Showing that the sum of the angles of a hyperbolic triangle are less than $\pi$

Question

Using the law of cosines for sides:

$\cosh(a) = \cosh(b) \, \cosh(c) - \sinh(b) \, \sinh(c) \, \cos(\alpha) $

I have to show that

$\alpha + \beta + \gamma < \pi$

Unfortunately I find no ansatz for this question.