Lesson 2

Lesson 2: Quadrilaterals

Focus

Shown is a photograph of an inukshuk, which is a stone landmark created by the Inuit people of the Arctic region of North America. The inukshuk may have been used as a point of reference or a marker for travel routes.

Commonly, an inukshuk is a single stone positioned in an upright manner. In the photograph here, the inukshuk resembles a human being. An inukshuk that resembles a human being is called an inunnguaq. Many of the stones that make up this figure resemble simpler geometric shapes. Can you identify the quadrilaterals in the photograph? In this lesson you will find out what characteristics make up a quadrilateral.

Lesson Outcome

At the end of this lesson you will be able to solve problems that involve quadrilaterals.

Lesson Questions

You will investigate the following questions:

• How can the properties of quadrilaterals be used to help solve problems?
• How do the properties of quadrilaterals affect quadrilaterals’ shape and size?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
  
• course folder submissions from Try This and Share activities
  
• additions to Glossary Terms

Materials and Equipment

• calculator
• ruler
• protractor

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Before beginning this lesson you should be able to understand the terms parallel, perpendicular, and diagonal.

Are You Ready?

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

Using the diagrams, answer the following questions.

1. Fill in the blanks with perpendicular, diagonal, or parallel.
   1. In the square, side AB is ______________ to side AD. Answer
   2. In the parallelogram, line GI is a ________________. Answer
   3. In the trapezoid, side LM is _________________ to side KN. Answer
2. In the parallelogram, name a set of sides that are parallel. Answer
3. In the square, diagonal BD is perpendicular to what? Answer
   

If you answered the Are You Ready? questions without difficulty, move to Discover.

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

To review the term parallel, visit Parallel at the Mathematics Glossary.

To review the term perpendicular, visit Perpendicular at the Mathematics Glossary.

To review the term diagonal, visit Diagonal at the Mathematics Glossary.

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

Take a look around and you will see that there are many different types of quadrilaterals. Are you aware of the properties that make each type of quadrilateral unique? Complete Try This 1 to explore some of the properties of quadrilaterals and discover the names given to specific quadrilaterals.

Try This 1

Open Quadrilateral Demonstration Applet. Follow the instructions given in the questions. Analyze each constructed quadrilateral. Record your findings in a table like the one shown after each set of instructions.

1. Click and drag the corners of the quadrilateral until all side lengths are equal and each enclosed angle is 90°. Look above the quadrilateral. What specific name is given to this shape? Record this name in the first column of the table, and complete the rest of the information about this quadrilateral.
2. Move the top right and bottom right corners to the right so each angle is 90°. Record the name and information in the table.
3. Move the top right corner left and then move the top left corner to the right, keeping the top and bottom lines parallel. Record the name and information in the table.
4. Move the corners so the top and bottom are the same length and parallel. Adjust the left and right sides so they are the same length and parallel. Make sure the enclosed angles are not 90°. Record the name and information in the table.
5. Move the corners so all sides are equal and the enclosed angles are not 90°. Record the name and information in the table.
6. Move a corner away from the centre of the quadrilateral until two sides are equal and the other two sides are equal. Record the name and information in the table.
   
   

   Type of Quadrilateral	Pairs of Equal Sides (opposite or adjacent)	Number of Equal Sides	Pairs of Parallel Sides	Number of Equal Angles (right angles or not right angles)

Save your responses in your course folder.

Share 1

With a partner or in a group, share your responses to Try This 1 and address the following questions:

1. In any given quadrilateral, what is the sum of the interior angles?
2. Trapezoids can be classified as right trapezoids or isosceles trapezoids. From your knowledge of triangles and the terms right and isosceles, determine when a trapezoid is right or isosceles.

If required, place a summary of your discussion in your course folder.

Explore

In Try This 1, you may have noticed that a quadrilateral is a polygon made up of four sides, four angles, and four vertices—hence the name quad, meaning “four,” and lateral, meaning “side.” The different types of quadrilaterals can be classified as a square, rectangle, trapezoid, parallelogram, rhombus, or kite. The four interior angles of any of these types of quadrilaterals will always add up to 360°.

Self-Check 1

Use your answers from Try This 1 to work through “Classifying Quadrilaterals Interactive.” As you complete this activity, make any changes necessary to your table from Try This 1.

In the next Try This, you will explore the properties of different quadrilaterals using the measurements of the sides, angles, and diagonals.

Try This 2

From Try This 1 and Try This 2, you should have noticed that quadrilaterals can be classified according to the properties of the quadrilateral.

Square		All sides are equal in length. All angles are 90°. Opposite sides are parallel. Diagonals are equal in length. Diagonals intersect at 90°.
Rectangle		Opposite sides are equal in length. All angles are 90°. Opposite sides are parallel. Diagonals are equal in length.
Trapezoid		One pair of opposite sides are parallel.
Parallelogram		Opposite sides are equal in length. Opposite sides are parallel. Opposite angles are equal.
Rhombus		All sides are equal in length. Opposite sides are parallel. Opposite angles are equal. Diagonals intersect at 90°.
Kite		Two shorter sides are equal in length. Two longer sides are equal in length. Opposite angles between one short side and one long side are equal. Diagonals intersect at 90°.
Right Trapezoid		One pair of opposite sides are parallel. Two angles measure 90°.
Isosceles Trapezoid		One pair of opposite sides are parallel. Sides that are not parallel are equal in length. Angles between equal sides and one parallel side are equal. Diagonals are equal in length.

Self-Check 2

1.  Identify the quadrilateral in each photograph by its specific name.
   
   1. 
      
      
      Adapted from iStockphoto/Thinkstock
   2. 
      
      Adapted from Hemera/Thinkstock
      
   3. 
      
      Adapted from iStockphoto/Thinkstock
      
   4. 
      
      Adapted from Hemera/Thinkstock
      
   
   Answer

You have explored the different properties of quadrilaterals and how to classify quadrilaterals based on the properties of the shape.

Quadrilaterals are extremely important in the construction of all building types. Without accurate measurements of lengths and angles, the structural integrity of a building can become extremely hazardous! Can you explain why?

first house: iStockphoto/Thinkstock; second house: iStockphoto/Thinkstock; third house: iStockphoto/Thinkstock

In the next Try This, you will explore how to solve problems that involve quadrilaterals.

Try This 3

Quaid has recently purchased a new one-ton pickup truck. He wants to install an extra fuel tank in the truck box so he won’t have to stop as frequently to refuel. He needs to get the tank custom-manufactured since he also wants the truck box to accommodate pulling a fifth-wheel trailer. The fuel tank needs to be in the shape of a prism with the specific dimensions shown.

1. What type of quadrilateral is figure ABCD (as shown in the diagram)? Explain your answer.
2. Calculate the area of figure ABCD.
3. Calculate the volume of the prism.
4. How many litres of fuel can the tank hold?

Connect

Lesson 2 Assignment

Lesson 2 Summary

In this lesson, you looked at quadrilaterals and different ways to classify quadrilaterals based on their properties. You studied that a quadrilateral is a polygon with four sides and that the interior angles add up to 360°. You studied the specific properties of squares, rectangles, trapezoids, parallelograms, rhombuses, and kites. You also looked at situations in construction where quadrilaterals might be used.