I've been writing a survey on Thurston's construction of a hyperbolic structure on the complement of the figure-eight knot. At one point it is important that the sum of the angles between three hyperbolic planes (find angles between each two, add them up) is $\pi$. I want to show this using Euclidean geometry and the Poincaré disk model in 3-space.
If the spheres $S_1$ and $S_2$ have centers $O_1$ and $O_2$, respectively and $X$ is a point on the intersection, then I can define the angle $\phi_{1,2}$ between $S_1$ and $S_2$ as the unique angle between $O_1X$ and $XO_2$. Is it true that if three spheres $S_1,S_2,S_3$ pairwise intersect, then $\phi_{1,2}+\phi_{2,3}+\phi_{1,3}=\pi$?