We study the interior angle sums of translation and geodesic triangles in the universal cover of real 2 × 2 matrices with unit determinant, as, a Thurston geometry denoted by P of SL 2 (R) geometry. We prove that the angle sum 3 i=1 (α i) ≥ π for translation triangles and for geodesic triangles the angle sum can be larger, equal or less than π.