parallelogram theorems

Theorem 1.  The opposite sides of a parallelogramMathworldPlanetmath are congruentMathworldPlanetmathPlanetmath.


In the parallelogram ABCD, the line BD as a transversal cuts the parallel linesMathworldPlanetmath AD and BC, whence by the theorem of the parent entry ( the alternate interior angles α and β are congruent. And since the line BD also cuts the parallel lines AB and DC, the alternate interior angles γ and δ 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 quadrilateralMathworldPlanetmath are congruent, the quadrilateral is a parallelogram.

Theorem 3.  If one pair of opposite sides of a quadrilateral are both parallelMathworldPlanetmath 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 geometryMathworldPlanetmath, 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