2. Applying the Gauss-Bonnet theorem for Euclidean space is much more simple than its nonzerical counterparts. Consider a triangle with angles $a, b, c$.
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 $\pi-a+\pi-b+\pi - c=3\pi-a-b-c$. So we end up with $2\pi-(3\pi-a-b-c)=0$ or $a+b+c=\pi$, a result old Euclid would agree with.