I know a polygon can be well defined by specifying its edge length. By well defined, I mean the polygon can be unambiguously determined. Loosely speaking, if there are $n$ vertices, $2n-3$ critical edges can define an unambiguous polygon in 2D (rigid body).
My question is: Is it possible to define a polygon by specifying the angles between the edges so that the graph can only be determined up to a scale?
Upper left figure: For a triangle , three properly interior angles can determine the graph up to a scale. Upper right figure: But for quadrangles, the same angles may lead to non-similar quadrangles. Lower figure: However, if I introduce two bisections and specify the eight interior angles, it seems the angles can determine the shape.
Can any one recommend a book or something on this topic?