# 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 $AD=BC$.
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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 |