# 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))

• Opposite sides don't need to have equal length. Commented Sep 10, 2011 at 2:32
• @André: For $n>2$. :)
– Blue
Commented Sep 10, 2011 at 3:04
• A countercounterexample. Commented Sep 10, 2011 at 3:15
• Did you mean to write \neg x ($\neg x$) instead of \not x ($\not x$)?
– user856
Commented Sep 10, 2011 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
Commented Sep 10, 2011 at 6:19

## 1 Answer

For (c), consider a non-square rectangle. For (b) if you define opposite sides as half the sides around the polygon, you should have them equal by the theorem that corresponding angles with a transversal of parallel lines are equal.

• A rectangle satisfies (a), (b), and (c), no? Commented Sep 10, 2011 at 5:03
• @Gerry Myerson: Not (c), that's why I said a non-square rectangle. Maybe you missed the prime in opposite sides are different lengths. Commented Sep 10, 2011 at 5:12
• (c) is "opposite sides have equal length". This is true of rectangles, square or not. (c') is "opposite sides have different lengths". Well, that's how I read it. But you are quite right that if opposite sides are parallel then opposite angles are equal, whether (c) holds or not. Commented Sep 10, 2011 at 5:48
• Yes, the negation of (c) is (c'), "not all opposite sides have equal lengths". Your comment re "if opposite sides are parallel then opposite angles are equal" becomes obviously true via another comment's suggestion to "draw the diagonal" (i.e. call upon Euclid's Proposition 29, "A straight line falling on parallel straight lines makes alternate angles equal to one another..."). Commented Sep 10, 2011 at 9:04
• You don't have to draw any diagonals, you can just extend the four sides in question (ignoring any other sides) until you get a parallelogram. Commented Sep 10, 2011 at 10:29