**Properties of parallelograms worksheet :**

The following are the three important properties of triangle.

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

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

3. The diagonals of parallelogram bisect each other.

- 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 parallels are equal in area.

1. In the parallelogram ABCD if ∠A = 65°, find ∠B, ∠C and ∠D

2. In the parallelogram ABCD if ∠A = 108°, find ∠B, ∠C and ∠D

3. ABCD is a parallelogram ∠BAO = 30°, ∠DAO = 45° and ∠COD = 105°. Calculate

(i) ∠ABO (ii) ∠ODC (iii)∠ACB (ii) ∠CBD

4. Find the measure of each angle of a parallelogram in which AB = 9 cm and it perimeter is 30 cm. Find the length of each side of a parallelogram.

5. In the given figure find the values of x , ∠A and ∠C.

6. In the given figure AO = x + 40 and OC = 2x + 18. Find the length of AO and OC.

7. In a parallelogram, one angle is four times greater than the other. Find the angles of the parallelogram.

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

9.In the given parallelogram, find the measures of x and y

10. In the given parallelogram, find the measures of x and y

**Problem 1 : **

In the parallelogram ABCD if ∠A = 65°, find ∠B, ∠C and ∠D

**Solution :**

Let ABCD be a parallelogram in which ∠A = 65°

Since AD ∥ BC we can treat AB as a transversal, so

∠A + ∠B = 180°

65° + ∠B = 180°

∠B = 180° - 65°

∠B = 115°

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

∠A = ∠C

∠B = ∠D

**Therefore ∠B, ∠C and ∠D are 115°, 65° and 115° respectively.**

Let us look at the next problem on "Properties of parallelograms worksheet"

**Problem 2 : **

In the parallelogram ABCD if ∠A = 108°, find ∠B, ∠C and ∠D

**Solution :**

Since AD ∥ BC we can treat AB as a transversal, so

∠A + ∠B = 180°

108° + ∠B = 180°

∠B = 180° - 108°

∠B = 72°

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

∠A = ∠C

∠B = ∠D

**Therefore ∠B, ∠C and ∠D are 72°, 108° and 72° respectively.**

Let us look at the next problem on "Properties of parallelograms worksheet"

**Problem 3 : **

ABCD is a parallelogram ∠BAO = 30°, ∠DAO = 45° and ∠COD = 105°. Calculate

(i) ∠ ABO (ii) ∠ ODC (iii)∠ ACB (ii) ∠ CBD

**Solution :**

(i) ∠ DOC = ∠ AOB (vertically opposite angles are equal)

∠ AOB = 105°

In triangle AOB,

∠ AOB + ∠ OAB + ∠ ABO = 180°

105° + 30° + ∠ ABO = 180°

∠ ABO = 180° - 135°

**∠ ABO = 45°**

(ii) Since AB ∥ DC we can treat BD as a transversal, so

∠ ABO = 45° then

**∠ ODC = 45°** (alternate angles are equal)

(iii) Since AD ∥ BC we can treat AC as a transversal, so

∠ DAC = 45° then

**∠ ACB = 45°** (alternate angles are equal)

(iv) In triangle DCB,

∠ BDC + ∠ DCB + ∠ CBD = 180°

45° + 75° + ∠ CBD = 180°

∠ CBD = 180° - 120°

**∠ CBD = 60°**

**Let us look at the next problem on "Properties of parallelograms worksheet"**

**Problem 4 : **

Find the measure of each angle of a parallelogram in which AB = 9 cm and it perimeter is 30 cm. Find the length of each side of a parallelogram.

**Solution :**

Perimeter of parallelogram = AB + BC + CD + DA (length of all sides)

Perimeter = 30

Since it is parallelogram,length of opposite sides are equal.

9 + BC + 9 + BC = 30

2 BC = 30 - 18

2 BC = 12

BC = 6 cm.

**Therefore length of AB and CD = 9 cm and length of BC and DA = 6 cm.**

**Let us look at the next problem on "Properties of parallelograms worksheet"**

**Problem 5 : **

In the given figure find the values of x , ∠A and ∠C.

**Solution :**

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

∠D = ∠B

87° = (x + 29)°

87 - 29 = x

x = 58°

∠D + ∠C = 180°

(Adjacent angles are supplementary)

87° + ∠C = 180°

∠C = 180° - 87° = 93°

∠A = 93°

**Therefore x = 58° , ∠A = 93° , ∠C = 93° **

Let us look at the next problem on "Properties of parallelograms worksheet"

**Problem 6 :**

In the given figure AO = x + 40 and OC = 2x + 18. Find the length of AO and OC.

**Solution :**

AO = x + 40 and OC = 2x + 18

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

AO = OC

x + 40 = 2 x + 18

2x - x = 40 - 18

x = 22

AO = 22 + 40 = 62

OC = 2(22) + 18 = 44 + 18 = 62

**Therefore AO and CO are 62 cm**

**Let us look at the next problem on "Properties of parallelograms worksheet"**

**Problem 7 :**

In a parallelogram, one angle is four times greater than the other. Find the angles of the parallelogram.

**Solution :**

Let "x" be one angle,

then "4x" be the other angle

x + 4 x = 180

5 x = 180 ==> x = 36

then 4 x = 4 (36) = 144

**Therefore the angles are 36°, 144°, 36° and 144°**

**Problem 8 :**

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

**Solution :**

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

Length of CJ = Length of HI

x + 44 = 55

x = 55 - 44

x = 11

Length of CJ and HI = 11 + 44 = 55

**Therefore CJ and HI are 55 cm**

**Let us look at the next problem on "Properties of parallelograms worksheet"**

**Problem 9 :**

In the given parallelogram, find the measures of x and y

**Solution :**

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

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

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

from (1)

x = 2y - 2 - y ===> x = y - 2

plug x = y - 2 in the second equation

3(y - 2) = 2 y ===> 3y - 6 = 2y ===> 3y - 2y = 6 ===> y = 6

plug y = 6 in x = y - 2

x = 6 - 2 ===> x = 4

**Therefore x = 4 and y = 6**

**Let us look at the next problem on "Properties of parallelograms worksheet"**

**Problem 10 :**

In the given parallelogram, find the measures of x and y

**Solution :**

∠EFY + ∠EYD + ∠EYF = 180°

7 x - 5 + 45 + 70 = 180°

7x + 110 = 180°

7 x = 180 - 110 ==> 7 x = 70 ==> x = 10

In a parallelogram opposite angles are equal

∠EYD = ∠EDY

7 x - 5 = 5 y

7(10) - 5 = 5 y

70 - 5 = 5 y ==> 65 = 5 y ==> y = 13

**Therefore x = 10 and y = 13**

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