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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.

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up vote 2 down vote accepted

You can rearrange $\alpha+\beta\lt\pi+\gamma$ as $\alpha+\beta+\gamma-\pi\lt2\gamma$. The left-hand side is the spherical excess, i.e. the area of the triangle. Two lines forming an angle $\gamma$ at a vertex meet again at the antipode of the vertex; the area they enclose is $2\gamma$. The triangle lies between those two lines, so its area is less than $2\gamma$.

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