Types of trapezoids

A general or 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.

*Illustration, symbols, and captions in a Word file*

• 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.

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

• 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 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.

*Illustration and notations in a Word file*

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.

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:
*Illustration and markings in the Word file*

Click here to learn more about an isosceles trapezoid and even practice some exercises on the topic.

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:
*Illustration in Word file*

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:

*Illustration and notations in a Word file*

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:
$\frac{15.5 * 5}{2}$