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# Parallelograms

Think of the markings in a zebra crossing. These are closed 4-sided shapes with opposite sides that are equal and parallel. Such shapes are known as parallelograms.

A four-sided plane figure is known as a quadrilateral. In this article, we are going to look at the special type of quadrilateral known as a parallelogram.

## Definition of a parallelogram

A quadrilateral with 2 pairs of opposite parallel sides is called a parallelogram.

We know that a quadrilateral has 4 sides. In a parallelogram, these 4 sides consist of 2 pairs of opposite parallel sides.

The following diagram illustrates a parallelogram.

Parallelogram illustration - StudySmarter Originals

In the above figure:

• AB // CD
• AC // BD

## Properties of parallelograms

In addition to the above, we can identify various properties of parallelograms.

We will use the following parallelogram ABDC with diagonals d1=BC and d2=AD to illustrate the properties.

Parallelogram with diagonals, d1 and d2 - StudySmarter Originals

• In a parallelogram, the opposite side are equal.

This means that in the above parallelogram, AB=CD and AC=BD.

• In a parallelogram, the opposite angles are equal.

This means that in the above parallelogram, ∠CAB=∠CDB and ∠ACD=∠ABD

• In a parallelogram, consecutive angles are supplementary.

In any parallelogram, you can identify 4 pairs of consecutive angles. These are always supplementary (which means the angles sum up to 180 degrees). In the above parallelogram:

∠CAB + ∠ABD = 180,

∠ABD + ∠BDC = 180,

∠BDC + ∠DCA = 180,

∠DCA + ∠CAB = 180.

• If any angle in a parallelogram is a right angle, that means all 4 internal angles are right angles.

This is a direct consequence of the above property. If any angle in a parallelogram is a right angle, then the adjacent angle is 180-90=90 (according to the above property). In turn, the next adjacent angle will be a right angle and so on. Therefore, in any parallelogram, if you identify any angle as a right angle, you can directly conclude that all 4 angles are right angles.

• The diagonals of a parallelogram bisect each other.

In the above parallelogram, the point O is the mid-point of both the diagonals d1 and d2.

• Each diagonal of a parallelogram separates the parallelogram into two congruent triangles.

In the above parallelogram, the diagonal d1 would divide the parallelogram into two congruent triangles, ΔABC and ΔBCD. Similarly, the diagonal d2 would divide the parallelogram into two congruent triangles, ΔABD and ΔACD.

## Area of parallelograms

Consider the following parallelogram:

Parallelogram with base b and height h - StudySmarter Originals

The area of a parallelogram is given by the formula:

Area = b × h

where b = base, h = height

Now the value, b, is the length of the side AB, which is considered the base here. Conventionally, one of the longer sides of the parallelogram is taken to be the base. The value h is the height of the parallelogram. It is also sometimes called the altitude. The height is the length of the line drawn from the base to its opposite side. The height is perpendicular to the base.

## Parallelogram: Example problems

In this section we explore examples of math problems that you may encounter about parallelograms and their properties.

A parallelogram with a base of 8 ft. has an area of 20 ft2. What is the height of the parallelogram?

In the following parallelogram, ∠ABD = 47°, ∠CBD = 72°. Find ∠CDA.

Parallelogram illustration - StudySmarter Originals

∠ABC = ∠ABD + ∠CBD

= 47 + 72 = 119.

Opposite angles of a parallelogram are equal. Hence,

∠CDA = ∠ABC = 119°

## Different types of parallelogram shapes

In this section, we will identify 3 special types of parallelograms, each with its own characteristics and properties:

1. Rhombus

2. Rectangle

3. Square

### Rhombus

A rhombus is a quadrilateral with all 4 of its sides equal in length (equilateral). It turns out that the opposite pairs of sides of a rhombus are always parallel. This makes every rhombus a parallelogram. Conversely, we can say that an equilateral parallelogram is a rhombus. The diagonals of a rhombus always bisect each other at right angles.

Since a rhombus is a special type of parallelogram, a rhombus exhibits all the properties of a parallelogram as well.

### Rectangle

A rectangle is a parallelogram with all of its internal angles as right angles. Since all angles in a rectangle are equal, it is equiangular.

Since a rectangle is a special type of parallelogram, it exhibits all the properties of a parallelogram as well.

### Square

A square is a quadrilateral with all 4 of its sides equal and with all of its angles as right angles. This makes a square a type of parallelogram, a type of rhombus, and a type of rectangle! Thus, a square demonstrates all the properties of parallelograms, rhombuses, and rectangles.

## Parallelograms - Key takeaways

• A quadrilateral with 2 pairs of opposite parallel sides is called a parallelogram.
• In a parallelogram, the opposite side are equal.
• In a parallelogram, the opposite angles are equal.
• In a parallelogram, consecutive angles are supplementary.
• If any angle in a parallelogram is a right angle, that means all 4 internal angles are right angles.
• The diagonals of a parallelogram bisect each other.
• Each diagonal of a parallelogram separates the parallelogram into two congruent triangles.
• The area of a parallelogram is given by the formula:

Area = b × h

where b = base, h = height

• A rhombus is a parallelogram with 4 equal sides.

• A rectangle is a parallelogram with all internal angles as right angles.

• A square is a parallelogram with 4 equal sides and all right angles.

A quadrilateral with 2 pairs of opposite parallel sides is called a parallelogram.

The area of a parallelogram is given by the formula:

Area = b × h

where b=base, h=height.

A rhombus is a type of parallelogram.

A trapezoid can have 1 pair of opposite sides that are not parallel, thus making it not a parallelogram.

A rectangle is a type of parallelogram.

## Final Parallelograms Quiz

Question

What is a parallelogram?

A quadrilateral with 2 pairs of parallel opposite sides is called a parallelogram.

Show question

Question

State the formula to find the area of a parallelogram.

Area = b × h

where b=base, h=height.

Show question

Question

What is a rhombus?

A rhombus is a quadrilateral whose all 4 sides are equal.

Show question

Question

What is a rectangle?

A rectangle is a parallelogram whose internal angles are all right angles.

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Question

What is a square?

A square is a quadrilateral whose 4 sides are all equal and whose angles are all right angles.

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Question

A parallelogram with an altitude of 4 has an area of 20. What is the length of the base?

5

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Question

A parallelogram with an altitude of 5 has an area of 45. What is the length of the base?

9

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Question

State whether the following statement is true or false:

In a parallelogram, the opposite angles are equal.

True

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Question

State whether the following statement is true or false:

In a parallelogram, the opposite angles are supplementary.

False

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Question

State whether the following statement is true or false:

In a parallelogram, consecutive angles are supplementary.

True

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Question

Which of the following are types of parallelograms?

Trapezoid

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Question

A parallelogram has a base 12 and altitude 8. What is its area?

96

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Question

The vertices of a parallelogram ABCD are A(3,6), B(7,7), C(5,1) and D(1,0). Find the coordinates of the intersection of the diagonals.

(4, 3.5)

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Question

We measure the internal angle of a parallelogram as 90. Can we conclude that it is definitely a rectangle?

Yes

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Question

We measure the internal angle of a parallelogram as 90. Can we conclude that it is definitely a square?

No

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