In the paper 'Examples of spherical tilings by congruent quadrangles' by Ueno and Agaoka, I came across the following claim (p.142): the sum of two angles in a spherical triangle is less than the sum of $\pi$ and the third angle.
They give this without any reference, and I can't find a proof for this by googling the net. The only thing I manage to proof is the following: the sum of two angles in a spherical triangle is greater than the $\pi$ minus the third angle.