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:
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
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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