Opposite Sides of a Parallelogram are Equal

Here we will discuss about the opposite sides of a parallelogram are equal in length.

In a parallelogram, each pair of opposite sides are of equal length.

Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS.

To prove: PQ = SR and PS = QR

Construction: Join PR

Opposite Sides of a Parallelogram are Equal

Proof:

          Statement

In ∆PQR and ∆RSP;

1. ∠QPR = ∠SRP

2. ∠QRP = ∠RPS

3. PR = PR

4. ∆PQR ≅ ∆RSP

5. PQ = SR and PS = QR. (Proved)

Reason


1. PQ ∥ RS and RP is a transversal.

2. PS ∥ QR and RP is a transversal.

3. Common side

4. By ASA criterion of congruency.

5. CPCTC


Converse of the above given theorem

A quadrilateral is a parallelogram if each pair of opposite sides are equal.

Given: PQRS is a quadrilateral in which PQ = SR and PS = QR

A Quadrilateral is a Parallelogram

To prove: PQRS is a parallelogram

Proof: In ∆PQR and ∆RSP, PQ = SR, QR = SP (given) and PR is the common side.

Therefore, by SSS criterion of congruency, ∆PQR ≅ ∆RSP

Therefore, ∠QPR = ∠PRS, ∠QRP = ∠RPS  (CPCTC)

Therefore, PQ ∥ SR, QR ∥ PS

Hence, PQRS is a parallelogram.


Solved examples based on the theorem of opposite sides of a parallelogram are equal in length:

1. In the parallelogram PQRS, Pq = 6 cm and SR : RQ = 2 : 1. Find the perimeter of the parallelogram.

Solution:

In the parallelogram PQRS, PQ ∥ SR and SP ∥ RQ.

The opposite sides of a parallelogram are equal. So, SR + PQ = 6 cm.

AS SR : RQ = 23 : 1, \(\frac{6 cm}{RQ}\) = \(\frac{2}{1}\)

⟹ RQ = 3 cm

Also, RQ = SP = 3 cm.

Therefore, perimeter = PQ + QR + RS + SP

                               = 6 cm + 3 cm + 6 cm + 3 cm

                               = 18 cm.


2. In the parallelogram ABCD, ∠ABC = 50°. Find the measures of ∠BCD, ∠CBA and ∠DAB.

Solution: 

AS AB ∥ DC, ∠ABC + ∠BCD = 180°

Therefore, ∠BCD = 180° - ∠ABC

                         = 180° - 50°

                         = 130°

As opposite angles in a parallelogram are equal, 

∠CDA = ∠ABC = 50° and

∠DAB = ∠BCD = 130°







9th Grade Math

From Opposite Sides of a Parallelogram are Equal to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

    Jul 12, 24 03:08 PM

    Fractions
    The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

    Read More

  2. Worksheet on Fractions | Questions on Fractions | Representation | Ans

    Jul 12, 24 02:11 PM

    Worksheet on Fractions
    In worksheet on fractions, all grade students can practice the questions on fractions on a whole number and also on representation of a fraction. This exercise sheet on fractions can be practiced

    Read More

  3. Fraction in Lowest Terms |Reducing Fractions|Fraction in Simplest Form

    Jul 12, 24 03:21 AM

    Fraction 8/16
    There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method. If numerator and denominator of a fraction have no common factor other than 1…

    Read More

  4. Conversion of Improper Fractions into Mixed Fractions |Solved Examples

    Jul 12, 24 12:59 AM

    To convert an improper fraction into a mixed number, divide the numerator of the given improper fraction by its denominator. The quotient will represent the whole number and the remainder so obtained…

    Read More

  5. Conversion of Mixed Fractions into Improper Fractions |Solved Examples

    Jul 12, 24 12:30 AM

    Conversion of Mixed Fractions into Improper Fractions
    To convert a mixed number into an improper fraction, we multiply the whole number by the denominator of the proper fraction and then to the product add the numerator of the fraction to get the numerat…

    Read More