A parallelogram is a type of quadrilateral with four sides, four vertices, and four angles. And it has two diagonals that bisect each other. We know that the opposite sides of a parallelogram are equal and parallel because they never intersect. In this article, we are going to prove that Pair of Opposite Sides of a Parallelogram are Equal and Parallel. Thoroughly read the entire page to find the proof of the given statement with suitable examples.

## Pair of Opposite Sides of a Parallelogram are Equal and Parallel Theorem

Let us discuss about the statement “Pair of Opposite Sides of a Parallelogram are Equal and Parallel” with step by step explanation here.

**Statement:**

Prove that, A quadrilateral is a parallelogram if a pair of opposite sides are equal and parallel.

**Proof:**

Given: ABCD is quadrilateral and AB║CD, AB=CD.

To prove: ABCD is a parallelogram

Proof: AC is a transversal and also AB║CD, therefore

∠BAC=∠DCA(Alternate angles)

In ΔADC and ΔCBA, we have

AB=CD(Given)

∠BAC=∠DCA(Alternate angles)

AC=CA(Common)

ΔADC≅ΔCBA by the SAS rule.

Hence, by CPCT, DA=BC

Thus, Both the pair of opposite sides are equal in the quadrilateral ABCD, therefore ABCD is a parallelogram.

Hence proved.

### Problems on Pair of Opposite Sides of a Parallelogram are Equal and Parallel

**Example 1.**

In the parallelogram PQRS ∠PQR = 60° find the measures of ∠QRS, ∠RSP and ∠SPQ

**Solution:**

Given that

∠PQR = 60°

In a parallelogram sum of any two opposite angles is equal to 180°

PQ//RS,∠PQR + ∠QRS = 180°

∠QRS = 180° – ∠PQR

= 180° – 60°

= 120°

In the parallelogram opposite angles are equal so

∠RSP = ∠PQR = 60° and

∠QRS = ∠SPQ = 120°

**Example 2.**

The two angles of a parallelogram have measures of (3x – 12)° and (2x + 16)° what is the measures of all the angles of this parallelogram

**Solution:**

Given that

Two angles are (3x – 12)° and (2x + 16)°

We know that

The two opposite angles of a parallelogram are equal to each other

Here

3x – 12 = 2x + 16

3x – 12 – 2x – 16

x = 28

(3x – 12)° = (3(28) – 12)

= 72°

(180 – 72) = 108°

Two opposite angles are 72° and the other two are 108°.

**Example 3.**

In the figure, PQRS is a parallelogram in which ∠P = 25° find the measures of each of the angles ∠Q, ∠R, and ∠S

**Solution:**

It is given that PQRS is a parallelogram in which ∠P = 85°

We know that

Sum of any two adjacent angles of a parallelogram is 180°

∠P + ∠Q = 180°

25° + ∠Q = 180°

∠Q = (180A° – 25°) = 165°

∠P + ∠Q = 165°

∠R + ∠S = 180°

25° + ∠S = 180°

∠S = (180° – 25°) =165°

Therefore

∠Q = 165°, ∠R = 25, and ∠S = 165°

**Example 4.**

In the parallelogram PQRS ∠PQR = 80° find the measures of ∠QRS, ∠RSP and ∠SPQ

**Solution:**

PQ//RS,∠PQR + ∠QRS = 180°

∠QRS = 180° – ∠PQR

= 180° – 80°

= 100°

In the parallelogram opposite angles are equal so

∠RSP = ∠PQR = 80° and

∠QRS = ∠SPQ = 100°

**Example 5.**

The two angles of a parallelogram have measures of (4x – 6)° and (3x + 13)° what is the measures of all the angles of this parallelogram

**Solution:**

Given that

Two angles are (4x – 6)° and (3x + 13)°

We know that

The two opposite angles of a parallelogram are equal to each other

Here

4x – 6 = 3x + 13

4x – 6 – 3x – 13

x = 19

(4x – 6)° = (4(19) – 12)

= 64°

(180 – 64) = 116°

The two opposite angles are 64° and the other two are 116°.

### FAQs on Pair of Opposite Sides of a Parallelogram are Equal and Parallel

**1. Are both pairs of opposite sides parallel in a parallelogram?**

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

**2. Why are opposite sides of a parallelogram equal?**

If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

**3. How many pairs of parallel sides does a parallelogram have?**

A parallelogram has two pairs of parallel sides.