Table of Contents

Last modified on August 3rd, 2023

A quadrilateral is any two-dimensional flat shape having four sides. A parallelogram, on the other hand, is a quadrilateral having two pairs of opposite parallel sides.

To prove whether a given quadrilateral is a parallelogram, there are six possible ways. Depending upon the information provided, you need to use any one of the below-given properties of a parallelogram to get to your conclusion.

- Proving that both pairs of opposite sides are parallel
- Proving that both pairs of opposite sides are congruent
- Proving that one pair of opposite sides is both congruent and parallel
- Proving that the diagonals bisect each other
- Proving that one angle is supplementary to both consecutive angles
- Proving that both the pairs of opposite angles are congruent

If we can prove one of the above properties to be true about the given quadrilateral, we can conclude that the given figure is a parallelogram. Also, it proves that all the six given properties are true for the given parallelogram.

Let us proof how a quadrilateral is a parallelogram.

Below given is a quadrilateral PQRS, whose opposite sides are parallel and congruent. We need to prove if the given quadrilateral is indeed a parallelogram. From the given information we understand that we need to prove the given quadrilateral is a parallelogram using property 1 and property 2.

Let us prove that the quadrilateral PQRS is a parallelogram. We will be using a two-column proof form (step by step) to get to our conclusion.

**To Prove**: The given quadrilateral PQRS, having one pair of opposite sides parallel and congruent, is a parallelogram.

**Given**: PS ≅ QR, PS ∥ QR

Let us draw two diagonal lines PR and QS

Steps | Statements | Reasons |
---|---|---|

1. | PS ≅ QR, PS ∥ QR | Given. |

2. | QS ≅ QS | Any real number is equal to itself (Reflexive property). |

3. | ∠PSQ ≅ ∠SQR | If two parallel lines are cut by a transverse line the alternate interior angles are congruent. |

4. | △ QPS ≅ △ SRQ | If two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, both the triangles are congruent (SAS). |

5. | QP ≅ SR | Corresponding parts of congruent triangles are congruent (CPCTC). |

6. | ▱PQRS | If a quadrilateral has opposite congruent sides, the quadrilateral is a parallelogram. Hence Proved |

Last modified on August 3rd, 2023