PROPERTIES OF PARALLELOGRAMS WORKSHEET

About "Properties of parallelograms Worksheet"

Properties of Parallelograms Worksheet :

Worksheet given in this section is much useful to the students who would like to practice problems on properties of parallelogram. 

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Properties of Parallelograms Worksheet - Problems

Problem 1 : 

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

Problem 2 : 

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

Problem 3 : 

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

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.

Problem 5 : 

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

Problem 6 :

In the parallelogram given below, 

AO  =  x + 40

OC  =  2x + 18

Find the length of AO and OC.

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. 

Problem 8 :

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

Problem 9 :

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

Problem 10 :

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

Properties of Parallelograms Worksheet - 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.

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

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.

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

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.

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

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. 

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

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.

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

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. 

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

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°. 

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

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. 

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

Problem 9 :

In the parallelogram given below, 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. 

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

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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