Types of trapezoids

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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.

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Given the trapezoid:

999121212555AAABBBCCCDDDEEE

What is the area?

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Types of trapezoids

General trapezoid

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*

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 basesHeight to the base2Sum~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.

*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.
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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:
*Illustration and markings in the Word file*

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

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:
*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 basesheight to the base2Sum~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 AA and BB are each equal to 9090 degrees.
It is also known that the leg on which angles AA and BB rest is equal to 55 cm.

Additionally, the sum of the bases in the trapezoid is 1515 and angle CC is equal to 6060.

  • Find the angle DD.
  • Calculate the area of the trapezoid.

Solution

We know it is a right-angled isosceles trapezoid based on the given information where both angle AA is 9090 degrees and angle BB is 9090 degrees.
Therefore, the sum of the other 22 angles is 180180 degrees.
It is given that C=60C = 60 degrees.
Therefore, D=120D = 120 degrees
18060=120180-60=120
We substitute the data into the area formula for a right-angled trapezoid and get:
15.552\frac{15.5 * 5}{2}

Do you know what the answer is?
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