Gauss-Bonnet help

We found out how to figure out Gauss-Bonnet for spherical geometry in our class, which was using the sum from 1 to n of (alpha + beta + gamma) - pi and turning this into 2piX(x).

We are supposed to try to find Gauss-Bonnet for hyperbolic and Euclidean. I think I can get it for hyperbolic, it seems very similar to spherical. However, Euclidean he said to work backwards? I'm confused with this, it seems like it is trickier and I don't know where to start.

Applying the Gauss-Bonnet theorem for Euclidean space is much more simple than its nonzerical counterparts. Consider a triangle with angles .
We know the Gaussian curvature is zero, so all we have to do is sum the jumps of the tangent at each vortex, which ofcourse is . So we end up with or , a result old Euclid would agree with.