# Types of trapezoids - Examples, Exercises and Solutions

## Types of trapezoids

Properties of a regular trapezoid
• A quadrilateral with only 2 parallel sides.
• Angles resting on the same leg are supplementary to 180 degrees, so the sum of all angles is 360 degrees.
• The diagonal of the trapezoid creates equal alternate angles between parallel lines.

Properties of a trapezoid that is a parallelogram
• A quadrilateral with 2 pairs of parallel sides – parallel bases and parallel legs.
• Its opposite sides are equal.
• Its opposite angles are equal.
• The diagonals bisect each other.

Properties of an Isosceles Trapezoid
• A quadrilateral with one pair of parallel sides and another pair of non-parallel but equal sides.
• The base angles are equal.
• The diagonals are equal.

Properties of a Right-Angled Trapezoid
• A quadrilateral with only one pair of parallel sides and 2 angles each equal to 90 degrees.
• The height of the trapezoid is the leg on which the two right angles rest.
• The other 2 angles add up to 180 degrees.

## Practice Types of trapezoids

### Exercise #1

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 #2

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 #3

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 #4

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 #5

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 #1

What is the area of the trapezoid in the figure?

### Step-by-Step Solution

We use the 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 #2

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 #3

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 #4

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 #5

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!

135°

### Exercise #1

The perimeter of the trapezoid in the diagram is 25 cm. Calculate the missing side.

### Step-by-Step Solution

We replace the data in the formula to find the perimeter:

$25=4+7+11+x$

$25=22+x$

$25-22=x$

$3=x$

$3$ cm

### Exercise #2

What is the area of the trapezoid in the figure?

### Step-by-Step Solution

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

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

Keep in mind that AD is the height of a trapezoid:

We replace the existing data in the formula:

$S=\frac{(2+9)\times7}{2}$

$S=\frac{11\times7}{2}=\frac{77}{2}=38.5$

$38.5$ cm².

### Exercise #3

Given the trapezoid:

What is the height?

### Step-by-Step Solution

Formula for the area of a trapezoid:

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

We substitute the data into the formula

$\frac{9+6}{2}\times h=30$

We solve:

$\frac{15}{2}\times h=30$

$7\frac{1}{2}\times h=30$

$h=\frac{30}{7\frac{1}{2}}$

$h=4$

4

### Exercise #4

The area of the trapezoid in the diagram is 1.375 cm².

Work out the length of the side marked in red.

### Step-by-Step Solution

The area of the trapezoids will be equal to: $S=\frac{(AB+DC)}{2}\times h$

We replace the data we have in the formula:

$1.375=\frac{AB+4}{2}\times0.5$

We multiply by 2 to get rid of the fraction:

$2.75=(AB+4)\times\frac{1}{2}$

We multiply by 2:

$5.5=AB+4$

$5.5-4=AB$

$1.5=AB$

$1.5$ cm

### Exercise #5

ABCD is a trapezoid.

AB = 4 cm

DC = 7 cm

BK = 6 cm

Can the trapezoidal area formula be applied? If so, apply it and calculate.

### Step-by-Step Solution

The formula for the area of a trapezoid is:

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

Since we are given AB and DC but not the height, we cannot calculate the area of the trapezoid.