**Properties of Parallelogram :**

The following are the four important properties of parallelogram.

1. The diagonals of a parallelogram bisect each other.

2. In a parallelogram, the opposite angles are congruent.

3. In a parallelogram, the opposite sides are equal.

4. In a parallelogram, the opposite sides are parallel.

- The diagonal of a parallelogram divides it into two triangles of equal area.
- A parallelogram is a rhombus if its diagonals are perpendicular
- Parallelograms are on same base and between same parallel lines are equal in area.

**Problem 1 : **

In the parallelogram given below, find ∠B, ∠C and ∠D.

**Solution :**

In a parallelogram, adjacent angles are supplementary.

In the above parallelogram, ∠A and ∠B are adjacent angles.

So, we have

∠A + ∠B = 180°

65° + ∠B = 180°

∠B = 180° - 65°

∠B = 115°

Because opposite angles are congruent, we have

∠C = ∠A ∠C = 65° |
∠D = ∠B ∠D = 115° |

Hence, the measures of ∠B, ∠C and ∠D are 115°, 65° and 115° respectively.

**Problem 2 : **

In the parallelogram ABCD given below, find ∠A, ∠B, ∠C and ∠D.

**Solution :**

In a parallelogram, adjacent angles are supplementary.

In the above parallelogram, ∠A and ∠B are adjacent angles.

So, we have

x + 2x = 180°

3x = 180°

x = 60°

The measure of angle ∠A is

= x

= 60°

The measure of angle ∠B is

= 2x

= 2 ⋅ 60°

= 120°

According to the properties of parallelogram, the opposite angles are congruent.

So, we have

∠C = ∠A ∠C = 60° |
∠D = ∠B ∠D = 120° |

Hence, the measures of ∠A, ∠B, ∠C and ∠D are 60°, 120°, 60° and 120° respectively.

**Problem 3 : **

In the parallelogram given below, find the measures of ∠ABO and ∠ACB.

**Solution :**

In the parallelogram given above ∠AOB and ∠COD are vertically opposite angles.

Because vertically opposite angles are equal, we have

∠AOB = ∠COD

∠AOB = 105°

In triangle ABO, we have

∠OAB + ∠AOB + ∠ABO = 180°

Plug ∠OAB = 30° and ∠AOB = 105°.

30° + 105° + ∠ABO = 180°

135° + ∠ABO = 180°

∠ABO = 45°

In the parallelogram given above, AD||BC, AC is transversal and ∠OCB and ∠OAD are alternate interior angles.

If two parallel lines are cut by a transversal, alternate interior angles are equal.

So, we have

∠OCB = ∠OAD

In the parallelogram given above, ∠OAD = 45°.

So, we have

∠OCB = 45°

Because ∠OCB ≅ ∠ACB, we have

∠ACB = 45°

Hence, the measures of ∠ABO and ∠ACB are 45° each.

**Problem 4 : **

The perimeter of the parallelogram ABCD is 30 units and the length of the side AB is 9 units, find the length of other sides of the parallelogram.

**Solution :**

**Given :** Perimeter of the parallelogram is 30 units.

That is,

AB + BC + CD + AD = 30 -----> (1)

Because it is parallelogram, length of opposite sides must be equal.

So, we have

AB = CD

AD = BC

Because AB = 9 units and AB = CD, we can have

Then, we have

(1)-----> 9 + BC + 9 + AD = 30

18 + BC + AD = 30

Subtract 18 from both sides.

BC + AD = 12

Because AD = BC, we can have

AD + AD = 12

2 ⋅ AD = 12

Divide both sides by 2.

AD = 6

Then, the length of CD is also 6 units.

Hence, the length of CD is 9 units, AD and BC are 6 units each.

**Problem 5 : **

In the parallelogram given below, find the measures of ∠A and ∠C.

**Solution :**

According to the properties of parallelogram, opposite angles are equal.

So, we have

∠B = ∠D

(x + 29)° = 87°

x + 29 = 87

x = 58

In a parallelogram, adjacent angles are supplementary.

So, we have

∠D + ∠C = 180°

87° + ∠C = 180°

∠C = 93°

In a parallelogram, opposite angles are equal

So, we have

∠A = ∠C

∠A = 93°

Hence, the measures of ∠A and ∠C are 93° each.

**Problem 6 :**

In the parallelogram given below,

AO = x + 40

OC = 2x + 18

Find the length of AO and OC.

**Solution :**

According to the properties of parallelogram, the diagonals bisect each other.

So, we have

AO = OC

x + 40 = 2 x + 18

2x - x = 40 - 18

x = 22

The length of AO is

AO = x + 40

AO = 22 + 40

AO = 62

The length of OC is

OC = 2x + 18

OC = 2⋅ 22 + 18

OC = 44 + 18

OC = 62

Hence, the lengths of AO and OC is 62 units each.

**Problem 7 :**

In two adjacent angles of a parallelogram, if one angle is four times of the other, then find the measures of the two angles.

**Solution :**

Let "x" be one of the angles.

Then, the adjacent angle of x is 4x.

In a parallelogram, adjacent angles are supplementary.

So, we have

x + 4 x = 180°

5x = 180°

Divide both sides by 5.

x = 36°

Then, the measure of the adjacent angle is

= 4x

= 4 ⋅ 36°

= 144°

Hence, the measures of the two adjacent angles are 36° and 144°.

**Problem 8 :**

In the parallelogram given above, find the lengths of the sides GJ and HI (in cm).

**Solution :**

According to the properties of parallelogram, the length of opposite sides are equal.

Length of GJ = Length of HI

x + 44 = 5x

44 = 4x

11 = x

The length of HI is

= 5x

= 5 ⋅ 11

= 55

Because opposite sides are equal, the length of GJ is also 55 units.

Hence, the lengths of GJ and HI is 55 units each.

**Problem 9 :**

In the given parallelogram, find the values of x and y.

**Solution :**

According to the properties of parallelogram, the diagonals of a parallelogram bisect each other.

From the one of the diagonals, we have

x + y = 2y - 2

x = y - 2 -----> (1)

From the other diagonal, we have

3x = 2y -----> (2)

Plug x = y - 2 in (2).

(2)-----> 3(y - 2) = 2 y

3y - 6 = 2y

y = 6

Plug y = 6 in (1).

(1)-----> x = 6 - 2

x = 4

Hence, the values of x is 4 and y is 6.

**Problem 10 :**

In the parallelogram given below, find the values of x and y.

**Solution :**

In the parallelogram given above, the measure of angle Y is

∠Y = 45° + 70°

∠Y = 115°

In a parallelogram, adjacent angles are supplementary.

Because ∠F and ∠Y are supplementary, we have

∠F + ∠Y = 180°

Plug ∠F = 7x - 5 and ∠Y = 115°

7x - 5 + 115 = 180

7x + 110 = 180

7x = 70

x = 10

The measure of angle ∠F is

= (7x - 5)°

= (7 ⋅ 10 - 5)°

= (70 - 5)°

= 65°

In a parallelogram, opposite angles are equal.

So, we have

∠D = ∠F

(5y)° = 65°

5y = 65

y = 13

Hence, the value of x is 10 and y is 13.

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