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!

Title	parallelogram theorems
Canonical name	ParallelogramTheorems
Date of creation	2013-03-22 17:15:37
Last modified on	2013-03-22 17:15:37
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Theorem
Classification	msc 51M04
Classification	msc 51-01
Synonym	properties of parallelograms
Related topic	Parallelogram
Related topic	TriangleMidSegmentTheorem