Quadrilaterals

Here you will learn about quadrilaterals, including their names, attributes and how to classify them.

Students first learn about quadrilaterals in early elementary school with their work in geometry. They expand upon the types of quadrilaterals as they progress through elementary school.

What are quadrilaterals?

Quadrilaterals are polygons with four straight sides, four angles, and four vertices.

Here are the special types of quadrilaterals.

Name	Shape	Attributes
Square		● Four right angles ● Four congruent sides ● Two pairs of opposite parallel sides Step-by-step guide: Square shape
Rectangle		● Two pairs of opposite congruent sides ● Two pairs of opposite parallel sides ● Four right angles Step-by-step guide: Rectangle
Parallelogram		● Two pairs of opposite congruent sides ● Two pairs of opposite congruent angles ● Two pairs of opposite parallel sides Step-by-step guide: Parallelogram
Rhombus		● Four congruent sides ● Two pairs of opposite parallel sides ● Two pairs of opposite congruent angles Step-by-step guide: Rhombus
Trapezoid		● One pair of opposite parallel sides called bases
Isosceles Trapezoid		● Non-parallel sides are congruent ● Two pairs of congruent angles Step-by-step guide: Trapezoid
Kite		● Two pairs of adjacent sides that are congruent (next to each other)

To help remember the properties of quadrilaterals and how they relate to one another. You can use a Venn diagram.

Step-by-step guide: Types of Quadrilaterals

Diagonals:

Quadrilaterals have two diagonals. Diagonals are line segments that connect non-consecutive vertices.

Other properties of quadrilaterals:

• The sum of the interior angles of a quadrilateral total 360 ^{\circ}.

• Quadrilaterals can be convex or concave.

⟶ A convex quadrilateral is a polygon with all interior angles less than 180^{\circ}. All quadrilaterals shown in the table above are convex.

⟶ A concave quadrilateral has at least one angle that is greater than 180^{\circ} (or a reflex angle). The quadrilateral shown below is concave.

What are quadrilaterals?

Common Core State Standards

How does this relate to 2 nd grade – 5 th grade math?

• Grade 2 – Geometry (2.G.A.1)
  Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

• Grade 3 – Geometry (3.G.A.1)
  Understand that shapes in different categories (example, rhombuses, rectangles, and others) may share attributes (example, 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 name quadrilaterals

In order to name a quadrilateral:

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

Quadrilaterals examples

Example 1: parallelogram

The quadrilateral has two pairs of parallel sides. Identify the type of quadrilateral that it is.

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

The quadrilateral has two pairs of parallel sides. It also appears that the quadrilateral has two pairs of opposite congruent sides and two pairs of opposite congruent angles.

2Identify the quadrilateral.

The quadrilateral is a parallelogram.

Example 2: rectangle

What is the best classification for the quadrilateral?

Example 3: regular quadrilateral

Select the regular quadrilateral and name it.

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

Example 4: rhombus

Kelly wants to classify the quadrilateral. What is the best classification?

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 quadrilaterals

• 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 hold 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.

The next lessons are

• Perimeter
• Symmetry
• Angles in polygons
• Surface area

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