Opposite Sides of a Parallelogram are Equal Theorem & Proof with Examples

Opposite Sides of a Parallelogram are Equal is one of the properties of the parallelogram. In the previous pages, we have seen some concepts of a parallelogram. Now in this article, you are going to learn about the opposite sides of a parallelogram are equal. Students who are lagging in the concept of parallelogram such as properties can check out this page.

Prove that Opposite Sides of a Parallelogram are Equal

In triangle ADC and ABC,
∠DAC=∠ACB(AB∥CD and taking AC as transversal)
AC=AC (since it is common)
∠DCA=∠BAC(AB is parallel to CD taking AC as transversal)
By ASA criterion of concurrency triangle, ADC is congruent to triangle ABC.
AD=BC
AB=CD

Also, Read Related Theorems

• Parallelograms on the Same Base and between Same Parallels
• Parallelograms and Rectangles on Same Base and between Same Parallels
• Triangle and Parallelogram on Same Base and between Same Parallels

Opposite Sides of a Parallelogram are Equal in Length Examples

Example 1.
In the parallelogram ABCD, AB = 3cm and DC : CB = 4: 2 find the perimeter of the parallelogram
Solution:
In the parallelogram ABCD, AB // DC and DA // CB
The opposite sides of parallelogram are equal
DC + AB = 8cm
DC : CB = 4:1
DC/CB = 4/1
8cm/CB = 4/1
CB = 8/4 = 2
CB = DA = 2cm
Therefore perimeter = AB + BC + CA + DA
= 3cm + 2cm + 3cm + 2cm
= 10cm

Example 2.
In the parallelogram PQRS, PQ = 10cm and SR : RQ = 5 : 1 find the perimeter of the parallelogram.
Solution:
In the parallelogram PQRS, PQ // SR and RP // RQ
The opposite sides of parallelogram are equal
SR + PQ = 10cm
SR : RQ = 5:1
SR/RQ = 5/1
10cm/RQ = 5/1
RQ = 10/5 = 2
RQ = SP = 2cm
Therefore perimeter = PQ + QR + RS + SP
= 10cm + 2cm + 10cm + 2cm
= 24cm

Example 3.
In a parallelogram ABCD AB = 2cm and BC = 5cm Find the CD and DA
Solution:
In a parallelogram opposite sides are equal
AB = CD and BC = DA
There AB = 2cm
BC = 5cm
So AB = CD = 2cm
BC = DA = 5cm

Example 4.
In the parallelogram PQRS, PQ = 2cm and SR : RQ = 16 : 2 find the perimeter of the parallelogram.
Solution:
In the parallelogram PQRS, PQ // SR and RP // RQ
The opposite sides of parallelogram are equal
SR + PQ = 2cm
SR : RQ = 16 : 2
SR/RQ = 16/2
2cm/RQ = 16/2
RQ = 16/2 × 2 = 4
RQ = SP = 4cm
Therefore perimeter = PQ + QR + RS + SP
= 2cm + 4cm + 2cm + 4cm
= 12cm

Example 5.
In a parallelogram ABCD AB = 8 cm Find the CD
Solution:
In a parallelogram opposite sides are equal
AB = CD
There AB = 8cm
So AB = CD = 8cm