The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.
2.
When " D " is external to the segment " BC ", directed line segments and directed angles must be used in the calculation.
3.
There are also 14 axioms, four giving the properties of the distance function, four describing properties of lines, four discussing angles ( which are directed angles in this treatment ), a similarity axiom ( essentially the same as Birkhoff's ) and a continuity axiom which can be used to derive the Crossbar theorem and its converse.
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