As we all know, the angles in all triangles in Euclidean geometry must add up to $180^\circ$. As some of us may know, this is not true in non-Euclidean geometries; for example, on the surface of a sphere, the angles in a triangle can be between $180^\circ$ and $540^\circ$, while on a hyperbolic paraboloid (a "saddle"), it can be anywhere between $0^\circ$ and $180^\circ$. This is just one property out of many that involve triangles in non-Euclidean geometries.
A Google search on this topic brought me too broad a range of references and sources. I understand that this has been studied on and worked on a lot. What other properties of triangles exist that are contrary to normal Euclidean geometry?