Geodesic Curvature of a curve on a flat surface

[sel]

Sorry if this ends up being a naive question, but I have just a little conundrum. I'm dealing with curves in R² and the Gauss-Bonnet theorem is a very useful result with what I'm currently doing, what with Gaussian curvature of a flat surface being zero, which is all fine

http://mathworld.wolfram.com/Gauss-BonnetFormula.html

To proceed with my problem, I need to show that the geodesic curvature of a curve in R² is the same as the standard curvature of it. I can sort of understand how it is the case from its definition but can't show it, if you know what I mean.

Thanks for all the help guys x

 

[pat_connell]

The main idea to prove this is to use the Gauss-Bonnet theorem. Since the Gaussian curvature of a flat surface is zero, the total curvature around any closed curve on the surface must also be zero. That is, the sum of the geodesic curvature and the standard curvature must be zero. Therefore, if we know the value of one of them, we can determine the value of the other.