# Parallelogram - Examples, Exercises and Solutions

## Parallelogram - Parallelogram Verification

Did you notice the quadrilateral that is formed at the intersection of 2 train tracks? What is it called? What are its characteristics? Let's take a look at the train tracks, why are train tracks 2 parallel tracks? For the train to not derail, there must be 2 tracks that always maintain the same distance apart. This is the definition of parallel lines that never meet because the distance between them is always equal. This is the definition of parallel lines that never meet because the distance between them is always equal. At the moment when 2 train tracks meet, a quadrilateral is formed between them, which has 2 pairs of opposite sides parallel, which is the parallelogram

If the data is:

• $AB ǁ CD$
• $AD ǁ BC$

Then: $ABCD$ is a parallelogram

## Practice Parallelogram

### Exercise #1

Is it possible that it is a parallelogram?

### 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:

$AO=OC=8$

$DO=OB=10$

Therefore, the quadrilateral is actually a parallelogram.

Yes

### Exercise #2

Is it possible that it is a parallelogram?

### 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:

$D=B=60$

$A=C=120$

Therefore, the quadrilateral is actually a parallelogram.

Yes

### Exercise #3

Is it possible that it is a parallelogram?

### 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.

No

### Exercise #4

Look at the parallelogram in the figure.

Its area is equal to 70 cm².

Calculate DC.

### 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!

$14$ cm

### Exercise #5

AB = DC.=

Is the shape below a parallelogram?

### 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:

$AB=DC=8$

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

We replace$x=8$for each of the sides:

$AD=2\times8+9$

$AD=16+9$

$AD=25$

$BC=8+5$

$BC=13$

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

$25\ne13$

Therefore, the quadrilateral is not a parallelogram.

No

### Exercise #1

Given $∢B+∢C=180$

Is it possible that it is a parallelogram?

### 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:

$D=B=140$

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:

$B+C=180$

We replace the existing data:

$140+4x=180$

$4x=180-140$

$4x=40$

Divide by 4:

$\frac{4x}{4}=\frac{40}{4}$

$x=10$

Now we replace X:

$C=4\times10=40$

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.

Yes

### Exercise #2

Look at the parallelogram in the figure below.

Its area is equal to 40 cm².

Calculate AE.

### Step-by-Step Solution

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

To find AE we will use the area given to us in the formula to find the area of the parallelogram:

$S=DC\times AE$

$40=8\times AE$

We divide both sides of the equation by 8:

$8AE:8=40:8$

$AE=5$

$5$ cm

### Exercise #3

AO = OC

Is it a parallelogram?

### 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:

$AO=OC$

$9x+1=10x$

We place like terms:

$1=10x-9x$

$1=x$

We replace:

$AO=9\times1+1=10$

$OC=10\times1=10$

Now we know that indeed$AO=OC$

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

$BO=3x-2$

$BO=3\times1-2=$

$BO=3-2=1$

$OD=5x+4$

$OD=5\times1+4$

$OD=5+4=9$

Now we find that: $BO\ne OD$

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

No

### Exercise #4

ABCD parallelogram, it is known that:

BE is perpendicular to DE

BF is perpendicular to DF

Calculate the area of the parallelogram in 2 different ways

### 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.

48 cm²

### Exercise #5

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

What is the area of the parallelogram?

### Step-by-Step Solution

To find the area,

first, the height of the parallelogram must be found.

To conclude, let's take a look at triangle EBC.

Since we know it is a right triangle (since it is the height of the parallelogram)

the Pythagorean theorem can be used:

$a^2+b^2=c^2$

In this case: $EB^2+EC^2=BC^2$

We place the given information: $2^2+EC^2=5^2$

We isolate the variable:$EC^2=5^2+2^2$

We solve:$EC^2=25-4=21$

$EC=\sqrt{21}$

Now all that remains is to calculate the area.

It is important to remember that for this, the length of each side must be used.
That is, AE+EB=2+7=9

$\sqrt{21}\times9=41.24$

41.24

### Exercise #1

AE is the height of the parallelogram ABCD.

AB is 3 cm longer than AE.

The area of ABCD is 32 cm².

Calculate the length of side AB.

### Step-by-Step Solution

Keep in mind that AB is 3 cm greater than AE, so we must pay attention to the data when we put the formula to calculate the parallelogram:

Height multiplied by the side of the height:

$AB\times AE=S$

We will mark AE with the letter a and therefore AB will be a+3:

$a\times(a+3)=32$

We open the parentheses:

$a^2+3a=32$

We use the trinomial/roots formula:

$a^2+3a-32=0$$(a+8)(a-5)=0$

That means we have two options:

$a=-8,a=5$

Since it is not possible to place a negative side in the formula to calculate the area$a=5$

Now we can calculate the sides:

$AE=5$

$AB=5+3=8$

8 cm

### Exercise #2

The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.

AE = 8

BC = 5

What is the area of the parallelogram?

### Step-by-Step Solution

In the first step, we must find the length of EC, which we will identify with an X.

We know that the perimeter of a rectangle is the sum of all its sides (AE+EC+CF+FA),

Since in a rectangle the opposite sides are equal, the formula can also be written like this: 2AE=2EC.

We replace the known data:

$2\times8+2X=24$

$16+2X=24$

We isolate X:

$2X=8$

and divide by 2:

$X=4$

Now we can use the Pythagorean theorem to find EB.

(Pythagoras: $A^2+B^2=C^2$)

$EB^2+4^2=5^2$

$EB^2+16=25$

We isolate the variable

$EB^2=9$

We take the square root of the equation.

$EB=3$

The area of a parallelogram is the height multiplied by the side to which the height descends, that is$AB\times EC$.

$AB=\text{ AE}+EB$

$AB=8+3=11$

And therefore we will apply the area formula:

$11\times4=44$

44

### Exercise #3

The area of trapezoid ABCD is X cm².

The line AE creates triangle AED and parallelogram ABCE.

The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.

Calculate the ratio between sides DE and EC.

### Step-by-Step Solution

To calculate the ratio between the sides we will use the existing figure:

$\frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}$

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

$A_{ADE}=\frac{h\times DE}{2}$

We know that the area of the parallelogram is equal to:

$A_{ABCD}=h\times EC$

We replace the data in the formula given by the ratio between the areas:

$\frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}$

We solve by cross multiplying and obtain the formula:

$h\times EC=3(\frac{1}{2}h\times DE)$

We open the parentheses accordingly:

$h\times EC=1.5h\times DE$

We divide both sides by h:

$EC=\frac{1.5h\times DE}{h}$

We simplify to h:

$EC=1.5DE$

Therefore, the ratio between is: $\frac{EC}{DE}=\frac{1}{1.5}$

$1:1.5$

### Exercise #4

ABCD is a parallelogram with a perimeter of 38 cm.

AB is twice as long as CE.

AD is three times shorter than CE.

CE is the height of the parallelogram.

Calculate the area of the parallelogram.

### Step-by-Step Solution

Let's call CE as X

According to the data

$AB=x+2,AD=x-3$

The perimeter of the parallelogram:

$2(AB+AD)$

$38=2(x+2+x-3)$

$38=2(2x-1)$

$38=4x-2$

$38+2=4x$

$40=4x$

$x=10$

Now it can be argued:

$AD=10-3=7,CE=10$

The area of the parallelogram:

$CE\times AD=10\times7=70$

70 cm²

### Exercise #5

ABCD is a parallelogram whose perimeter is equal to 24 cm.

The side of the parallelogram is two times greater than the adjacent side (AB>AD).

CE is the height of the side AB

The area of the parallelogram is 24 cm².

Find the height of CE

### Step-by-Step Solution

The perimeter of the parallelogram is calculated as follows:

$S_{ABCD}=AB+BC+CD+DA$ Since ABCD is a parallelogram, each pair of opposite sides is equal, and therefore, AB=DC and AD=BC

According to the figure that the side of the parallelogram is 2 times larger than the side adjacent to it, it can be argued that$AB=DC=2BC$

We inut the data we know in the formula to calculate the perimeter:

$P_{ABCD}=2BC+BC+2BC+BC$

We replace the given perimeter in the formula and add up all the BC coefficients accordingly:

$24=6BC$

We divide the two sections by 6

$24:6=6BC:6$

$BC=4$

We know that$AB=DC=2BC$We replace the data we obtained (BC=4)

$AB=DC=2\times4=8$

As ABCD is a parallelogram, then all pairs of opposite sides are equal, therefore BC=AD=4

To find EC we use the formula:$A_{ABCD}=AB\times EC$

We replace the existing data:

$24=8\times EC$

We divide the two sections by 8$24:8=8EC:8$

$3=EC$