Table of Contents

Last modified on August 3rd, 2023

This article will help us learn how to prove something is a parallelogram. More precisely, how to prove a quadrilateral is a parallelogram.

There are 5 basic ways to prove a quadrilateral is a parallelogram. They are as follows:

- Proving opposite sides are congruent
- Proving opposite sides are parallel
- Proving the quadrilateral’s diagonals bisect each other
- Proving opposite angles are congruent
- Proving consecutive angles are supplementary (adding to 180Â°)

Let us now prove the above statements one by one.

Prove that opposite sides of a parallelogram are congruent

**To prove:** AB â‰… CD, AD â‰… BC

**Proof:**

Given: AB âˆ¥ CD, AD âˆ¥ BC

Draw in a diagonal AC

Statement | Reason |
---|---|

âˆ ACB â‰… âˆ CAD | Alternate interior angles |

âˆ BAC â‰… âˆ ACD | Alternate interior angles |

AC â‰… AC | Common side (identity) |

Î”ADC â‰… Î”ABC | Angle-Angle-Side (AAS) postulate |

AD â‰… BC AB â‰… CD | Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Hence Proved |

Prove that opposite sides of a parallelogram are parallel

**To prove:** AB âˆ¥ CD, AD âˆ¥ BC

**Proof:**

Given: AB â‰… CD, AD â‰… BC

Draw in a diagonal AC

Statement | Reason |
---|---|

AC â‰… AC | Common side (identity) |

Î”ADC â‰… Î”ABC | SSS postulate |

âˆ ACB â‰… âˆ CAD | CPCTC |

âˆ BAC â‰… âˆ ACD | CPCTC |

AB âˆ¥ CD | Alternate interior angles Hence Proved |

AD âˆ¥ BC | Alternate interior angles Hence Proved |

Prove that the diagonals of a parallelogram bisect each other.

This is an “if and only if” proof, so there are two things to prove.

**1. To prove:** AE â‰… EC, BE â‰… ED

**Proof:**

Given: ABCD is a parallelogram

Statement | Reason |
---|---|

AD âˆ¥ BC | Definition of parallelogram |

âˆ DAE â‰… âˆ BCE | Alternate interior angles |

ADE â‰… CBE | Alternate interior angles |

AD â‰… BC | Opposite sides of parallelogram |

Î”ADC â‰… Î”ABC | AAS postulate |

AE â‰… EC | CPCTC Hence Proved |

BE â‰… ED | CPCTC Hence Proved |

And the converse

2. **To prove:** ABCD is a parallelogram

**Proof:**

Given: AE â‰… EC, BE â‰… ED

Statement | Reason |
---|---|

âˆ AED â‰… âˆ BEC | Vertical angles |

âˆ AEB â‰… âˆ CED | Vertical angles |

Î”ADC â‰… Î”ABC | AAS |

AD â‰… BC | CPCTC |

AB â‰… CD | CPCTC |

ABCD is a parallelogram | Theorem 2 Hence Proved |

Prove that the opposite angles of a parallelogram are congruent.

**To prove:** âˆ ADC â‰… âˆ ABC, âˆ BAD â‰… âˆ BCD

**Proof:**

Given: AB âˆ¥ CD, AD âˆ¥ BC

Draw in a diagonal AC

Statement | Reason |
---|---|

âˆ ACB â‰… âˆ CAD | Alternate interior angles |

âˆ ADB â‰… âˆ CBD | Alternate interior angles |

âˆ AED â‰… âˆ BEC | Vertical angles |

âˆ AEB â‰… âˆ CED | Vertical angles |

Î”ADC â‰… Î”ABC | AAS postulate |

Î”BAD â‰… Î”BCD | AAS postulate |

âˆ ADC â‰… âˆ ABC | CPCTC postulate Hence Proved |

âˆ BAD â‰… âˆ BCD | CPCTC postulate Hence Proved |

Prove that the consecutive angles of a parallelogram are supplementary (add up to 180Â°).

**To prove:** âˆ ABC + âˆ BCD = 180Â°

**Proof:**

Given: AB âˆ¥ CD, AD âˆ¥ BC

Extend BC till P

Statement | Reason |
---|---|

âˆ ABC â‰… âˆ DCP | Corresponding angle |

âˆ BCD + âˆ DCP = 180Â° | Linear angle |

âˆ ABC + âˆ BCD = 180Â° | âˆ ABC â‰… âˆ DCP Hence Proved |

Last modified on August 3rd, 2023