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.
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. | â–±PQRS | If a quadrilateral has opposite congruent sides, the quadrilateral is a parallelogram. Hence Proved |
Last modified on August 3rd, 2023
