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 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 even make sense in spherical geometry because there are no parallelograms!