6. Polygons

6.02 Identify similar figures

6.03 Find unknown measures in similar figures

6.04 Dilations in the coordinate plane

6.05 Properties of quadrilaterals

Lesson

Practice

6.06 Find unknown measures in quadrilaterals

Book a Demo

United States of America Virginia

Grade 7

6.05 Properties of quadrilaterals

Lesson

Practice

Ideas

Compare and contrast properties of quadrilaterals
Classify quadrilaterals

Compare and contrast properties of quadrilaterals

A polygon is a closed plane figure composed of at least three line segments that do not cross.

Quadrilateral

A polygon with exactly four sides and four vertices

A set of different kinds of four-sided polygons.

Properties of quadrilaterals include the number of parallel sides, angle measures, number of congruent sides, lines of symmetry, and the relationship between the diagonals.

Diagonal

A diagonal is a segment in a polygon that connects two vertices but is not a side

Line of symmetry

A line of symmetry divides a figure into two congruent parts, each of which are mirror images of the other

Exploration

Check the boxes to change the type of quadrilateral.

Drag the blue vertices to change the size and orientation of the shape.

Loading interactive...

Explore each shape. Move the vertices to adjust its size and orientation and write down what you notice about the:

Sides
Angles
Diagonals
Lines of symmetry

A parallelogram with markings. Ask your teacher for more information.

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

The properties of parallelograms are:

Opposite sides are parallel and congruent
Opposite angles are congruent
Diagonals bisect each other (cut each other equally in half)
Each diagonal divides the figure into two congruent triangles

Parallelograms, have no lines of symmetry (except for rectangles and rhombi).

A rectangle with markings. Ask your teacher for more information.

A rectangle is a special type of parallelogram with four right angles.

Rectangles have the following properites:

Opposite sides are parallel and congruent
All four angles are 90\degree and congruent.
Diagonals are congruent and bisect each other
Have two lines of symmetry

Notice a rectangle has all of the properites of a parallelogram, plus some of its own properties.

A rhombus with markings. Ask your teacher for more information.

A rhombus is a special type of parallelogram with four congruent sides.

The following are properties of a rhombus:

All sides are congruent
Opposite sides are parallel
Opposite angles are congruent
Diagonals bisect each other at right angles
Has two lines of symmetry

Notice a rhombus has all of the properites of a parallelogram, plus some of its own properties.

A square with markings. Ask your teacher for more information.

A square is a regular polygon with four congruent sides and four right angles. This makes a square not only a parallelogram but also a rectangle and a rhombus. Its properties are:

Opposite sides are congruent and parallel
All four angles are congruent and each angle measures 90\degree
Diagonals are congruent and bisect each other at right angles
Has four lines of symmetry

Trapezoid with markings of parallel lines on the bases.

A trapezoid is a quadrilateral with exactly one pair of parallel sides. This means it is not a parallelogram.

The parallel sides of a trapezoid are called bases.

The nonparallel sides are called the legs.

An isosceles trapezoid with markings of parallel lines on the bases and arc markings to the angles. Ask your teacher for more information.

An isosceles trapezoid is a type of trapezoid whose legs are equal in length. Its properties are:

Legs are congruent
Base angles are congruent
Bases are parallel
Has one line of symmetry

Examples

Example 1

Place each shape in its appropriate place in the Venn diagram.

A Venn Diagram showing three overlapping circles labeled Diagonals always bisct each other, are perpendicular, and are equal in length.

trapezoid

rectangle

square

parallelogram

rhombus

Worked Solution

Create a strategy

We need to understand the properties of each shape and how they relate to one another in the Venn diagram.

Apply the idea

Square - Diagonals are congruent and are bisect each other at right angles (are perpendicular)

Rectangle - Diagonals are congruent and bisect each other

Rhombus - Diagonals bisect each other at right angles (are perpendicular)

Parallelogram - Diagonals bisect each other

Trapezoid - Diagonals have none of the listed properties

A Venn Diagram showing three overlapping circles labeled Diagonals always bisct each other, are perpendicular, and are equal in length. Ask your teacher for more information.

Example 2

Place the shapes in order from smallest to greatest number of lines of symmetry.

A table with four shapes. The first column shows a square, the second column shows an isosceles trapezoid, the third column shows a parallelogram, and the fourth column shows a rhombus.

Worked Solution

Create a strategy

First, we need to draw and count the number of lines of symmetry each shape has. Then, we can order them by the number of their lines of symmetry.

Apply the idea

A table with four shapes and dashed lines showing their lines of symmetry. The first column shows a square with a dashed vertical line, a dashed horizontal line, and two dashed diagonals. The second column shows an isosceles trapezoid with a vertical dashed line. The third column shows a parallelogram. The fourth column shows a rhombus with two dashed diagonals.

A square has 4 lines of symmetry, an isosceles trapezoid has 1, a parallelogram has none, and a rhombus has 2. In order from smallest to great:

	\text{Smallest}	\to	\to	\text{Greatest}
Lines of symmetry	0	1	2	4
Shape	\text{Parallelogram}	\text{Isosceles trapezoid}	\text{Rhombus}	\text{Square}

Idea summary

The properties of sides and angles of the different properties of quadrilaterals can be summarized as follows:

A table of the different properties of quadrilaterals in terms of sides and angles. Ask your teacher for more information.

The properties of diagonals can be summarized as follows:

A table of the different properties of quadrilaterals in terms of diagonals. Ask your teacher for more information.

Classify quadrilaterals

Exploration

We've just learned all about some different properties of quadrilaterals. Let's explore whether there is any overlap.

Review the properties of all of the different types of quadrilaterals. Are there any shapes that satisfy all of the criteria of a different shape?
Are there any quadrilaterals that do not satisfy the the criteria for a different type of shape?

All of the shapes we've discussed are quadrilaterals because they have 4 sides. Quadrilaterals all share some properties, but they can be divided in to different subgroups based on additional properties that a given shape has.

Let's take a look at the quadrilateral hierachy:

A quadrilateral hierachy. Ask your teacher for more information.

Each shape belongs to all of the groups connected above it.

A Venn diagram titled Quadrilateral is composed of 3 overlapping circles labeled Parallelograms, Rhombi, and Rectangles. The overlapping area has label of Squares. A circle outside has a label of Trapezoids.

The Venn diagram is another way to visualize which groups a figure belongs to.

Some figures belong to multiple groups. When a shape belongs to multiple groups, the most precise classification is the group that is the most restrictive one (the one with the most properties).

For example, we see that a square is a rectangle, a rhombus, a parallelogram, and a quadrilateral. The most precise classification is the most detailed one, in this case a square.

It is also important to note that all squares are rectangles, but not all rectangles are squares.

A rhombus is a parallelogram and a quadrilateral, but the most precise classification is a rhombus.

A rectangle is also a parallelogram and a quadrilateral, but the most precise classification is a rectangle.

Examples

Example 3

Consider what you know about parallelograms.

Which of the following quadrilaterals are not parallelograms?

A rectangle where it shows 4 right angles and each pair of opposite sides have the same markings.

A square where it shows 4 right angles and all sides have the same markings.

A parallelogram where it shows opposite sides congruent.

A rhombus where it shows all sides have the same markings.

A trapezoid with the top base shorter than the bottom base and parallel markings on the bases

Worked Solution

Create a strategy

Remember that a quadrilateral is a parallelogram if one of the following is true:

Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent

Apply the idea

Options A and B show that both pairs of opposite sides are congruent and both pairs of opposite angles are congruent. So they must be parallelograms.

Options C and D show both pairs of opposite sides are congruent. So they must be parallelograms.

The quadrilateral in option E does not satisfy any of the properties of a parallelogram. Only one pair of opposite sides is parallel. So the correct answer is option E.

Reflect and check

Since option E has exactly one pair of opposite sides parallel, it is a trapezoid.

Example 4

Patricia draws a quadrilateral, and covers it up. She tells Glen that the quadrilateral consists of right angles only. From this information, Glen knows that the quadrilateral is definitely a:

rhombus

square

trapezoid

rectangle

Worked Solution

Create a strategy

Identify which types of quadrilaterals that have four right angles. Then choose the type that doesn't have any additional requirements.

Apply the idea

From the choices, square and rectangle both have four right angles. A square is a special type of rectangle with the additional property that all four sides must be congruent.

Since Patricia said nothing about the side-lengths Glen cannot say that the quadrilateral is definitely a square.

So the quadrilateral is a rectangle. The correct answer is option D.

Example 5

For each of the shapes below, choose the most precise classification.

Trapezoid

Rectangle

Square

Worked Solution

Create a strategy

Use the markings to identify the properties of the shape.

Apply the idea

The shape has:

opposite sides congruent
four right angles

The quadrilateral that has these properties is a rectangle. So the correct answer is option B.

Reflect and check

The most precise classification here is rectangle. A rectangle is also a parallelogram and a quadrilateral.

We could not say this rectangle is a square because all four sides are not congruent.

Square

Trapezoid

Rectangle

Worked Solution

Create a strategy

Use the markings to identify the properties of the shape.

Apply the idea

The shape has:

all four sides congruent
four right angles

The quadrilateral that has these properties is a square. So the correct answer is option A.

Reflect and check

This quadrilateral is a rectangle, a rhombus, a parallelogram, and a quadrilateral but we classify a shape by choosing the most precise classification. This is a square.

Trapezoid

Rectangle

Square

Worked Solution

Create a strategy

Use the markings to identify the properties of the shape.

Apply the idea

The shape has:

one pair of opposite sides parallel
two right angles

The quadrilateral that could satisfy these properties is a trapezoid. So the correct answer is option A.

Reflect and check

A quadrilateral does not need to have two right angles to be considered a trapezoid. But having two right angles also does not exclude a shape from being a trapezoid. Having exactly one pair of parallel sides was enough to classify this shape. The information about the angles was not needed.

A four-sided shape shows all sides have the same markings and angles are congruent.

Rhombus

Square

Rectangle

Worked Solution

Create a strategy

Identify the properties that are shown by the markings.

Apply the idea

The shape has:

all four sides are congruent
opposite angles are congruent

The quadrilateral that has this property is a rhombus. So the correct answer is option A.

Example 6

A quadrilateral has exactly two lines of symmetry. Using only this information, which shape could it be? Select all that apply.

Rectangle

Square

Rhombus

Isosceles trapezoid

Worked Solution

Create a strategy

Determine the number of lines of symmetry each shape has and select the shape(s) that has exactly two lines of symmetry.

Apply the idea

A rectangle has two lines of symmetry.
A square has four lines of symmetry.
A rhombus has two lines of symmetry.
An isosceles trapezoid has one line of symmetry.

The correct options are A and C.

Idea summary

Quadrilateral Hierarchy:

Outcomes

7.MG.3

The student will compare and contrast quadrilaterals based on their properties and determine unknown side lengths and angle measures of quadrilaterals.

7.MG.3a

Compare and contrast properties of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid: i) parallel/perpendicular sides and diagonals; ii) congruence of angle measures, side, and diagonal lengths; and iii) lines of symmetry.

7.MG.3b

Sort and classify quadrilaterals, as parallelograms, rectangles, trapezoids, rhombi, and/or squares based on their properties: i) parallel/perpendicular sides and diagonals; ii) congruence of angle measures, side, and diagonal lengths; and iii) lines of symmetry.

6.05 Properties of quadrilaterals

Ideas

Compare and contrast properties of quadrilaterals

Exploration

Examples

Example 1

Example 2

Classify quadrilaterals

Exploration

Examples

Example 3

Example 4

Example 5

Example 6

Outcomes

7.MG.3

7.MG.3a

7.MG.3b

What is Mathspace

About Mathspace