Given 3 angle of spherical triangle we could find a solution for arclength of each side with the cosine rule
So, given 4 arbitrary angles, is it possible to find 4 arclength for each side in the same way? What is the formula for this solution
I try to work around by cutting a quad into 2 triangles. But I can't find a relation between diagonal cut and angle that it would be splitted. I now wonder is it actually need more argument to solve or it have a related property that I don't know about
Before doing anything, check the planar case. Suppose you specify four sides of a quadrilateral. To define the quadrilateral precisely you need to specify an angle as well, or some other piece of information that enables you to identify an angle (such as saying the quadrilateral is cyclic). There are five degrees of freedom, not four.
In the spherical case you must also specify five degrees of freedom. Unlike the planar case you can use four angles independently, as long as they sum to more than 360°; but you still need to specify a side, or some other equivalent information, as well.
Say you specify angles $A, B, C, D$ in rotational order plus side $\overline {AB}$. Extend $\overline {AD}$ beyond $D$ and $\overline {BC}$ beyond $C$ until the extensions intersect at $E$. From the angles of the quadrilateral at $A$ and $B$ and the given side you can solve $\triangle ABE$, getting its other two sides $AE, BE$ and the angle at $E$. Given the latter angle and the supplements of the quadrilateral angles at $C$ and $D$, solve for the sides of $\triangle CDE$. One side of the latter triangle is side $\overline {CD}$ of the quadrilateral. The remaining two sides of the quadrilateral are differences between collinear pairs of sides in the two solved triangles.
Good luck!