# parallelogram theorems

###### 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 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!

Title parallelogram theorems ParallelogramTheorems 2013-03-22 17:15:37 2013-03-22 17:15:37 pahio (2872) pahio (2872) 11 pahio (2872) Theorem msc 51M04 msc 51-01 properties of parallelograms Parallelogram TriangleMidSegmentTheorem