I am trying to characterize the set of angles in a (convex) quadrilateral that distinguishes it from any other quadrilateral that is not similar to it. Such a set will be said to uniquely define a quadrilateral up to similarity. For the rest of the question, this will be the definition of "uniquely defines". Also, "quadrilateral" will mean a convex one.
Absolutely no assumptions may be made on any of the angles, they are completely generic, as in the image below. Of course general truths may be deduced, for example that $y=180-a-b$. I have marked all $12$ (or $16$ with the vertices angles $A,B,C,D)$ angles for ease of reference.
For example, the set $\{ A,B,C,D\}$ does not uniquely define a quadrilateral. This MSE link Shows an example where this set is identical, but $2$ non-similar quadrilaterals can be formed from the exact same set.
Considering instead the $4$-tuple $(A,B,C,D)$ (where the order of angles matters), rather than the set, does define uniqueness in the above link's example, but still does not uniquely define a quadrilateral. Take any square with the tuple $(90,90,90,90)=\{90\}$. Then any rectangle with exactly $2$ different sides will have the exact same tuple/set, however it will not be similiar to that square. So the set/tuple $(A,B,C,D)/\{A,B,C,D\}$ do not uniquely define the quadrilaterals (up to similarity).
Just to further explain the meaning of this in terms of the sketch below, this means of course that the set $\{e+d,c+b,a+h,g+f\}$ does not define a quadrilateral. Obviously, the set $\{w,x,y,z\}=\{x,y\}=\{x,180-x\}$ neither defines a quadrilateral.
- It is important to note once again, that I am looking for uniqueness up to similarity
- The quadrilateral to start with has no special traits, it is completely generic. Separation to cases may be considered.
- I am only considering euclidean geometry here in the plane
- It may be assumed that all angles in the figure below are known
- It is best if angles only from the $16$ marked below will be included. What combination of these angles will define a quadrilateral?
I am really insisting on a set (or multiple sets) here. What I mean by that, is that given this set of angles of the quadrilateral, then another quadrilateral may not be similar to it unless it possesses the exact same set. That way, the order of the elements does not matter, because any set that differs by even one angle (or by the size of the set), cannot be similar to it anyway. If no such set exists, I'd like to know why, and rather than that I'd like to know what $n$-tuple of angles uniquely defines a quadrilateral. If non exists, then what algorithm/set of examinations will distinguish any $2$ quadrilaterals?
