## Parallelogram

Definition: A quadrilateral having its opposite sides parallel is called a parallelogram. In the adjoining figure, AB∥CD and AD∥BC. So ABCD is a parallelogram.

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Properties of parallelogram

1. Opposite sides of a parallelogram are equal

2. Opposite angles of a parallelogram are equal.

3. Diagonals of a parallelogram bisect each other.

### Proofs:

1. Prove theoretically that the opposite sides of a parallelogram are equal.

Given: ABCD is a parallelogram where, AB∥DC and BC∥AD.

To prove: AB=CD and AD=BC

Construction: B and D joined.

Proof:

Statements Reasons

1. In ΔABD and ΔBCD

i. ∠ABD = ∠BDC (A) ———> Alternate angles

ii. BD = BD (S) —————–> Common side

iii. ∠ADB = ∠CBD (A) ———> Alternate angles

2. ΔABD ≅ ΔBCD ——————-> By A.S.A. axiom

3. AD = CD and AD = BC ———> Corresponding sides of congruent triangles

Proved.

2. Prove theoretically that the opposite angles of a parallelogram are equal.

Given: ABCD is a parallelogram where, AB∥DC and BC∥AD.

To prove: ∠BAD = ∠BCD and ∠ABC = ∠ADC

Construction: B and D joined.

Proof:

Statements Reasons

1. In ΔABD and ΔBCD

i. ∠ABD = ∠BDC (A) ——-> Alternate angles

ii. BD = BD (S) —————> Common side

iii. ∠ADB = ∠CBD (A) ——-> Alternate angles

2. ΔABD ≅ ΔBCD —————–> By A.S.A. axiom

3. ∠BAD = ∠BCD ——————> Corresponding angles of congruent triangles

4. ∠ABC = ∠ADC ——————> Similarly by joining A and C

Proved.

3. Prove theoretically that the diagonals of a parallelogram bisect each other.

Given: ABCD is a parallelogram. Diagonals AC and BD intersect at O.

To prove: AO=CO and BO=DO

Proof:

Statements Reasons

1. In ΔAOD and ΔBOC

i. ∠OAD = ∠OCB (A) ————> Alternate angles

ii. AD = BC (S) ——————–> Opposite sides of a parallelogram

iii. ∠ODA = ∠OBC (A) ————> Alternate angles

2. ΔABD ≅ ΔBCD ———————-> By A.S.A. axiom

3. AO = CO and BO = DO ———–> Corresponding sides of congruent triangles

Proved.

### Workout Examples

Example 1: Find the values of unknown angles in the given parallelogram.

Solution: From the figure,

a + 120° = 180° —————-> Co-interior angles.

or, a = 180° – 120°

or, a = 60°

b = 120° ————————–> Opposite angles of a parallelogram.

c = a —————————–> Opposite angles of a parallelogram.

= 60°

∴ a = 60°

b = 120°

c = 60°

Example 2: Find the values of unknown angles in the given parallelogram.

Solution: From the figure,

4a + 5a = 180° —————-> Co-interior angles.

or, 9a = 180°

or, a = 180°/9

or, a = 20°

b = 4a ——————–> Corresponding angles.

= 4 × 20°

= 80°

c = b ——————–> Alternate angles.

= 80°

d = 5a ——————–> Opposite angles of a parallelogram.

= 5 × 20°

= 100°

∴ a = 20°

b = 80°

c = 80°

d = 100°

Example 3: Find the values of unknown angles in the given parallelogram.

Solution: From the figure,

a + 35° + 60° = 180° —————-> Sum of angles of ΔABC.

or, a + 95° = 180°

or, a = 180° – 95°

or, a = 85°

b = 35° ————————–> Alternate angles.

c = a —————————–> Alternate angles.

= 85°

d = 60° ————————–> Opposite angles of a parallelogram.

∴ a = 85°

b = 35°

c = 85°

d = 60°

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