It seems easy to show by Gauss-Bonnet theorem that a surface which has two families of geodesics with fixed angle $\theta$ must be locally flat, i.e. its Gauss curvature $K=0$. But to show it is in fact a developing surface, we must to show it is also a ruled surface, which, on the other hand, seems nonsense for me.
My quesion is that is there any counter-example?