PROPERTIES OF PARALLELOGRAMS WORKSHEET

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.

Solutions

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 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. 

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.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6