# Types of Trapezoids

🏆Practice trapeze

## 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.

## Test yourself on trapeze!

True OR False:

In all isosceles trapezoids the bases are equal.

## Types of trapezoids

### Regular Trapezoid

A regular trapezoid is a quadrilateral that:

Two of its opposite sides are parallel and are called the bases of the trapezoid.

• The other two sides are not parallel and face different directions – they are called the legs of the trapezoid.

#### Properties of the basic trapezoid –

• Two sides are parallel to each other
• Angles that rest on the same leg (one from the smaller base and the other from the larger base) add up to 180 degrees.
• If we draw a diagonal that cuts through both bases, it will create equal alternate angles between parallel lines.
• The sum of all angles in a trapezoid will be 360 degrees.

#### Area of the trapezoid:

$Sum~of~the~bases \cdot Height~to~the~base \over 2$

• If we draw a segment that passes exactly in the middle of the two legs of the trapezoid, we will get a segment that is parallel to the bases and equal to half their sum. This segment is called the "midsegment".

### A trapezoid that is also a parallelogram

A trapezoid that is also a parallelogram is essentially a trapezoid that:

• 2 of its bases are parallel.
• 2 of its legs are parallel to each other and face the same direction.

Properties of the trapezoid that is also a parallelogram:

In a parallelogram, there are 2 pairs of sides that are parallel to each other.

• The opposite sides are equal to each other.
• The opposite angles are equal to each other.
• The diagonals bisect each other.
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### Isosceles trapezoid

An isosceles trapezoid is a trapezoid where the two non-parallel sides are equal in length.

The properties of an isosceles trapezoid include all the properties of a regular trapezoid plus the following properties:

The legs are equal to each other

• The base angles are equal to each other
• The diagonals in the trapezoid are equal to each other

Let's see this in the diagram:

### Right-angled trapezoid

A right trapezoid is a trapezoid that has 2 right angles, each equal to 90 degrees.

The properties of a right trapezoid are:

• The leg adjacent to the two right angles is also the height of the trapezoid.
• The sum of the other angles (the non-right angles) is 180 degrees.

Let's see this in the illustration:

How do you calculate the area of a right-angled trapezoid?

Just like calculating the area of a regular trapezoid, according to the formula:

$Sum~of~the~bases \cdot height~to~the~base \over 2$

Practice:
Given the following trapezoid:

It is known that angles $A$ and $B$ are each equal to $90$ degrees.
It is also known that the leg on which angles $A$ and $B$ rest is equal to $5$ cm.

Additionally, the sum of the bases in the trapezoid is $15$ and angle $C$ is equal to $60$.

• Find the angle $D$.
• Calculate the area of the trapezoid.

Solution

We know it is a right-angled isosceles trapezoid based on the given information where both angle $A$ is $90$ degrees and angle $B$ is $90$ degrees.
Therefore, the sum of the other $2$ angles is $180$ degrees.
It is given that $C = 60$ degrees.
Therefore, $D = 120$ degrees
$180-60=120$
We substitute the data into the area formula for a right-angled trapezoid and get:
$\frac{15.5 * 5}{2}$

Do you know what the answer is?

## Examples with solutions for Trapeze

### 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:

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

$x=30$