Theorem: For a spherical triangle with angles $\alpha, \beta, \gamma$ on a unit sphere: $A = \alpha + \beta + \gamma - \pi$.

Proof:

Consider the triangle $\triangle ABC$ with angles $\alpha$ (at $A$), $\beta$ (at $B$), $\gamma$ (at $C$).

Step 1: Extend each side to a full great circle, dividing the sphere into regions. The triangle $\triangle ABC$ and its antipodal triangle $\triangle A'B'C'$ together with four additional triangles tile the sphere.

Step 2: Consider the lune formed by great circles through $A$, $B$, and $C$. The lune with angle $\alpha$ at $A$ has area $2\alpha$ (since total sphere area is $4\pi$ and angle $2\pi$ gives area $4\pi$).

Step 3: Three lunes (one for each vertex) cover the sphere, with the triangle counted multiple times. Careful accounting:

Step 4: Solving for $A$: $A = \alpha + \beta + \gamma - \pi$. ∎