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. 

Some Other Properties of Parallelogram

• 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.

Practice Problems

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

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

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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