346 NEL

NEL

GOALS

--

You will be able to• name and sort polygons, including

triangles, based on their sidelengths and angles

• draw triangles

• identify and describe polygons inthe environment

• identify congruent polygons

• communicate about properties of polygons

347

How are the triangles inthis photo the same?How are they different?

All sidelengths equal

At least one pairof parallel sides

Polygons

A

DP

I

G

J

H

M

K

N

O

E

B

C

L

F

348 NEL

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You will need• a Venn diagram• Polygons 1

(blackline master)• scissors

Polygon PuzzlersCalvin is organizing a game for his class. Each student willchoose five polygons and score points based on twoproperties: • A polygon with one of the properties is worth 3 points. • A polygon with both of the properties is worth 6 points.

Calvin wants to determine the highest possible score. He sorts polygons using a Venn diagram.

What is the greatest number of pointspossible for five different polygons?

NEL 349

Do you agree or disagree with each statement? Explain your thinking.

1. A triangle can have only one obtuse angle.

2. A triangle has a right angle. One side of thetriangle is 9 cm long and another side is 12 cmlong. The length of the third side must be about15 cm.

3. You can make a triangle with these side lengths:

4. A quadrilateral with two sides that are 4 cm longand two sides that are 6 cm long must be arectangle.

4 cm

6 cm

10 cm

A. Complete the sorting using polygons like Calvin’s.

B. Choose any five of Calvin’s polygons. Figure out thetotal number of points.

C. Choose two other sets of five of Calvin’s polygons.Calculate the total number of points for each set.

D. Describe a set of five polygons that will earn thegreatest number of points.

E. What is the highest possible score for five polygons?

Reading StrategySynthesizingThink about all theproperties ofpolygons. Pick twopolygons. Whatproperties do bothpolygons have?

11

Zoe builds model sailboats as a hobby. She is experimentingwith different shapes of sails.

350 NEL

Classifying Triangles bySide Lengths

Chapter 11 Classifying Triangles byClassifying Triangles bySide LengthsSide Lengths

You will need• blue pipe cleaners

(10 cm)• yellow pipe cleaners

(7 cm)• red pipe cleaners

(5 cm)• pencil crayons • a ruler

GOAL

Use side lengths to classify triangles.

Zoe’s Triangles

I’ll model the shapes of the sails using pipe cleaners. I can classify the model triangles by the lengths of their sides.

My first model has three equal sides. It is an equilateral triangle.

equilateraltriangleA triangle with allsides equal in length

How can you classify triangles by thelengths of their sides?

NEL 351

isosceles triangleA triangle with twosides equal in length

scalene triangleA triangle with nosides equal in length

A. Make as many different pipe-cleaner triangles as youcan. Use only one pipe cleaner on each side.

B. Sketch your triangles. Colour the sides to match thepipe cleaners you used.

C. The triangle with all sides blue is an equilateraltriangle. Which of your other triangles are equilateral?

D. This two-coloured triangle is an isosceles triangle.Which of your other triangles are isosceles?

E. This three-coloured triangle is scalene. Are any of your other triangles scalene?

F. Why are there only three ways to classifytriangles by their side lengths?

G. Is the triangle at the left scalene, isosceles, orequilateral? How do you know?

My second model has two equal sides. It is an isosceles triangle.

This model has sides that are all different lengths. It is a scalene triangle.

Communication Tip

Line segments thatare equal length areoften marked withthe same number of ticks.

Checking1. Classify these triangles by their side lengths.

a) b)

2. Are these triangles classified correctly? How do you know?

a) b)

Practising3. Measure to classify these triangles by their side lengths.

a) c)

b) d)

4. One side of a triangle is 10 cm. What might the lengths ofthe other two sides be for each kind of triangle? a) scaleneb) isoscelesc) equilateral

5. What information do you need to determine whether atriangle is scalene, isosceles, or equilateral?

isoscelesscalene

352 NEL

City planners are designing a 12 km walking path in theshape of a triangle. Each side length will be a wholenumber of kilometres. Shaun and Félix are making modelsof possible paths.

What triangles with a perimeter of 12 kmcan you make?

NEL 353

Chapter 11

Exploring TrianglesExploring TrianglesExploring TrianglesYou will need• a ruler• drinking straws• scissors

GOAL

Determine the possible side lengths of triangles.

Scale

1 cm

represents

1 km.

Leela is making samosas in the shape of triangles for theinternational festival at her school. She is making differenttypes of triangles for variety.

How can you classify Leela’s triangles?

A

E

D

F

G

B

C

354 NEL

Classifying Triangles byInterior Angles

Chapter 11Classifying Triangles byClassifying Triangles byInterior AnglesInterior Angles

You will need• a square corner• a ruler

GOAL

Use side lengths and interior angles to classifytriangles.

A. Which of Leela’s triangles is a scalene triangle but notan obtuse triangle?

B. Which of Leela’s triangles is an isosceles triangle?

C. Which of Leela’s triangles is a right triangle?

D. Classify all of Leela’s triangles by both their interiorangles and side lengths.

NEL 355

E. What visual clues can you use to determinewhether a triangle is right, obtuse, or acute?

F. Why is it enough to look at only the largestangle to classify any triangle?

Calvin’s Classifying

right triangle A triangle with oneinterior angle thatmeasures 90º (a right angle)

obtuse triangle A triangle with oneinterior anglegreater than 90º(obtuse)

acute triangle A triangle with onlyacute interiorangles; all angles areless than 90º

I’ll look at triangle A first. I’ll measure the side lengths. They’re alldifferent, so triangle Ais a scalene triangle.

I can also classify triangle A using interior angles. It couldbe a right triangle, anobtuse triangle, or an acute triangle.

The largest angle intriangle A is an obtuse angle because it is larger than a right angle. Triangle A is an obtuse scalene triangle.

A

A

Checking1. Classify triangles A, B, and C using both side lengths

and interior angles.

Right triangles Isosceles triangles

Triangles

A

B C

G

D

E

F

H

356 NEL

Practising 2. Classify triangles D to H using side lengths and

interior angles.

4. This isosceles triangle has two equal sides. Trace andcompare the two interior angles that are formedwhere the two equal sides meet the third side of thetriangle. What do you notice?

5. Jenna said that two of the angles in a right isoscelestriangle will be 45�. Do you agree or disagree?Explain.

6. Is it possible to have a triangle that is not obtuse,acute, or right? Explain your thinking.

Compare these angles.

NEL 357

3. a) Identify an obtuse triangle, an acute triangle, anda right triangle on this gable. Trace the triangles.

b) Classify each triangle according to its side lengths.

Ariana paddled 3 km across the lake from her campsite. Sheturned her kayak 90� and paddled 4 km along the shore.Then she paddled back to her starting point.

About how far will Ariana have topaddle to return to her campsite?

358 NEL

Drawing TrianglesChapter 11

Drawing TrianglesDrawing TrianglesYou will need• a protractor• a ruler

GOAL

Draw triangles to solve problems.

NEL 359

A. Why did Grace need both a protractor and a ruler to draw her triangle?

Grace’s Diagram

I can draw a diagram. I’ll use a scale in which 1 cmrepresents 1 km. So, 4 cm will represent 4 km.

I’ll draw a line segment with my ruler. Then I’ll place my protractor so that the centre is on an endpoint. I’ll make a mark at 90 to show an angle of 90�.

I’ll draw a line segment from the endpoint through the 90� mark.

I’ll measure 3 cm from the endpoint on the line segment and make a mark. I’ll measure 4 cm on the other line segment and make a mark.

I’ll draw the third side of the triangle by joining the marks. Then I’ll measure the third side. The length is about 5 cm.

Since 1 cm represents 1 km, Ariana will have to paddle about 5 km to return to her campsite.

360 NEL

Checking1. Jordan and Max skated away from each other at a

60� angle. After 3 s, Jordan had skated 7 m, while Max had skated 5 m. If Jordan has the puck, how farwill he have to pass it to get it to Max? Draw a scale diagram to solve the problem.

Practising2. Use a protractor and a ruler to draw each triangle,

using the measures given.a) b)

3. Draw these triangles.a) any right scalene triangleb) any obtuse isosceles trianglec) any acute scalene triangle

4. Adrian drew a right triangle like this one. Draw a right triangle with one 6 cm side and one 8 cm side that looksdifferent from Adrian’s triangle.

5. In Canada, a wheelchair ramp must have an 85° angleat the top.a) Draw a scale diagram of this ramp using a ruler

and a protractor. Make the rise 1 cm.b) Measure the length of the ramp.

6. Which triangle in Question 2 was easier to draw? Explain your thinking.

Crise

A

B

40° 75°4 cm

4 cm 4 cm

60˚

Max

Jordan

6 cm

8 cm

Triangles in Circles Thales, a Greek astronomer and mathematician, discovered an interesting property of circles.

1. Trace a jar lid to draw a circle. Cut out the circle.

2. Fold the circle in half and mark the line of symmetry. This will be one side of a triangle.

3. Pick any point on the circle. Join thispoint to the endpoints of the line of symmetry to create a triangle. Classify the triangle by its angles.

4. Repeat Step 3 for several different points on the circle. What do you think Thales discovered?

NEL 361

You will need

• jar lid or can• scissors • a ruler• a protractor

Frequently Asked Questions Q: How can you classify triangles?

A1: You can classify triangles by their side lengths. • A scalene triangle has three different side lengths. • An isosceles triangle has two equal side lengths. • An equilateral triangle has three equal side lengths.

180

5 cm

60˚50˚

5 cm

1

2

3

4

5

0 cm

6

1808

90 10080

11070

12060 130

50 14040 15030

16020

17010

18000 180

10 170

2016

030

150

40140

50

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60

120

70

110

80

100

362 NEL

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scalene isosceles equilateral

rightacute obtuse

A2: You can classify triangles by their interior angles. • An acute triangle has three acute angles. • A right triangle has one right angle. • An obtuse triangle has one obtuse angle.

Q: How can you draw a triangle?

A: You can use a protractor and a ruler to draw atriangle.

For example, if you know the length of one side andthe two interior angles at either end of this side, youcan draw this side with a ruler and use a protractor tocreate the angles.

PracticeLesson 1

1. Classify each triangle by its side lengths.a) b)

KENDRAMAX IAN

PAIGEROBIN

NEL 363

Lesson 2

2. The perimeter of a triangle is 15 cm. Identify somepossible side lengths for each type of triangle.a) scaleneb) isoscelesc) equilateral

Lesson 3

3. Walid made triangle name tags for the open house at his school.a) Classify the name tags using their side lengths.b) Classify the name tags using their interior angles.

Lesson 4

4. Draw this triangle using the measures given. What isthe missing side length?

5. Draw the following triangles.a) a triangle with a 6 cm side that has a 60� angle

and a 45� angle at its endpointsb) a triangle with 2 cm and 5 cm sides that form a

right anglec) any right isosceles triangle

110˚

50˚

? cm

7 cm

D E

F

Reading StrategySummarizingWhat do you knowabout side lengthsand interior anglesof triangles? Defineeach type oftriangle.

364 NEL

Chapter 11

Sorting PolygonsSorting PolygonsSorting PolygonsYou will need• a Venn diagram • Polygons 2

(blackline master)• scissors• a protractor• a ruler

GOAL

Use rules or properties to sort polygons.

regular polygonA polygon with allsides the same andall interior anglesthe same

For example, this is aregular hexagon.

A

B C

G

DE

K

F

H I

ML

J

N

Félix is using regular polygons to make stained-glass designs. He issorting the pieces to figure out which pieces he should choose.

Which stained-glasspieces should Félixchoose?

A. How do you think Félix decided where to put thesethree pieces?

B. Sketch the rest of the pieces in a Venn diagram likeFélix’s.

C. Which piece is the only regular polygon that is aquadrilateral? How do you know?

D. Which stained-glass pieces should Félix choose?

NEL 365

Félix’s Sorting

I’ll sort the pieces using a Venn diagram with a circleinside a larger loop.

Regularpolygons

Polygons

Shapes

F

H

N

Communication Tip

We use Greekprefixes to namepolygons with morethan 4 sides.

Penta means 5, so apentagon has 5 sides.

Hexa means 6, so ahexagon has 6 sides.

Octa means 8, so anoctagon has 8 sides.

E. Why do you think Félix used a Venn diagramwith a circle inside a larger loop, instead of twooverlapping circles?

F. Both of the shapes at the left have four equalside lengths. How do you know that one ofthem is not a regular polygon?

Checking1. Sort Félix’s stained-glass pieces using a Venn diagram

like this one.

Practising2. a) Measure one interior angle in each of these

regular polygons. Record your answers in a chartlike the one below.

QuadrilateralsRegular

polygons

Polygons

366 NEL

Regular Polygon Angles

Polygon

Interior

angle

(degrees)

equilateral

triangle square

regular

pentagon

regular

hexagon

regular

octagon

b) How does the angle size in a regular polygonchange as the number of sides increases?

3. Is it possible to draw a regular polygon with aninterior angle of 115�? Why or why not?

4. Is each shape below a regular polygon? Explain your thinking.

a) c)

b) d)

5. If you know the number of sides of a polygon, do youknow the number of interior angles? Explain yourthinking.

6. Which traffic signs in your neighbourhood are regularpolygons? Explain how you know.

7. Identify some examples of regular polygons in yourclassroom. Describe them.

8. How do you know that a loonie is a regular polygon?

9. Do you have to measure every side length or everyinterior angle to determine whether a polygon isregular or not regular? Use examples to explain.

NEL 367

On a class trip to a textile museum, Abby and Shaun saw ablanket with parts that were made by sewing togetherpatches shaped like parallelograms. They wonderedwhether the two halves of a parallelogram are congruent.

How can you decide whether twotriangles are congruent?

368 NEL

Chapter 11

Congruent PolygonsCongruent PolygonsCongruent Polygons

You will need• Polygons 3

(blackline master)• tracing paper• a protractor• a ruler

GOAL

Use different methods to identify congruent polygons.

NEL 369

Shaun’s Test

I’ll measure the side lengths of the triangles.

Then I’ll measure the interior angles of the triangles.

The side lengths and interior angles of the two triangles are the same. So the triangles are congruent.

Abby’s Test

I’ll use tracing paper to copy one triangle.

Then I’ll rotate and translate the traced triangle to cover the other triangle.

The angles and side lengths match.

So the triangles are congruent.

6.5 cm

4.5 cm

4.5 cm

3.0 cm3.0 cm

120˚

120˚

22˚

22˚

38˚

38˚

A. Is the image of a translation, reflection, or rotationalways congruent to the original shape? How doyou know?

B. How are Abby’s and Shaun’s tests alike? How arethey different?

Checking1. a) Do you think the two parts of the hexagon at the

left are congruent? b) Explain how you could check whether the two

parts are congruent without measuring.

Practising2. Which of these quadrilaterals are congruent?

How do you know?

D E

370 NEL

A B C

3. a) Do you think square D is congruent to square E?b) Test to find out if the squares are congruent.

4. a) Draw a regular polygon with three sides. Explain what you did.

b) Draw a polygon that looks like it might becongruent to the polygon you drew in part a), but isn’t. How did you draw it?

A B

C

FE

D

NEL 371

5 cm

F

7. Explain how you can use transformations to show that these two triangles are congruent.

8. Two polygons have the same angles and the samenumber of sides. Do they have to be congruent?

5. a) Transform triangle F so that the 5 cm side is at the topof the image triangle. Label the new triangle “G”.

b) How can you tell without measuring that the twotriangles are congruent?

6. Which triangles in this design are congruent? Explain how you found out.

372 NEL

Communicating aboutPolygons

Chapter 11Communicating aboutCommunicating aboutPolygonsPolygons

You will need• a ruler• a protractor

GOAL

Describe polygons using side lengths and anglemeasures.

Samara’s Description

My polygon sticker is a quadrilateral. The opposite sides are equal in length.

A polygon sticker has been placed on the underside of eachskateboard in Scully’s Skateboard Emporium. Anyone whocan accurately describe the polygon sticker in writing willwin tickets to a skateboard competition.

How can you describe the polygon sticker?

A. Can you think of other polygons that match Samara’sdescription? Explain.

B. Improve Samara’s description. Use the CommunicationChecklist to help you.

NEL 373

✔ Did you use mathlanguage?

✔ Did you explain yourthinking?

✔ Did you check thatyour descriptionmatches the polygon?

CommunicationChecklist

3. a) Describe each polygon.

A B

b) How is your description of polygon A differentfrom your description of polygon B?

4. Why might you use a diagram as well as words todescribe a polygon accurately?

Checking1. Describe the shape of the STOP sign.

Practising2. Describe the polygon sticker on Matt’s

skateboard. Use the Communication Checklist to help you.

C. What are the important details to include if youwant to describe a polygon accurately?

Matching CardsNumber of players: 2 to 4How to play: Match the Shape Cards with the

Property Cards that describe them.

• Step 1 Give an equal number of Shape Cards and Property Cards to each player. Use as many cards as possible.

• Step 2 On your first turn, find all the pairs of matching ShapeCards and Property Cards in your hand. Place the pairsface up in front of you and explain why they match.

• Step 3 On your next turn, choose a card from the person atyour right, without looking. If the card matches one of your cards, place the pair face up in front of you. Explain why the cards match.

The first player to use all of her or his cards wins.

374 NEL

You will need

• Shape Cards(blackline master)

• Property Cards(blackline master)

Grace’s TurnI chose a Property Card from Calvin’shand. One of my cards matches it, so Iput the two cards on the table andexplained why they match.

A regular polygon with

four sides

Frequently Asked Questions Q: How can you identify a regular polygon?

A: You can compare the side lengths and the angles.For example, the yellow pattern block is a regularhexagon because all its side lengths are equal and all its interior angles are equal.

The orange polygon is not a regular hexagon becauseits side lengths and angles are not all equal.

Q: How can you determine if two polygons arecongruent?

A1: You can transform one polygonand place it on top of the otherpolygon to see if they are aperfect match.

For example, the two polygons at the right have equal sides andequal interior angles, so they are congruent.

A2: You can measure the side lengths of both polygonswith a ruler and measure the angles of both polygonswith a protractor. If the side lengths are the same andthe angles between the matching side lengths areequal, then the polygons are congruent.

90 10080

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NEL 375

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PracticeLesson 1

1. Classify triangles A to D according to their side lengths.

A B

DC

376 NEL

Lesson 3

2. Classify the triangles in Question 1 according to theirinterior angles.

3. a) Which triangle is an obtuse scalene triangle?b) Which triangles are equilateral triangles?c) Which triangles are acute scalene triangles?

I

HE

G

F

J

Lesson 4

4. a) Draw each triangle actual size. Label the measures.

4 cm40° 80°

L

5 cm

3 cmK

A B

C D

NEL 377

b) Name each triangle using the side lengths andangle measures.

5. Jan’s house and Keith’s house are 20 m apart. Jan’s house is 30 m from Lara’s house.a) How far apart might Keith’s house and Lara’s house

be if the three houses form a scalene triangle? b) How far apart might Keith’s house and Lara’s house

be if the three houses form an isosceles triangle?

6. Draw a triangle to match each description.a) an isosceles triangle with two equal angles of 30�

b) a triangle with three 60� anglesc) an obtuse scalene triangle

Lesson 5

7. a) Which shapes at the left are regular polygons? How do you know?

b) How can you change the other polygons to make them regular polygons with the samenumber of sides?

8. Sketch and describethree differentpolygons made bycombining the tiles inthis pattern.

Lesson 6

9. Are the two triangles below congruent? How do you know?

378 NEL

Lesson 7

10. Grace has marked an area in her schoolyard to buildan ecology garden. How can she communicate the sizeand shape of the marked area?

Look back at What Do You Think? on page 349. How have your answers and explanations changed?

NEL 379

111111

✔ Did you usemeasurements tocheck that yourpolygons are regularand congruent?

✔ Are yourexplanations clear?

✔ Did you use mathlanguage?

Polygon ContestA math club is holding a polygon contest. Students cancreate an entry by following the rules below. If they followall the rules correctly, they will qualify for a prize draw.

Task Checklist

acutescalene

1

2

3

4

5

0cm

How can you create a qualifying entry?

A. Make two congruent polygons by following the rulesfor Part 1 of the contest.

B. Draw and colour your polygons on pattern blockpaper. How do you know your polygons are congruentand regular?

C. Trace one of your polygons onto plain paper. Draw eighttriangles by following the rules for Part 2 of the contest.

D. Explain how you classified your triangles.

Part 1• Use at least eight pattern blocks to make a regular polygon.• Make a congruent polygon using another combination of any number of pattern block pieces.

Part 2• Draw eight different triangles to fill up an outline of one of your polygons.• Classify each triangle by its side length and by its interior angles.

1. What is the measure of angle A on the elephant’s ear?Use a protractor.A. 30° C. 135°B. 65° D. 100°

2. Calculate the measure of angle B on theelephant’s ear.A. 35° C. 90°B. 60° D. 95°

3. Which estimate makessense for angle C on the elephant’s head?A. 30° C. 100°B. 45° D. 195°

4. What is the area of the TV screen at the left?A. 144 cm2 C. 8464 cm2

B. 2704 cm2 D. 4784 cm2

5. What is the volume of the butter?A. 396 cm3

B. 121 cm3

C. 66 cm3

D. 500 cm3

6. Which product is between 40 and 50?A. 12.9 � 3 C. 8.002 � 8B. 6 � 7.56 D. 9 � 5.99

7. Erica hung four hockey posters side by side with nospaces between them. Each poster is 19.8 cm wide.What is the combined width of the four posters?A. 198 cm C. 23.8 cmB. 108 cm D. 79.2 cm

380 NEL

C

A

B

45˚

92 cm

52 cm

6 cm

6 cm

11 cm

19.8 cm

8. A sandwich is 30.48 cm long. If three people share itequally, how long will each piece be?A. 3.48 cm C. 10.16 cmB. 11.6 cm D. 10.12 cm

9. What is the theoretical probability of rolling an evennumber on a die numbered from 1 to 12?

A. 2:12 B. C. 1.0 D.

10. Liam tested the spinner at theright to see how often he would spin red. The graph at the left shows his experimentalprobabilities. Which statement is not true?A. No trials matched the

theoretical probability.B. Trial 1 was close to the theoretical probability.C. Trial 2 was less than the theoretical probability.D. Trial 3 was less than the theoretical probability.

11. Which name best describes the shape of this moth?A. right triangleB. obtuse triangleC. acute triangleD. scalene triangle

12. Which window is a regular polygon?A. C.

B. D.

1012

612

NEL 381

Trial

3210

Spinner Experiment

Pro

ba

bili

ty o

f sp

inni

ng r

ed

0.30.4

0.10.2

0.70.80.91.0

0.50.6