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I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral?

More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?

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No, when we refer to an n-gon, we can't assume that any case of a 180 degree "angle" is an angle, as any side of an n-gon would have infinitely many such "angles" - it would be a nonsense definition. We're interested in angles smaller than 180 degrees.

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An $\aleph_1$-gon? ;) –  Rex Oct 26 '12 at 0:06
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Here are a few sources that agree: mathworld.wolfram.com/Polygon.html. Here's one that seems to disagree: proofwiki.org/wiki/Definition:Polygon. –  Rex Oct 26 '12 at 0:16
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@Rex: Even going by your second link, a triangle isn't a quadrilateral, because it's composed of different line segments. It may occupy the same points on the plane as certain quadrilaterals, but it's still a distinct object... –  Micah Oct 26 '12 at 0:44
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If the $n$-gons are allowed to flex (like a planar linkage) it is better to allow degenerate cases with two or more consecutive sides collinear to count as having $n$ sides, because these will be intermediate positions in a motion between convex and concave positions of the linkage. Else you are forced into odd phrases like the flexing polygon having $n$ sides at times $t < t_0$ and instantaneously transforming into an $(n-1)$-gon at time $t_0$ before changing back to its $n$-gonal self at times $t > t_0$.

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