Parallelogram

Here you will learn about parallelograms, including their properties, and special cases.

Students first learn about parallelograms in 3 rd grade with their work in geometry. They expand upon their knowledge of parallelograms as they progress through 5 th grade.

What are parallelograms?

Parallelograms are quadrilaterals with opposite sides being parallel to each other.

Parallelograms have other unique properties.

Properties of a Parallelogram

★ Two pairs of opposite congruent sides

• Side AB and Side DC are congruent
• Side AD and Side BC are congruent

★ Two pairs of opposite congruent angles

• Angle A and Angle C are congruent
• Angle D and Angle B are congruent

★ Diagonals bisect each other

• Line segment BD
• Line segment AC

★ Adjacent angles are supplementary meaning they sum to \bf{180^{\circ}}

• angle A + angle B = 180^{\circ}
• angle A + angle D = 180^{\circ}
• angle C + angle B = 180^{\circ}
• angle C + angle D = 180^{\circ}

★ One diagonal divides the parallelogram into two congruent triangles

• Triangle ABC is congruent to triangle CDA
• Triangle DAB is congruent to triangle BCD

Parallelograms also have diagonals. Diagonals are line segments that connect opposite vertices. For example, in the diagram below, line segments AC and BD are diagonals.

There are three types of special parallelograms.

Name	Shape	Attributes
Rectangle		● Four right angles ● Congruent diagonals
Rhombus		● Four congruent sides ● Perpendicular diagonals
Square		● Four right angles ● Congruent diagonals ● Four congruent sides ● Perpendicular diagonals

To help understand how the parallelograms relate to one another, you can use a Venn diagram.

What are parallelograms?

Common Core State Standards

How does this relate to 3 rd grade math – 5 th grade math?

• Grade 3 – Geometry (3.G.A.1)
  Understand that shapes in different categories (for example, rhombuses, rectangles, and others) may share attributes (like, having four sides), and that the shared attributes can define a larger category (example, quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

• Grade 4 – Geometry (4.G.A.2)
  Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

• Grade 5 – Geometry (5.G.3)
  Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

• Grade 5 – Geometry (5.G.4)
  Classify two-dimensional figures in a hierarchy based on properties.

How to classify parallelograms

In order to classify a parallelogram:

1. Recall the definition and properties of parallelograms.
2. Identify the parallelogram.

Parallelogram examples

Example 1: parallelogram

Select the quadrilateral that is a parallelogram.

1. Recall the definition and properties of parallelograms.

Parallelograms are quadrilaterals that have two pairs of opposite parallel sides.

2Identify the parallelogram.

Quadrilateral A is the parallelogram because, by the definition of a parallelogram, there are two pairs of opposite parallel sides.

Example 2: parallelograms

Kelly drew the quadrilateral below and said it was only a parallelogram. Is she correct?

Example 3: parallelogram

Lucas’s teacher asked him to draw a parallelogram that has four congruent sides but not four congruent angles. Lucas draws a rhombus. Is he correct?

Example 4: square

Name the special parallelogram.

Example 5: trapezoid

Name the quadrilateral.

Example 6: quadrilaterals

Describe how the quadrilaterals are the same and how they are different. Name them.

Examine the properties of the quadrilateral, including side and angle relationships.

Quadrilateral A has two pairs of opposite parallel sides and two pairs of opposite congruent sides. Quadrilateral A also has two pairs of opposite congruent angles.

Quadrilateral B has one pair of opposite parallel sides and two pairs of congruent angles. Quadrilateral B also has one pair of opposite congruent sides.

The quadrilaterals are the same because they have a pair of opposite parallel sides, a pair of opposite congruent sides, and two pairs of congruent angles.

They are different because Quadrilateral A has two pairs of opposite parallel sides and Quadrilateral B only has one pair. Quadrilateral A has two pairs of opposite congruent sides, and quadrilateral B only has one.

Also, Quadrilateral A has two pairs of opposite angles that are congruent, whereas Quadrilateral B has adjacent angles (next to each other) that are congruent.

Teaching tips for parallelogram

• Have students create a hierarchy of quadrilaterals on their own so that they can investigate the relationships between the sides of a quadrilateral and the angles of a quadrilateral.

• Instead of worksheets, have students engage with geometric digital platforms such as desmos to investigate the properties of quadrilaterals.

• Incorporate project based learning activities which include ways for students to see how the concepts are applied in a real world setting.

Easy mistakes to make

• Assuming all quadrilaterals with four \bf{90}^{\circ} angles are squares
  Rectangles and squares are quadrilaterals with four right angles. If the four sides are equal, then it is a square. If the four sides are not equal, then it is a rectangle.

• Mistaking a trapezoid for a parallelogram
  Trapezoids are not classified as parallelograms because they do not have the properties of parallelograms. Trapezoids have only one pair of opposite parallel sides. Parallelograms have two pairs of opposite parallel sides.

• Mistaking a rhombus for a square
  A rhombus has four equal sides, but its angles are not right angles like in a square.

Related quadrilateral lessons

• Quadrilaterals
• Types of quadrilaterals
• Rectangle
• Rhombus
• Square shape
• Trapezoid

The next lessons are

• Perimeter
• Symmetry
• Angles in polygons
• Surface area

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