Answer
The maximum sum for the angles in a triangle on a sphere is $540^{\circ}$.
Work Step by Step
The largest sum is achieved for an equilateral triangle whose vertices each lie just a hair above the equator. The angle at each vertex of the triangle has a limiting value of $180^{\circ}$.
The three $180^{\circ}$ angles in the triangle would have a maximum sum of $540^{\circ}$.
