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9.4: Classifying Quadrilaterals

Created by: CK-12

Introduction

The Grind Box

Isabelle is a friend of Marc and Isaac’s at school. She overheard them talking at lunch about the skatepark and is excited to help. Isabelle has been skateboarding for a few years and loves the sport.

“She is really good,” Marc says to Isaac.

“Yes, and she is coming over to help us,” Isaac says.

He and Marc have a snack and soon the doorbell rings and it is Isabelle. She has brought her notebook and pencil.

“Hi guys, I have a great idea for the grind box,” Isabelle says, as Isaac’s mom comes into the room.

“What is a grind box?” Isaac’s mom asks.

“It’s a box designed for grinding and sliding tricks,” Marc explains. “We are going to design one today for our park.”

“We can probably build it ourselves,” says Isabelle.

Isaac’s Mom smiles and leaves the room. The three get to work. All is going well until they begin drawing the actual grind box. All three of them have different ideas about the shape the box should be.

Here is Isaac’s box.

Isabelle looks at the drawing and shakes her head.

“Those won’t work guys, the angles are all wrong. You have to have right angles to make this work. The box is only a box if you use rectangles and squares in the design.”

“The angles are alright on this one,” Isaac argues, looking at his design.

“No they aren’t. This is a parallelogram-the angles are not right angles and this box could fall over if you landed incorrectly.”

“We have to design a grind box using squares and rectangles.”

“The angle thing makes sense,” Marc says. “but are you sure that only squares and rectangles will work?”

Before Isabelle can answer, stop right there-this is where you come in. This lesson will teach you all about quadrilaterals. By the end of the lesson, you will be able to answer this question yourself and help the trio with their dilemma.

What You Will Learn

By the end of this lesson, you will be able to apply the following skills:

  • Classify quadrilaterals by angles
  • Classify quadrilaterals by sides
  • Draw specified quadrilaterals using ruler and protractor
  • Find unknown angle measures in given quadrilaterals

Teaching Time

In our last lesson you learned all about triangles. In this lesson, you will learn about quadrilaterals. Let’s begin by learning about identifying a quadrilateral.

What is a quadrilateral?

A quadrilateral is a closed figure with four sides and four vertices. Remember that the vertex is a point where line segments meet. The points where the sides of a quadrilateral meet are called the vertices (which is the plural form of "vertex"). The prefix “quad” means four. You can always remember that a quadrilateral is a four sided figure because of this prefix.

Each quadrilateral has four sides and four angles. Let’s look at an example.

A quadrilateral has four angles. We can name the angles by using the angle symbol and the letter of each vertex. Here we have \angle{A}, \angle{B}, \angle{C}, \angle{D}.

A quadrilateral also has four vertices. They are named by letter, A, B, C and D. Naming the quadrilateral uses a small quadrilateral symbol and the four letters of the vertices, \Box \ ABCD.

A quadrilateral has four sides. The sides are named by the endpoints of each line segment. The sides are \overline{AB}, \overline{BD}, \overline{CD}, \overline{AC}.

There are several different kinds of quadrilaterals. We can classify and identify quadrilaterals based on angles and side lengths. While each four sided figure is a quadrilateral, sometimes there is a better name for the figure. A name that is more specific can tell us things about the figure.

Let’s learn about classifying quadrilaterals.

I. Classify Quadrilaterals by Sides and Angles

There are several different types of quadrilaterals. We can figure them out by looking at the relationship between the sides of the quadrilateral and the angles.

Parallelogram – a quadrilateral with opposite sides which are parallel and congruent.

Congruent is a word that you will see a lot in geometry. Congruent means exactly the same. In this case, a parallelogram has opposite sides that are parallel and congruent meaning that they have the same length.

Here is an example of a parallelogram.

What about the angles of a parallelogram?

Notice that the opposite angles in this parallelogram are the same. In this figure, one pair of angles is acute and one pair of angles is obtuse. The arrows indicate the obtuse angles. The other two angles are acute. This is sometimes true but not always. Let’s look at a rectangle.

Rectangle is a parallelogram with four right angles. This means that a rectangle has opposite sides parallel and congruent and it also has four right angles.

Square is a rectangle with four congruent sides and four right angles. Notice that the properties of a parallelogram applies to squares too.

Rhombus is a parallelogram with four congruent sides. Notice that a rhombus does not ALWAYS have right angles like a square, but it does always have congruent sides.

Trapezoid is a parallelogram with one pair of opposite sides parallel. Here is an example of trapezoid. Notice that this trapezoid has two acute angles and two obtuse angles. Be sure to check out the angles when you look at a trapezoid.

Identify each of the following quadrilaterals. Be as specific as you can.

1.

2.

3.

Check your work with a neighbor.

II. Draw Specified Quadrilaterals Using a Ruler and a Protractor

What if you wanted to draw a specific quadrilateral? How could you do it?

You can draw specific quadrilaterals using a ruler and a protractor. We use the protractor to be sure that our work is accurate. This is especially important when drawing squares or rectangles or any figure with a right angle.

How would we draw a square?

We can start by using a protractor to draw in each of the four right angles. By using a ruler and a protractor, our lines will be straight and we will be able to determine that we have drawn the square correctly.

Drawing it freehand may seem easier, but it does not assure accuracy! The best way to be sure that your work is accurate is to use a protractor and a ruler.

Here is the first angle of my square. Now I can turn my protractor upside down and draw the other angle.

Here is my final figure.

Here we have a square that is accurate. To check our work, we can re-measure each angle using the protractor.

You can use a protractor to measure angles when drawing any of the quadrilaterals.

IV. Find Unknown Angle Measures in Given Quadrilaterals

In our last lesson, you learned that the sum of the interior angles of a triangle is equal to 180 degrees. What about a quadrilateral? This section will teach you about the sum of the interior angles of a quadrilateral. We will use this information in problem solving.

What is the sum of the interior angles of a quadrilateral?

To best understand this, let’s look at a square.

A square has four right angles. Each right angle is 90^\circ. We can add up the sum of the interior angles of a square and see how this is related to all quadrilaterals.

90 + 90 + 90 + 90 = 360^\circ

The sum of the interior angles of all quadrilaterals is 360^\circ.

How can we use this information to find the measure of missing angles?

Example

We can write an equation using the variable and given measurements and figure out the measure of the missing angle.

80 + 75 + 105 + x & = 360 \\260 + x & = 360 \\360 - 260 & = x \\100 & = x

The missing angle is equal to 100^\circ.

You can use this information to help you when figuring out missing angle measures in different quadrilaterals.

Real Life Example Completed

The Grind Box

Here is the original problem once again. Reread it and then see if you can answer the questions posed about the shapes used in the box.

Isabelle is a friend of Marc and Isaac’s at school. She overheard them talking at lunch about the skatepark and is excited to help. Isabelle has been skateboarding for a few years and loves the sport.

“She is really good,” Marc says to Isaac.

“Yes, and she is coming over to help us,” Isaac says.

He and Marc have a snack and soon the doorbell rings and it is Isabelle. She has brought her notebook and pencil.

“Hi guys, I have a great idea for the grind box,” Isabelle says as Isaac’s mom comes into the room.

“What is a grind box?” Isaac’s mom asks.

“It’s a box designed for grinding and sliding tricks,” Marc explains. “We are going to design one today for our park.”

“We can probably build it ourselves,” says Isabelle.

Isaac’s Mom smiles and leaves the room. The three get to work. All is going well until they begin drawing the actual grind box. All three of them have different ideas about the shape the box should be.

Here is Isaac’s box.

Isabelle looks at the drawing and shakes her head.

“Those won’t work guys, the angles are all wrong. You have to have right angles to make this work. The box is only a box if you use rectangles and squares in the design.”

“The angles are alright on this one,” Isaac argues, looking at his design.

“No they aren’t. This is a parallelogram-the angles are not right angles and this box could fall over if you landed incorrectly.”

“We have to design a grind box using squares and rectangles.”

“The angle thing makes sense,” Marc says. “but are you sure that only squares and rectangles will work?”

First, underline all of the important information.

Next, let’s think about the characteristics of the three figures in the problem.

First, Isaac draws a grind box parallelogram which has opposite sides parallel and congruent. The sum of the angles of a parallelogram equals 360^\circ, but the angles are not necessarily right angles.

A rectangle and a square are also parallelograms, but they have angles that are 90^\circ.

Isabelle is correct that a parallelogram might tip over because the angles are not right angles.

The grind box pictured uses squares and rectangles because of the stability of the right angle. Isabelle, Marc and Isaac decide to build their grind box using squares and rectangles, that way they can be sure that it is a stable construction.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Quadrilateral
closed figure with four sides and four vertices.
Trapezoid
Quadrilateral with one pair of opposite sides parallel.
Rectangle
Parallelogram with four right angles.
Parallelogram
Quadrilateral with opposite sides congruent and parallel.
Square
Four congruent sides and four congruent angles.
Rhombus
Parallelogram with four congruent sides.
Parallel
lines that are equidistant and will never intersect
Congruent
exactly the same, having the same measure

Technology Integration

Khan Academy Quadrilateral Properties

  1. http://www.teachertube.com/members/viewVideo.php?video_id=158697&title=Know_Your_Quadrilaterals – This is a music video on different quadrilaterals and how to identify them.
  2. http://www.teachersdomain.org/resource/vtl07.math.geometry.pla.skateboard/ – This is a video from PBS on using parallelograms to reach different things.

Time to Practice

Directions: Look at each image and name the quadrilateral pictured.

1.

2.

3.

4.

5.

Directions: Name the geometric figure described below.

6. Has four sides and four angles

7. Has one pair of opposite sides that are parallel

8. Has four right angles and four congruent sides

9. A parallelogram with four right angles.

10. A parallelogram with four congruent sides

Directions: Answer each of the following questions about quadrilaterals.

11. True or false. A quadrilateral will always have only four sides.

12. The interior angles of a quadrilateral add up to be _________ degrees.

13. A square will have four ___________ degree angles.

14. A rectangle will have four ___________ degree angles.

15. True or false. A rhombus will also always have four right angles.

16. If the sum of three of the angles of a quadrilateral is equal to 300^\circ, it means that the measure of the missing angle is ____________.

17.

What is the value of x?

18.

19.

20. What are all four angles of this rectangle equal to?

21. If the sum of the interior angles of a quadrilateral is equal to 360^\circ, how many triangles can you draw inside a quadrilateral?

22. How many degrees are in a triangle?

23. Write an equation to show how the angles of the two triangles are equal to 360 degrees.

Directions: Identify the following figures.

24.

25.

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