PROPERTIES OF PARALLELOGRAM

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.

Some other properties of triangle

• 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.

Properties of parallelogram - Practice problems

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°

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

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