# Conditions on polygon to ensure equal interior angles or opposite sides

In following, ($x$) denotes condition $x$, and ($x$') denotes condition $\neg x$.

For a polygon of $2n$ sides, let it be given that (a) opposite sides are parallel. It is not difficult to find examples where (b') opposite interior angles differ and (c') opposite sides are of different lengths.

Example: Octagon (1,0), (2,1), (0,1), (0,2), (3,2), (4,3), (5,3), (5,0), (1,0).

I am tempted to conjecture that if the polygon is convex and opposite sides are parallel, then (b) opposite interior angles match and (c) opposite sides have equal length. But I don't see how to prove it and don't have a counterexample.

1. Does anyone know of a counterexample? Update 1: Following the first remark by André Nicolas, here's a construction of convex polygon Q where (a) and (b) hold but not (c): Let P be a regular hexagon, with one edge extending from (-5,0) to (+5,0). Intersect P with all points where -6 =< x =< 7 to form an octagon Q.

2. If the conjecture is false, what conditions would be sufficient to enforce (b) or (c) ? (It seems obvious that (a) and (b) imply (c), while (a) and (c) imply (b))

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Opposite sides don't need to have equal length. –  André Nicolas Sep 10 '11 at 2:32
@André: For $n>2$. :) –  Blue Sep 10 '11 at 3:04
A countercounterexample. –  André Nicolas Sep 10 '11 at 3:15
Did you mean to write \neg x ($\neg x$) instead of \not x ($\not x$)? –  Rahul Sep 10 '11 at 4:36
( Convex + (a) ) $\implies$ (b): draw the diagonal and consider the "Z"s (or are they "N"s?) made with the sides. –  Blue Sep 10 '11 at 6:19