Are there geodesic triangles on surfaces with non-constant curvature with angle sum 180? I've been reading about differential geometry and the Gauss-Bonnet theorem to write a paper for my geometry class and am interested specifically in geodesic triangles on surfaces. 
I was wondering if it is possible to create a geodesic triangle on a surface with non-constant curvature such that the interior angles sum to $π$, $$\theta _{1} + \theta_{2}+\theta_{3}= \pi,$$ even though the Gaussian curvature $K \neq 0.$
By this I mean, is it possible to place part of the triangle on a positive curvature section of the surface and part of it on a negative curvature section of the surface so that one or two vertices are affected by the positive curvature and the other two or one vertices are affected by the negative curvature, causing the interior angle sum to still be $π$? 
For example, here's a surface with positive and negative curvature from Kristopher Tapp's Differential Geometry of Curves and Surfaces. Can the top triangle be moved downward so its bottom two angles, $\theta_{1}$ and $\theta_{2}$, are on positive curvature and the top angle $\theta_{3}$ is on negative curvature so that $\theta_{1} + \theta_{2} + \theta_{3} = \pi$ and its sides are still geodesics?
If so, does anyone have any resources they know of that I could read that specifically talk about this?

Just consider your favorite flat triangle T in a euclidian plane. the consider a $C^\infty$ fonction $f: T\to \bf R$ wih is $0$ in the neighbourhood of the boundary of $T$. Now the triangle which is the graph of this fonction ha $>0$ curvature near the point where $f$ is maximal or minimal. It cannot be everywhere positive , because due to Gauss Bonnet the integral of the curvature is $0$. The boundary is still geodesic as you did not change the metric in near it, and the angles did not change.
