I think you're saying something a little too strong. I think such a measurement does provide some information about the dimensions of the quadrilateral being me

I think you're saying something a little too strong. I think such a measurement does provide some information about the dimensions of the quadrilateral being measured. I think it would be possible with a bounded number of observations to measure the lengths purely by measuring the angles. You would have to shift your perspective each time, but it shouldn't take too many. So, each such measurement must contribute a non-zero amount of information about the dimensions. Either that, or i'm misinterpreting your first sentence.