## Table of Contents

- What Is a Parallelogram?
- Types of Parallelogram Shapes
- Properties of Parallelograms
- Area of a Parallelogram
- Lesson Summary

What is a parallelogram? Learn the definition of the parallelogram shape in geometry. Read about the properties of parallelograms and see examples.
Updated: 08/27/2021

- What Is a Parallelogram?
- Types of Parallelogram Shapes
- Properties of Parallelograms
- Area of a Parallelogram
- Lesson Summary

**Parallelograms** are found everywhere in nature and in geometry. They are a specific classification for shapes with four sides. Notebooks, cell phones, and keyboards are all real-world examples of shapes that lie within the parallelogram classification.

The definition of a parallelogram is a geometric shape with four sides, and, as the name suggests, two sets of parallel lines. All parallelograms have exactly four corners, four angles, and four sides. Since they have four sides, it is considered a type of quadrilateral. **Quadrilaterals** are shapes with four total sides. Parallelograms will have either the same length for every side or two different side lengths (one length for each pair of parallel lines). Parallelograms will never have more than two length values for the sides, or they wouldn't be considered a parallelogram. If three of the four sides had different lengths, then one of the line pairs wouldn't be parallel. Therefore, it is not possible for there to be more than two side lengths if there are two sets of parallel lines.

Notice that the illustration has two sets of parallel sides and two side length values. It also has exactly four corners, angles, and sides. It wouldn't be possible for a shape to contain four sides with two sets of parallel lines and three different lengths.

There are a few different quadrilateral shapes that are considered to be parallelograms. These shapes include squares, rectangles, rhombuses, and rhomboids. All of these shapes have four sides, four corners, and four angles. They also all have two sets of parallel lines, making them parallelograms.

A **square** is a shape with four sides of equal length. They also have four 90° angles (or four right angles); One right angle located at each corner. Squares are categorized as being parallelograms because they are quadrilaterals with two sets of parallel lines.

The shape in the image has four sides, making it a quadrilateral. Since it has four right angles and four sides of equal lengths, it is a square. It also has two sets of parallel lines opposite of one another, making it a parallelogram.

**Rectangles** are a type of quadrilateral with two different side lengths. Since they are quadrilateral, they have four sides, four corners, and four angles. All of the angles in a rectangle are right angles, similar to squares. Unlike squares, however, not all of the side lengths are equal. Two of the sides will be a different length than the other two.

Notice how not all the sides are the same length, but there are still four right angles at each corner. A rectangle is still considered a parallelogram because it has two sets of lines parallel opposite one another. The sides opposite of another also have the same length values.

A **rhombus** is a type of shape with four sides, making it a quadrilateral. Similar to a square, a rhombus is classified by having four sides of the same length. The difference, however, is that a rhombus doesn't have four right angles or any right angles at all. Rhombuses instead have two acute and two obtuse angles, each directly across from each other. This makes the shape appear like a slanted square since the sides are slanted at an angle. A diamond is an example of a rhombus, as it has four sides of equal length, but they are slanted. Since a diamond is a rhombus, and a rhombus is a parallelogram, then a diamond is also a parallelogram.

The illustration displays a shape with four sides of the same length, but no right angles at the corners. This makes the shape a rhombus since there are four sides of equal length but slanted. Notice how the acute and obtuse angles are across from one another. There are also two separate sets of parallel lines, making this shape a parallelogram.

The last type of parallelogram is called a **rhomboid**. As the name hints, a rhomboid is very similar to a rhombus. It's also very similar to a rectangle. Rhomboids have four sides, making them quadrilateral. However, unlike a rhombus, a rhomboid doesn't have four sides of the same length. Two of the sides have a different length than the other two (similar to a rectangle). They don't have any right angles though, making them different than rectangles. They have two acute and two obtuse angles opposite of each other.

A rhomboid can be thought of like a rectangle with slanted sides. Since they consist of two pairs of parallel lines, they are classified as parallelograms. Notice how this rhomboid has two acute and two obtuse angles opposite from one another. This is one of the characteristics of rhomboids.

This chart illustrates how each of these shapes relates to the others. All of the shapes have four sides, four corners, and four angles. This makes all of the shapes quadrilaterals. A rhombus and a square are similar because they both have four sides of equal length. A rectangle and a rhomboid are similar because they have two different side lengths. All of the shapes are parallelograms because they all have two sets of parallel lines and four total sides.

There are specific properties of parallelograms that make them geometrically unique. When followed, these guidelines can determine whether or not a shape is considered a parallelogram.

- Quadrilateral: A quadrilateral consists of four sides. In order for a shape to be considered a parallelogram, it must have exactly four sides.
- Parallel sides: A parallelogram must have two sets of parallel lines opposite from one another.
- Bisecting lines: Lines drawn diagonally from each corner of a parallelogram will create bisecting lines. These are lines that intersect halfway from each line.
- Equal side length: All parallelograms have at least one pair of sides of equal length.
- Congruent triangles: When a single line is drawn diagonally from one corner to another on a parallelogram, two congruent triangles are formed. Congruent triangles are triangles with all sides having the same length.
- Opposite but equal angles: Angles opposite from each other in a parallelogram will have the same measure.
- Opposite but equal lines: Parallelograms always have sides with the same length (congruent) as the side opposite from them.

Notice how in the diagram, the diagonal lines all meet at the halfway point from each line. This satisfies the third property, which states that parallelograms must form bisecting lines when a diagonal is drawn from each corner. This shape also has four sides, two sets of parallel lines, at least one pair of lines of equal length, equal length lines opposite of one another, equal angles opposite of one another, and two congruent triangles from one diagonal line. Therefore, the shape in this image is considered a parallelogram.

The parallelogram featured in the illustration forms two congruent triangles when a single diagonal line is drawn from one corner to another. This is one of the guidelines for a parallelogram.

To test if a shape is considered a parallelogram, all of these requirements must be met. For example; Although a kite has four sides and can appear to have parallel lines, a kite is not a parallelogram because the lines drawn at each diagonal don't intersect at the halfway points of each line. They also don't have opposite sides of equal measure, therefore, a kite is not a parallelogram.

Parallelograms are shapes with two sets of parallel lines and four total sides. Like any shape in geometry, parallelograms contain an area that can be calculated given enough information. The area of a parallelogram can be determined by multiplying the base by the height of the shape.

{eq}A = B \times H {/eq}

The area (A) is equal to the base length (B) multiplied by the height (H).

For example; A rectangle has a base of 4cm and a height of 2cm. To find its area, multiply the base by the height. The resulting area is 6{eq}cm^{2} {/eq} since 4 x 2 = 6.

What is the area of a rhomboid with a base of 5cm and a height of 4cm?

Since the area is equal to the base times the height, multiple 5cm by 4cm to get 20{eq}cm^{2} {/eq}. Therefore, the area of the rhomboid is 20{eq}cm^{2} {/eq}.

**Parallelograms** are a type of geometric shape classified by having two sets of parallel lines, and they are often found in nature and in geometry. They are considered to be quadrilaterals since they have four total sides. There are a few different types of shapes that fall under the category of parallelograms, including **squares**, **rectangles**, **rhombuses**, and **rhomboids**. These shapes are parallelograms because they meet all of the criteria for parallelograms. These criteria include the following:

- Four sides
- Two sets of parallel lines
- Bisecting lines formed when drawing diagonals from each corner
- At least one pair of the sides have equal length
- Sides opposite of each other will be the same length
- Angles opposite of each other will be the same measurement
- Two congruent triangles are formed when a single diagonal is drawn from one corner to another

If a shape meets all of the listed requirements, then it is a parallelogram. If a shape fails even just one of the guidelines, then it is not considered a parallelogram. The area of a parallelogram can be determined by multiplying the base length by the height.

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- Activities
- FAQs

In the following activity, students will verify the properties of parallelograms through measurements. The hands-on measuring will help learners remember the properties of parallelograms.

- Various parallelograms on paper - include rectangles, squares, rhombuses, and rhomboids.
- Ruler
- Protractor
- Pencil or pen

1. For each parallelogram, measure the opposite sides with the ruler. What do you observe?

2. For each parallelogram, measure the opposite angles with the protractor (opposite angles diagonally - that is if the upper left angle is measured, the lower right angle should be measured.) What do you observe?

3. For each parallelogram, measure the angles which share a side and write down the measurements (there should be 4 pairs of angles to write down). What do you observe?

4. Use the ruler to draw the two diagonals of the parallelograms. Then, measure the length of each diagonal from a corner to the point of intersection in the middle. What do you observe?

Students should observe that the opposite sides of the parallelograms are congruent - even if all four sides are not. Similarly, they should observe that the opposite angles in a parallelogram are congruent - even if all four angles are not congruent. When measuring angles which share a side, if students do not notice any relationship, suggest they try different operations on the angles - the goal is to note that the sum of the angles which share a side is always equal to 180 degrees. When measuring the pieces of the diagonals, students should observe that the diagonals are divided perfectly in half.

Parallelograms must have two sets of parallel lines, with four lines total. At least one pair of the lines will have equal length. Lines drawn diagonally from the corners will create bisecting lines.

A rectangle is considered a parallelogram. It has two pairs of parallel lines, and at least one of the parallel lines is the same length, making it a parallelogram.

Although they can, not all parallelograms have four equal sides. They will, however, have at least two sides of the same length.

There are four types of parallelograms and include squares, rectangles, rhombuses, and rhomboids. These shapes are considered to be parallelograms because they have two sets of parallel lines.

A square is considered a parallelogram because it consists of two sets of parallel lines, and at least one pair of parallel lines are the same length.

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