# Can we define the shape of a polygon in the plane using its interior angles?

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?

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I'm not sure what meaning of "graph" you're using here; perhaps a planar straight line graph?

If we're speaking of polygons in the plane, as your figure suggests, any polygon can be triangulated, and specifying the interior angles of the triangles in the triangulation determines the polygon up to scaling. In particular, you only need to insert one diagonal into your quadrilateral.

In fact, I'm guessing triangulation is precisely the construction that gives your $2n-3$ figure when dealing with edge lengths too.

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