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

**Proof:**

In ∆PQR and ∆RSP; 1. ∠QPR = ∠SRP 2. ∠QRP = ∠RPS 3. PR = PR 4. ∆PQR ≅ ∆RSP 5. PQ = SR and PS = QR. (Proved) |
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

**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°**From ****Opposite Sides of a Parallelogram are Equal**** to HOME PAGE**

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