parallelogram theorems

Theorem 1.  The opposite sides of a parallelogram are congruent.

Proof.

In the parallelogram $ABCD$, the line $BD$ as a transversal cuts the parallel lines $AD$ and $BC$, whence by the theorem of the parent entry (http://planetmath.org/CorrespondingAnglesInTransversalCutting) the alternate interior angles $\alpha$ and $\beta$ are congruent. And since the line $BD$ also cuts the parallel lines $AB$ and $DC$, the alternate interior angles $\gamma$ and $\delta$ are congruent. Moreover, the triangles $ABD$ and $CDB$ have a common side $BD$. Thus, these triangles are congruent (ASA). Accordingly, the corresponding sides are congruent:  $AB=DC$  and  $AD=BC$.∎

Theorem 2.  If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

Theorem 3.  If one pair of opposite sides of a quadrilateral are both parallel and congruent, the quadrilateral is a parallelogram.

Theorem 4.  The diagonals of a parallelogram bisect each other.

Theorem 5.  If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

All of the above theorems hold in Euclidean geometry, but not in hyperbolic geometry. These theorems do not make sense in spherical geometry because there are no parallelograms!