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

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