I am looking for a generalization of the result in plane geometry that triangles are determined up to isometry by 3 parameters in the following arrangements: side-side-side, side-angle-side, and angle-side-angle. This consists of the following sub-questions:
(1) For a given n, is the minimum number of parameters (side lengths and face angles) required to uniquely identify an $n$-simplex up to isometry equal to $\binom{n+1}{2}$?
(2) For a given $n$, let $m$ be the minimum number of side lengths and face angles required. How many unique arrangements of $m$ side lengths and face angles satisfy this property? For example, for $n=2$, $m=3$, and there are 3 unique arrangements.
(3) What restrictions on the arrangement of $m$ side lengths and face angles are sufficient to ensure an $n$-simplex is uniquely determined?
I proved (very clunkily) a special case of this for an n-simplex with $\binom{n}{k}$ edge lengths and $n$ face angles under particular restrictions for a larger problem, but if this is a known result in a field such as rigidity theory or combinatorial geometry I would much rather cite that.
I'm also curious if there is a formulation of this for side lengths and dihedral angles or vertex angles.