Parallelogram - Examples, Exercises and Solutions

Parallelogram

Parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length. A key feature of parallelograms is that they have two sets of parallel lines, which gives them their name. Examples of parallelograms include squares, rectangles, and rhombuses, which are all specific types of parallelograms with additional unique properties.

Characteristics of the Parallelogram

  • Sides opposite in a quadrilateral: are sides that do not have a common meeting point.
  • Adjacent sides in a quadrilateral: are sides that have a common meeting point.
  • Adjacent angles: are 2 angles that have a common vertex and side.
  • Opposite angles in the quadrilateral are angles that do not have common sides.
  • Diagonal: is a section that connects 2 non-adjacent vertices (and is not a side)

If the data is:

  • ABǁCD AB ǁ CD
  • ADǁBC AD ǁ BC

Then: ABCD ABCD is a parallelogram

Parallelogram

A1 - Parallelogram KLMN

Practice Parallelogram

Examples with solutions for Parallelogram

Exercise #1

Calculate the area of the following parallelogram:

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Video Solution

Step-by-Step Solution

To solve the exercise, we need to remember the formula for the area of a parallelogram:

Side * Height perpendicular to the side

In the diagram, although it's not presented in the way we're familiar with, we are given the two essential pieces of information:

Side = 6

Height = 5

Let's now substitute these values into the formula and calculate to get the answer:

6 * 5 = 30

Answer

30 cm²

Exercise #2

In front of you the next quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD12070

Step-by-Step Solution

To determine if the quadrilateral is a parallelogram, we need to verify the properties of the angles. A key property of parallelograms is that consecutive angles are supplementary, meaning their sum equals 180 180^\circ .

The problem provides the measures of two consecutive angles: B=70 \angle B = 70^\circ and C=120 \angle C = 120^\circ .

Next, let's calculate the sum of these angles:
B+C=70+120=190 \angle B + \angle C = 70^\circ + 120^\circ = 190^\circ

The sum of B \angle B and C \angle C is 190 190^\circ , which is not equal to 180 180^\circ .

This indicates that the quadrilateral cannot be a parallelogram because two consecutive angles do not add up to 180 180^\circ .

Therefore, the given quadrilateral is not a parallelogram.
Thus, the correct answer is No.

Answer

No

Exercise #3

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDDOOO108810

Step-by-Step Solution

According to the properties of the parallelogram: the diagonals intersect each other.

From the data in the drawing, it follows that diagonal AC and diagonal BD are divided into two equal parts, that is, the diagonals intersect each other:

AO=OC=8 AO=OC=8

DO=OB=10 DO=OB=10

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #4

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD1206012060

Step-by-Step Solution

Let's review the property: a quadrilateral in which two pairs of opposite angles are equal is a parallelogram.

From the data in the drawing, it follows that:

D=B=60 D=B=60

A=C=120 A=C=120

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #5

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD711811

Step-by-Step Solution

According to the properties of a parallelogram, any two opposite sides will be equal to each other.

From the data, it can be observed that only one pair of opposite sides are equal and therefore the quadrilateral is not a parallelogram.

Answer

No

Exercise #6

Calculate the area of the parallelogram according to the data in the diagram.

101010777AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

We know that ABCD is a parallelogram. According to the properties of parallelograms, each pair of opposite sides are equal and parallel.

Therefore: CD=AB=10 CD=AB=10

We will calculate the area of the parallelogram using the formula of side multiplied by the height drawn from that side, so the area of the parallelogram is equal to:

SABCD=10×7=70cm2 S_{ABCD}=10\times7=70cm^2

Answer

70

Exercise #7

A parallelogram is shown below.

Calculate the length of the side DC.

AAABBBCCCDDD7x-45x+12y

Step-by-Step Solution

To solve this problem, let's begin by identifying the relationship between the sides:

Since AB=7x4 AB = 7x - 4 and DC=5x+12 DC = 5x + 12 , and we know that opposite sides of a parallelogram are equal, we have:

7x4=5x+12 7x - 4 = 5x + 12

First, let's solve for x x :

7x5x=12+4 7x - 5x = 12 + 4
2x=16 2x = 16

Divide both sides by 2:

x=8 x = 8

Now that we have the value of x x , substitute it back into the expression for DC DC to find its length:

DC=5x+12=5(8)+12 DC = 5x + 12 = 5(8) + 12
DC=40+12=52 DC = 40 + 12 = 52

Therefore, the length of side DC DC is 52\mathbf{52}.

Answer

52

Exercise #8

AB = DC.=

Is the shape below a parallelogram?

AAABBBCCCDDDx+582x+9x

Step-by-Step Solution

In a parallelogram, we know that each pair of opposite sides are equal to each other.

The data shows that only one pair of sides are equal to each other:

AB=DC=8 AB=DC=8

Now we try to see that the additional pair of sides are equal to each other.

We replacex=8 x=8 for each of the sides:

AD=2×8+9 AD=2\times8+9

AD=16+9 AD=16+9

AD=25 AD=25

BC=8+5 BC=8+5

BC=13 BC=13

That is, we find that the pair of opposite sides are not equal to each other:

2513 25\ne13

Therefore, the quadrilateral is not a parallelogram.

Answer

No

Exercise #9

Below is a quadrilateral:

Given B+C=180 ∢B+∢C=180

Is it possible that it is a parallelogram?

AAABBBCCCDDD4x14040140

Step-by-Step Solution

Remember that in a parallelogram each pair of opposite angles are equal to each other.

The data shows that only one pair of angles are equal to each other:

D=B=140 D=B=140

Therefore, we will now find angle C and see if it is equal to angle A, that is, if angle C is equal to 40:

Let's remember that a pair of angles on the same side are equal to 180 degrees, therefore:

B+C=180 B+C=180

We replace the existing data:

140+4x=180 140+4x=180

4x=180140 4x=180-140

4x=40 4x=40

Divide by 4:

4x4=404 \frac{4x}{4}=\frac{40}{4}

x=10 x=10

Now we replace X:

C=4×10=40 C=4\times10=40

That is, we found that angles A and C are equal to each other and that the quadrilateral is a parallelogram since each pair of opposite angles are equal to each other.

Answer

Yes

Exercise #10

Look at the parallelogram in the figure.

Its area is equal to 70 cm².

Calculate DC.

555AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

The formula for the area of a parallelogram:

Height * The side to which the height descends.

We replace in the formula all the known data, including the area:

5*DC = 70

We divide by 5:

DC = 70/5 = 14

And that's how we reveal the unknown!

Answer

14 14 cm

Exercise #11

Look at the quadrilateral below.

AO = OC

Is it a parallelogram?

AAABBBCCCDDDOOO5x+49x+110x3x-2

Step-by-Step Solution

Let's pay attention to the diagonals, remember that in a parallelogram the diagonals intersect each other.

Therefore, we will find AO, OC, BO, DO and check if they are equal and intersect each other.

We refer to the figure:

AO=OC AO=OC

9x+1=10x 9x+1=10x

We place like terms:

1=10x9x 1=10x-9x

1=x 1=x

We replace:

AO=9×1+1=10 AO=9\times1+1=10

OC=10×1=10 OC=10\times1=10

Now we know that indeedAO=OC AO=OC

Now we establish that X=1 and see if BO is equal to OD:

BO=3x2 BO=3x-2

BO=3×12= BO=3\times1-2=

BO=32=1 BO=3-2=1

OD=5x+4 OD=5x+4

OD=5×1+4 OD=5\times1+4

OD=5+4=9 OD=5+4=9

Now we find that: BOOD BO\ne OD

Since the diagonals do not intersect each other, the quadrilateral is not a parallelogram.

Answer

No

Exercise #12

ABCD is a parallelogram.

Its perimeter is 47 cm.

What is its area?

666888AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

First, let's remember that the perimeter of a parallelogram is the sum of its sides,

which is

AB+BC+CD+DA

We recall that in a parallelogram, opposite sides are equal, so
BC=AD=6

Let's substitute in the formula:

2AB+12=47

2AB=35

AB=17.5

Now, after finding the missing sides, we can continue to calculate the area.

Remember, the area of a parallelogram is side*height to the side.

17.5*8= 140

Answer

140 140 cm²

Exercise #13

Look at the parallelogram in the figure below.

Its area is equal to 40 cm².

Calculate AE.

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Video Solution

Step-by-Step Solution

We are told that ABCD is a parallelogram,AB=CD=8 AB=CD=8 According to the properties of a parallelogram, each pair of opposite sides are equal and parallel.

Hence to find AE we will need to use the area given to us in the formula in order to determine the area of the parallelogram:

S=DC×AE S=DC\times AE

40=8×AE 40=8\times AE

We divide both sides of the equation by 8:

8AE:8=40:8 8AE:8=40:8

AE=5 AE=5

Answer

5 5 cm

Exercise #14

ABCD parallelogram, it is known that:

BE is perpendicular to DE

BF is perpendicular to DF

BF=8 BE=4 AD=6 DC=12

Calculate the area of the parallelogram in 2 different ways

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Video Solution

Step-by-Step Solution

In this exercise, we are given two heights and two sides.

It is important to keep in mind: The external height can also be used to calculate the area

Therefore, we can perform the operation of the following exercise:

The height BF * the side AD

8*6

 

The height BE the side DC
4
*12

 The solution of these two exercises is 48, which is the area of the parallelogram.

 

Answer

48 cm²

Exercise #15

In front of you the next quadrilateral:

What should the value of x be for the quadrilateral ABCD to be a parallelogram?

AAABBBCCCDDD12060120x

Step-by-Step Solution

In a parallelogram, consecutive angles are supplementary, which means they add up to 180 180^\circ . Given that C=120 \angle C = 120^\circ , the angle D=x \angle D = x must fulfill the equation:

180=C+D=120+x 180^\circ = \angle C + \angle D = 120^\circ + x

Solving for x x :

x=180120=60 x = 180^\circ - 120^\circ = 60^\circ

Therefore, the value of x x that makes ABCD ABCD a parallelogram is x=60 x = 60^\circ .

Answer

60