I'm trying to prove that the interior angles on a spherical triangle sum to [itex] \pi + (A)/(a)^2 [/itex] where A is the area of the triangle and a is the radius of the spherical space.(adsbygoogle = window.adsbygoogle || []).push({});

I think I know how to prove it, but there is one part that has me stumped.

I'm using the General Relativity book by James Hartle which states that any geometry can be built using line elements. The book claims that you can use, in Euclidean geometry, the definition that [itex] \theta = (\delta C / r) [/itex] ( [itex]\delta C[/itex] is the arc length)to prove that the sum of the angles of a triangle = [itex] \pi [/itex]. I reason that I can use that proof with the modifications for spherical space to prove that the interior angles on a spherical triangle sum to [itex] \pi + (A)/(a)^2 [/itex]. My problem is, I haven't been able to prove that a planar triangle's angles sum to 180 using the definition for an angle!

So my questions are:

1. Is this how I should be trying to make my proof?

2. If so, how do I prove that the angles of a triangle sum to pi on a plane just using [itex]C = 2 \pi r [/itex] and [itex] \theta = (\delta C / r) [/itex]?

Thanks

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# Non-Euclidean Geometry question

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