with $k$ sides is $(k-2)\pi$. I looked up the phrase "interior angle" and found this non-rigorous article.

I am hoping that someone will either rigorously develop the requisite ideas, or if complicated, mention their names/resources where I could understand this. Since I think it may be relevant, I have all the knowledge of a first year graduate course in differential/algebraic topology, and some differential geometry knowledge as well, but only passing. (To be more precise, I understand some basic Riemannian geometry like Riemmanian metrics, geodesics, and some basics about the curvature tensor, but not more. Certainly not a first course equivalent in Riemannian geometry.)

I was told by a peer that there are many ways of rigorizing angles when it becomes insufficient to define angle in $\mathbb{R}^k$ by $\cos^{-1}(\langle u, v\rangle /\|u\|\|v\|)$. This is also what I've considered angle to mean in the context of manifolds, where one has lengths and angles via a Riemannian metric. If someone could give some of the other definitions of angle (or length if any exists other than the notion of arclength for curves in a manifold, and the notion of minimal distance similarly, as well as the obvious algebraic notion of length for tangent space vectors) I would greatly appreciate it.

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Since you put it in the context of Riemannian manifolds, note that the statement "the sum of the angles of a polygon in 2D with k sides is (k−2)π" is not true for a general manifold. It is true in the Euclidean space, though. Indeed, the sum of internal angles is closely connected to the concept of (inner) curvature of a manifold. – celtschk Jun 2 '13 at 1:41
@celtschk Yes, thank you for clarifying. I was meaning to suggest that perhaps the rigorization of the claim for flat Euclidean space is done by specializing certain ideas from manifold theory. I was unaware there were multiple notions of curvature. (presumably inner, outer, and maybe other?) I'm curious about this. – Jeff Jun 2 '13 at 1:54
There's indeed inner and outer curvature (I'm not aware of any other, but that might just be a lack of knowledge on my side). Basically, the inner curvature is what you can detect by only considering quantities in the manifold itself, while the outer curvature is dependent on its embedding in a outer space (and doesn't exist if there's no such embedding defined). Think of a piece of paper that you bend in space without ripping or folding it. Any lengths or angles on the paper are of course unaffected, so you get no inner curvature. But it's curved in space, so there's an outer curvature. – celtschk Jun 2 '13 at 2:06
“If someone could give some of the other definitions of angle”: Are you interested in angle and length definitions using Cayley-Klein metrics? These come from projective geometry and not differential geometry, so they won't apply to manifolds except for those with constant Gaussian (i.e. inner) curvature. – MvG Jun 3 '13 at 9:06
See e.g. Euclid's Elements by Byrne p. 33 for angle sum in a triangle. Similar deductions based on Euclid's axioms are possible for other polygons as well. So instead of transferring the problem to some algebraic setup, one could well base this on geometric axioms alone. I guess there should be even formal proofs out there, although they might be difficult to read. Perhaps a formal system for Euclid's Elements might be a good starting point, although I haven't read that. – MvG Jun 3 '13 at 9:34

If you want to define angles purely axiomatically, it is done quite nicely for instance here (freely available):

A. Petrunin, Euclidean Plane and its Relatives; a minimalistic introduction

Other sources:

R. Hartshorne, Geometry: Euclid and Beyond. Springer Verlag, (Undergraduate Texts in Mathematics), 1997.

E. Moise, Elementary Geometry From An Advanced Standpoint, 3rd ed. Boston: Addison-Wesley. 1990.

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