You only need elementary trigonometry and scalar products
Given any right-angled triangle $\Delta ABC$ on a sphere of unit radius, right angled at $A$, and with lengths of sides $a, b$ and $c$, then Pythagoras' Theorem in Spherical Geometry is $$\cos a = \cos b \cos c.$$ Prove this result.
Find a triangle containing three right angles on the surface of a sphere of unit radius. What are the lengths of the sides of your triangle?
Use the Pythagoras' Theorem result above to prove that all spherical triangles with three right angles on the unit sphere are congruent to the one you found.