Last modified on August 27th, 2021

chapter outline

 

Proving That a Quadrilateral is a Parallelogram

In geometry, we are frequently given to prove that a certain shape is indeed a certain shape or not. Say for example you are given a quadrilateral and asked to prove whether it is a parallelogram. By looking at the shape you may think that it is a parallelogram. But unless you prove your assumption you can’t be sure that it is indeed a parallelogram. This sums up the importance of mathematical proof.

How to Prove a Quadrilateral is a Parallelogram

Remember, that 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 five 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.

 

  1. Proving that both pairs of opposite sides are parallel
  2. Proving that both pairs of opposite sides are congruent
  3. Proving that one pair of opposite sides is both congruent and parallel
  4. Proving that the diagonals bisect each other
  5. Proving that one angle is supplementary to both consecutive angles
  6. 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.

Example

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.

Figure 1: Heading: no heading, filename: Proving a Quadrilateral is a Parallelogram

Given: PS ≅ QR,

              PS ∥ QR

Figure 2: Heading: no heading, filename: How to Prove a Quadrilateral is a Parallelogram

Let us draw two diagonal lines PR and QS

StepsStatementsReasons
1.PS ≅ QR, PS ∥ QRGiven.
2.QS ≅ QSAny real number is equal to itself (Reflexive property).
3.∠PSQ ≅ ∠SQRIf two parallel lines are cut by a transverse line the alternate interior angles are congruent.
4.△ QPS ≅ △ SRQIf 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 ≅ SRCorresponding parts of congruent triangles are congruent (CPCTC).
6.▱PQRSIf a quadrilateral has opposite congruent sides, the quadrilateral is a parallelogram.
Hence Proved

The above proof can also be done using the paragraph proof form where the statements and the explanations are written in a paragraph form. But you are not encouraged to follow this technique as there is a possibility that you may skip a step or leave out an explanation for one or more of your statements.

Last modified on August 27th, 2021

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