Well, I think the title already explains my question. Given a sphere and an ordered sequence of inner angles ($\alpha$, $\beta$, $\gamma$, $\delta$) how many spherical quadrangles do there exist that have that sequence as angles and the added property that three of the edges need to have the same size and the fourth edge needs to have a different size.?
I was told that this on the sphere there might be several quadrangles with quite different appearances, but I can't find any reference explaining this in more detail or bounding the number of possible quadrangles.
You can assume that the angles satisfy the conditions necessary to be the angles of a spherical quadrangle.