Do I have enough information to solve for the angle of a plane intersecting two cones?

Question

I've got a problem where I need to determine the angle of a plane intersecting two cones, and all I have are the measurements shown, a, b, c, and d. Is that enough information to determine the angle?

Answer 1

Think about it this way: Draw an arbitrary angle $\theta$. Use dice or some other random means to choose $a$ through $d$ (potentially satisfying some implicit inequalities), and mark these lengths on the legs of that angle. Connect them and intersect those connections with the vertical line through the apex of the angle. Does the resulting figure satisfy all the constraints from your given problem?

Do the randomized dice rolls convey any information about the angle that you picked independently? Of course not. So it would seem that unless there is something you didn't mention (or I failed to read), the information provided can be consistent with a (possibly bounded) range of angles, and is not enough to pick a single angle.