The trapezoid is considered one of the most intimidating shapes for students, therefore we have decided to provide a summary of the general idea behind the trapezoid and explain its properties to them and introduce some types of trapezoids.

## Characteristics of the Trapezoid

A trapezoid is a quadrilateral based on 4 sides like any other,
but special in that it will always have two parallel sides also called bases, which we can call the larger base and the smaller base
and it will also have two opposite sides also called legs.

## Examples with solutions for Trapezoids

### Exercise #1

True OR False:

In all isosceles trapezoids the bases are equal.

### Step-by-Step Solution

True: in every isosceles trapezoid the base angles are equal to each other.

True

### Exercise #2

$∢D=50°$

The isosceles trapezoid

What is $∢B$?

### Step-by-Step Solution

Let's recall that in an isosceles trapezoid, the sum of the two angles on each of the trapezoid's legs equals 180 degrees.

In other words:

$A+C=180$

$B+D=180$

Since angle D is known to us, we can calculate:

$180-50=B$

$130=B$

130°

### Exercise #3

Given the trapezoid:

What is the area?

### Step-by-Step Solution

Formula for the area of a trapezoid:

$\frac{(base+base)}{2}\times altura$

We substitute the data into the formula and solve:

$\frac{9+12}{2}\times5=\frac{21}{2}\times5=\frac{105}{2}=52.5$

52.5

### Exercise #4

The trapezoid ABCD is shown below.

AB = 2.5 cm

DC = 4 cm

Height (h) = 6 cm

Calculate the area of the trapezoid.

### Step-by-Step Solution

First, let's remind ourselves of the formula for the area of a trapezoid:

$A=\frac{\left(Base\text{ }+\text{ Base}\right)\text{ h}}{2}$

We substitute the given values into the formula:

(2.5+4)*6 =
6.5*6=
39/2 =
19.5

$19\frac{1}{2}$

### Exercise #5

Given: $∢C=2x$

$∢A=120°$

isosceles trapezoid.

Find x.

### Step-by-Step Solution

Given that the trapezoid is isosceles and the angles on both sides are equal, it can be argued that:

$∢C=∢D$

$∢A=∢B$

We know that the sum of the angles of a quadrilateral is 360 degrees.

Therefore we can create the formula:

$∢A+∢B+∢C+∢D=360$

We replace according to the existing data:

$120+120+2x+2x=360$

$240+4x=360$

$4x=360-240$

$4x=120$

We divide the two sections by 4:

$\frac{4x}{4}=\frac{120}{4}$

$x=30$

30°

### Exercise #6

Look at the trapezoid in the diagram.

What is its perimeter?

### Step-by-Step Solution

To calculate the perimeter, we'll add up all the sides of the trapezoid:

7+10+7+12 =

36

And that's the solution!

36

### Exercise #7

The trapezoid ABCD is shown below. Given in cm:

Base AB = 6

Base DC = 10

Height (h) = 5

Calculate the area of the trapezoid.

### Step-by-Step Solution

Let's recall that the area of a trapezoid is:

Now let's substitute the given data into the formula:

(10+6)*5 =
2

16*5 = 80

80/2=40

40 cm²

### Exercise #8

What is the perimeter of the trapezoid in the figure?

### Step-by-Step Solution

To find the perimeter we will add all the sides:

$4+5+9+6=9+9+6=18+6=24$

24

### Exercise #9

Calculate the area of the trapezoid.

### Step-by-Step Solution

We use the formula (base+base) multiplied by the height and divided by 2.

Note that we are only provided with one base and it is not possible to determine the size of the other base.

Therefore, the area cannot be calculated.

Cannot be calculated.

### Exercise #10

What is the area of the trapezoid in the figure?

### Step-by-Step Solution

We use the following formula to calculate the area of a trapezoid: (base+base) multiplied by the height divided by 2:

$\frac{(AB+DC)\times BE}{2}$

$\frac{(7+15)\times2}{2}=\frac{22\times2}{2}=\frac{44}{2}=22$

$22$ cm².

### Exercise #11

Do the diagonals of the trapezoid necessarily bisect each other?

### Step-by-Step Solution

The diagonals of an isosceles trapezoid are always equal to each other,

but they do not necessarily bisect each other.

(Reminder, "bisect" means that they meet exactly in the middle, meaning they are cut into two equal parts, two halves)

For example, the following trapezoid ABCD, which is isosceles, is drawn.

Using a computer program we calculate the center of the two diagonals,

And we see that the center points are not G, but the points E and F.

This means that the diagonals do not bisect.

No

### Exercise #12

The perimeter of the trapezoid equals 22 cm.

AB = 7 cm

AC = 3 cm

BD = 3 cm

What is the length of side CD?

### Step-by-Step Solution

Since we are given the perimeter of the trapezoid and not the length of CD, we can calculate:

$22=3+3+7+CD$

$22=CD+13$

$22-13=CD$

$9=CD$

9

### Exercise #13

Look at the trapezoid in the figure.

The long base is 1.5 times longer than the short base.

Find the perimeter of the trapezoid.

### Step-by-Step Solution

First, we calculate the long base from the existing data:

Multiply the short base by 1.5:

$5\times1.5=7.5$

Now we will add up all the sides to find the perimeter:

$2+5+3+7.5=7+3+7.5=10+7.5=17.5$

17.5

### Exercise #14

Given the trapezoid in front of you:

Given h=9, DC=15.

Since the area of the trapezoid ABCD is equal to 126.

Find the length of the side AB.

### Step-by-Step Solution

We use the formula to calculate the area: (base+base) times the height divided by 2

$S=\frac{(AB+CD)\times h}{2}$

We input the data we are given:

$126=\frac{(AB+15)\times9}{2}$

We multiply the equation by 2:

$252=9AB+135$

$252-135=9AB$

$117=9AB$

We divide the two sections by 9

$13=AB$

13

### Exercise #15

In an isosceles trapezoid ABCD

$∢B=3x$

$∢D=x$

Calculate the size of angle $∢B$.

### Step-by-Step Solution

To answer the question, we must know an important rule about isosceles trapezoids:

The sum of the angles that define each of the trapezoidal sides (not the bases) is equal to 180

Therefore:

∢B+∢D=180

3X+X=180

4X=180

X=45

It's important to remember that this is still not the solution, because we were asked for angle B,

Therefore:

3*45 = 135

And this is the solution!