I need to find the red angles: $\angle(AOB)$, $\angle(BOC)$, $\angle(COD)$, $\angle(AOD)$.
The green angles are known: $\angle(ABC)$, $\angle(BCD)$, $\angle(CDA)$, $\angle(DAB)$.
Is there any general approach to find the red angles? I assume it should be.
If it's impossible, then I'm ready to consider the general approach for the other more detailed case, when I know a little more angles: $$ \angle(ABC), \angle(BCD), \angle(CDA), \angle(DAB), \angle(OAD), \angle(OAB), \angle(OCB), \angle(OCD) $$
The picture is pretty same:
If it's still not enough, I can add the limitation to convex quadrilaterals and maybe some other restrictions.
The preferable result is a general formula for the angles without any trigonometry manipulations.
As anonymous points out, your problem cannot be solved. Consider this example: