**Properties of parallelogram :**

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.

**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 parallelogram" **

**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 parallelogram"

**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°**

**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.**

**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° **

**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**

**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**

**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**

**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**

After having gone through the stuff given above, we hope that the students would have understood "Properties of parallelogram".

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