🎯 What will you learn in this lesson?

👉 Understand the definition of a parallelogram.

👉 Learn the properties of opposite sides and opposite angles.

👉 Understand how the diagonals of a parallelogram behave.

👉 Prove the main properties using congruent triangles.

👉 Use different tests to prove that a quadrilateral is a parallelogram.

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

So, for parallelogram ABCD,

AB ∥ DC and AD ∥ BC.

Properties of a Parallelogram

⭐ A diagonal of a parallelogram divides it into two congruent triangles. Hence, its opposite sides and opposite angles are equal.

ABCD is a parallelogram, so AB ∥ DC and AD ∥ BC

Proof of the theorem+

Given:	ABCD is a parallelogram and AC is a diagonal.
To prove:	(1) △ABC ≅ △CDA, (2) AB = DC and BC = AD, (3) ∠ABC = ∠ADC and ∠BAD = ∠BCD.
Proof:	Since AD ∥ BC and AC is a transversal, ∠ACB = ∠CAD. ...(i)
	Since AB ∥ DC and AC is a transversal, ∠BAC = ∠ACD. ...(ii)
	Also, AC is common to both triangles.
	Therefore, △ABC ≅ △CDA [ASA congruence].
	Hence, AB = DC and BC = AD [corresponding sides of congruent triangles].
	Also, ∠ABC = ∠ADC, and by adding the equal parts in (i) and (ii), ∠BAD = ∠BCD.
	Thus the opposite sides and opposite angles of a parallelogram are equal.

⭐ The diagonals of a parallelogram bisect each other.

The diagonals AC and BD of parallelogram ABCD intersect at O

Proof of the theorem+

Given:	Diagonals AC and BD of parallelogram ABCD intersect at O.
To prove:	AO = OC and BO = OD.
Proof:	Since AD ∥ BC, ∠OAD = ∠OCB.
	∠AOD = ∠BOC [vertically opposite angles].
	AD = BC [opposite sides of a parallelogram].
	Therefore, △AOD ≅ △COB [AAS/ASA congruence].
	Hence, AO = OC and DO = BO.
	Therefore, the diagonals of a parallelogram bisect each other.

⭐ If both pairs of opposite sides of a quadrilateral are equal, then it is a parallelogram.

In quadrilateral ABCD, AB = DC and AD = BC

Proof of the theorem+

Given:	In quadrilateral ABCD, AB = DC and AD = BC.
To prove:	ABCD is a parallelogram.
Construction:	Join B and D.
Proof:	In △ABD and △CDB, AB = DC, AD = BC and BD is common.
	Therefore, △ABD ≅ △CDB [SSS congruence].
	So, ∠ADB = ∠CBD. Hence AD ∥ BC.
	Also, ∠ABD = ∠CDB. Hence AB ∥ DC.
	Both pairs of opposite sides are parallel, so ABCD is a parallelogram.

⭐ If both pairs of opposite angles of a quadrilateral are equal, then it is a parallelogram.

In quadrilateral ABCD, ∠BAD = ∠BCD and ∠ABC = ∠ADC

Proof of the theorem+

Given:	∠BAD = ∠BCD and ∠ABC = ∠ADC.
To prove:	ABCD is a parallelogram.
Proof:	The sum of the four angles of a quadrilateral is 360°.
	∠BAD + ∠ABC + ∠BCD + ∠ADC = 360°.
	Using the given equalities, 2(∠BAD + ∠ABC) = 360°.
	Therefore, ∠BAD + ∠ABC = 180°.
	Hence AD ∥ BC, because the interior angles on the same side of transversal AB are supplementary.
	Similarly, AB ∥ DC.
	Therefore, ABCD is a parallelogram.

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

In quadrilateral ABCD, AB = DC and AB ∥ DC

Proof of the theorem+

Given:	AB = DC and AB ∥ DC.
To prove:	ABCD is a parallelogram.
Construction:	Draw diagonal AC.
Proof:	Since AB ∥ DC and AC is a transversal, ∠BAC = ∠ACD.
	In △ABC and △CDA, AB = DC, ∠BAC = ∠ACD and AC is common.
	Therefore, △ABC ≅ △CDA [SAS congruence].
	Thus ∠ACB = ∠DAC, so BC ∥ AD.
	Now AB ∥ DC and BC ∥ AD. Therefore, ABCD is a parallelogram.

⭐ If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

The diagonals AC and BD of quadrilateral ABCD bisect each other at O

Proof of the theorem+

Given:	AO = OC and BO = OD.
To prove:	ABCD is a parallelogram.
Proof:	In △AOD and △COB, AO = OC and OD = OB.
	Also, ∠AOD = ∠COB [vertically opposite angles].
	Therefore, △AOD ≅ △COB [SAS congruence].
	Hence, AD = BC and ∠OAD = ∠OCB.
	Since ∠OAD and ∠OCB are alternate interior angles, AD ∥ BC.
	Thus one pair of opposite sides is equal and parallel. Therefore, ABCD is a parallelogram.

✏️ Solved examples

Example 1

In parallelogram ABCD, AB = 8 cm and BC = 5 cm. Find DC and AD.

Solution: Opposite sides of a parallelogram are equal.

DC = AB = 8 cm and AD = BC = 5 cm.

Example 2

In parallelogram PQRS, ∠P = 68°. Find the other three angles.

Solution: Opposite angles are equal and adjacent angles are supplementary.

∠R = 68° ∠Q = ∠S = 180° - 68° = 112°

Example 3

The diagonals AC and BD of parallelogram ABCD meet at O. If AO = 6.5 cm and BO = 4 cm, find AC and BD.

Solution: The diagonals bisect each other.

AC = 2 × AO = 13 cm BD = 2 × BO = 8 cm

📝 Practice questions

In parallelogram ABCD, AB = 12 cm and AD = 7 cm. Find DC and BC.
One angle of a parallelogram is 125°. Find the other three angles.
The diagonals of a parallelogram meet at O. If AC = 18 cm and BD = 14 cm, find AO, OC, BO and OD.
In quadrilateral PQRS, PQ = RS and QR = PS. What can you conclude? Give the theorem used.
In quadrilateral ABCD, AB ∥ DC and AB = DC. Prove that ABCD is a parallelogram.
The diagonals of quadrilateral WXYZ bisect each other. Is WXYZ necessarily a parallelogram? Explain.

✅ Answers

1. DC = 12 cm, BC = 7 cm.

2. The angles are 125°, 55°, 125° and 55°.

3. AO = OC = 9 cm; BO = OD = 7 cm.

4. PQRS is a parallelogram because both pairs of opposite sides are equal.

5. A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.

6. Yes. A quadrilateral whose diagonals bisect each other is a parallelogram.

📚 Important points to remember

👉 Opposite sides of a parallelogram are equal and parallel.

👉 Opposite angles of a parallelogram are equal.

👉 Adjacent angles of a parallelogram add up to 180°.

👉 The diagonals of a parallelogram bisect each other.

👉 A diagonal divides a parallelogram into two congruent triangles.

👉 To prove that a quadrilateral is a parallelogram, you may use any one of these tests: both pairs of opposite sides equal, both pairs of opposite angles equal, one pair of opposite sides equal and parallel, or diagonals bisecting each other.