CHAPTER 3 Geometry I - cbv.ns.ca 8_TR/Teachers Resource...CHAPTER 3 Geometry I ... has a rotational...

CHAPTER 3 Geometry I

S p e c i f i c Cu r r i c u l u m O u t co m e s

M a j o r O u t c o m e s

E1 make and apply informal deductions about the minimum sufficient condi-

tions to guarantee the uniqueness of a triangle and the congruence of two

triangles

E2 make and apply generalizations about the properties of rotations and dilata-

tions, and use dilatations in perspective drawings of various 2-D shapes

E4 perform various 2-D constructions and apply the properties of transforma-

tions to these constructions

E5 make and apply generalizations about properties of regular polygons

C o n t r i b u t i n g O u t c o m e s

D10 apply the Pythagorean relationship in problem situations

C h a p t e r Pro b l e m

A chapter problem is introduced in the chapter opener. This chapter problem begins

with highlighting the geometric properties presented in a quilt and encourages

students to think about constructing their own quilt pattern. The quilt square shown

has a rotational symmetry of 4 but no reflective symmetry because of the colours

used. Ask students to bring in examples of quilts and discuss the patterns. Consider

partnering with a quilting group in the community where quilters can present a brief

presentation of the patterns in their quilts.

The chapter problem is revisited in section 3.1, question 11, section 3.2, ques-

tion 12, section 3.3, question 10, and section 3.4, question 11. You may wish to have

students complete the chapter problem revisits that occur throughout the chapter.

These simpler versions provide scaffolding for the chapter problem and offer strug-

gling students some support. The revisits will assist students in preparing their

response for the Chapter Problem Wrap-Up on page 141.

Alternatively, you may wish to assign only the Chapter Problem Wrap-Up

when students have completed Chapter 3. The Chapter Problem Wrap-Up is a

summative assessment.

Key Wordsuniquecongruentrotationtranslationreflectionorientationtransversalregular polygonexterior angle

Get Ready Wordsregular polygonreflective symmetryrotational symmetryimagepre-image

84 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Planning Chart

Section Suggested Timing

Teacher’s ResourceBlackline Masters Assessment Tools Adaptations

Materials andTechnology Tools

Chapter Opener• 10 min (optional)

Get Ready• 60 min

• BLM 3GR Parent Letter• BLM 3GR Extra Practice

• grid paper

3.1 Unique Triangles• 180 min

• BLM 3.1 Extra Practice Formative Assessment:• BLM 3.1 AssessmentQuestion, #10

• Geostrips®• protractors• Bullseye compasses • pattern block trianglepaper

3.2 Prove Triangles areCongruent • 180 min

• BLM 3.2 Extra Practice • rulers (millimetres)• protractors• Bullseye compasses

3.3 Properties ofTransformations• 240 min


• BLM 3.3 Alternate DTMPart A Activity

• centimetre square dotpaper• Bullseye compasses• protractors • rulers (millimetres)• transparent mirrors• tracing paper

3.4 Regular Polygons• 180 min


• BLM 3.4 RegularPolygonsOptional:• BLM 3.4 DTM PolygonTable• BLM 3.4 CYU Table

• BLM 3.4 Alternate DTMActivity

• Geostrips®• rulers (millimetres)• protractors• Bullseye compasses• tracing paper• transparent mirrorsOptional:• The GeometryTemplate®

Chapter 3 Review• 60 min

• BLM 3R Extra Practice • Bullseye compasses• square dot paper• protractors• tracing paper

Chapter 3 Practice Test• 60 min

Summative Assessment:• BLM 3PT Chapter 3Test

• tracing paper

Chapter Problem Wrap-Up• 30 min

• BLM 3CP ChapterProblem Wrap-UpRubric

Chapters 1–3 Review• 120 min

• square dot paper

Chapter 3 • MHR 85

Get Ready

W A R M - U P

Which benchmark is each fraction closest to: 0, 1–2

, or 1?

1. <0> 2. < >

Use >, <, or = to compare each pair of fractions.

3. � (>) 4. � (<)

5. Is 9 � 5 greater than or less than 4? <greater than 4>

Estimate the sum. (Estimates may vary.)

6. 5 � 2 � 4 � 1 <15>

Add.

7. � < >

Try using the Make “1” strategy to solve each problem.

8. � < or > 9. � <1 or 1 >

Subtract.

10. 6 � 1 <4 > 11. 10 � 4 <5 >

Multiply.

12. � 63 <7> 13. � 56 <32>

Divide.

14. � <5> 15. 3 � <7>

A S S E S S M E N T F O R L E A R N I N G

Explain to students that Chapter 3 is about 2-D geometry. The chapter involves

studying triangles and regular polygons, and the transformations (translations,

reflections, and rotations) of polygons that produce congruent images.

Discuss with students where they have used 2-D geometry in in their everyday

lives. You may wish to brainstorm and develop a mind map for each topic or start the

development of a graphic organizer to be used throughout the chapter.

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Materials• grid paper

Related Resources• BLM 3GR Parent Letter• BLM 3GR Extra Practice

Suggested Timing60 min


After students have discussed 2-D geometry, have them complete the assessment

suggestions below in pairs or individually. This assessment is designed to provide you

and your students with information about their readiness for the chapter. After

strengths and weaknesses have been identified, students can work on appropriate

sections of the Get Ready.

Method 1: Have students develop a journal entry to describe as many real world

examples as they can of how triangles are used in design and construction.

Method 2: Challenge students to show how much they know about each topic.

Encourage them to use words, numbers, and diagrams to show what they know.

R e i n fo rce t h e Co n ce p t s

Have those students who need more reinforcement of the prerequisite skills

complete BLM 3GR Extra Practice.

T E A C H I N G S U G G E S T I O N S

The Get Ready provides students with the skills they require to fully understand the

topics developed in Chapter 3. You can have students complete all of the Get Ready

before starting the chapter. Or you may wish to have students complete each part of

the Get Ready before they work on the related section.

You could also review the classifications of triangles by sides and by angles; that

is, equilateral, isosceles, and scalene, and right, acute, and obtuse. To help students

recall these classifications, post well labelled diagrams of these triangles and other

shapes in the classroom. Since there will be many opportunities to find the third

angle of a triangle, you could include mental math strategies for subtracting from

180, such as back through 100 and up through 100 for situations where two angles add

to less than 100°.

Co m m o n E r ro r s

• Students say that a figure with no rotational symmetry has rotational

symmetry of order 1.

Rx Remind students that if a figure is rotated 360° before it produces an image

that coincides with itself, it has no rotational symmetry, and therefore

cannot have an order value. The lowest value the order of rotational

symmetry can have is 2 (when a figure produces an identical image when

rotated 180° and another identical image when rotated 360°).

Te c h n o l o g y

Use Internet resources to research information and activities on 2-D geometry. There

are activities available online for The Geometer’s Sketchpad® as well as interactive sites

that examine topics such as classifying triangles and geometry in art, architecture, and

nature. Go to www.mcgrawhill.ca/books/math8NS for some interesting Web sites.

L i t e ra c y Co n n e c t i o n s

Active Readers: There are Active Readers (AR) resources for grade eight called

Infused Resources. In this group of resources, there are resources that are mathe-


matics related. Check the AR resources in your classroom or with your librarian. The

books are as follows:

• How Many Ants in the Anthill?

• Crunching Numbers

• Puzzling Out Patterns

• Decoding Data

• Thinking It Through

• Sizing Up Shapes

• What’s the Chance?

• Numbers Know How

While the books are not based on the outcomes, they can be used for interest

reading or as ideas for mini-assignments when students have completed the in-class

work. You might work with the language arts teacher for teaching text features or

comprehension activities to support the language arts program.

Report Writing: In science, students are required to write procedural steps for lab

reports. When students write journal entries, such as the one in Section 3.4, they are

writing procedure and they often write in an “and … then” format. Students can use a

graphic organizer to put the procedures into sequence. It is important to model the

thinking with the class so that students are able to apply the process to their own reports.

G e t R e a d y An s we r s1. Number of Sides in

Regular Polygon Drawing Common Name(s) n-gon Name

3equilateral triangle

or regular trigon3-gon

4square, regular

tetragon, or regularquadrilateral

4-gon

5 regular pentagon 5-gon

6 regular hexagon 6-gon

8 regular octagon 8-gon

12 regular dodecagon 12-gon

In this space students areto brainstorm alternativewords to use in place of“and . . .then”.

In this space, students write their paragraph about theprocedure. Remind students that they need a beginning,middle, and end for proper paragraph formation.Suggest they start with an attention grabbing sentenceand end with a conclusion of what they have discovered.

In this space, students put the steps in sequential order in a numbered list.


2. a)

b) Each regular polygon has the same number of lines of symmetry and the

same order of rotational symmetry as its number of sides.

3. a), c) not congruent b), d) congruent

4. to 6.

4. RS and R�S�, RT and R�T�, ST and S�T�, �R and �R�, �S and �S�, �T and �T�

5. RS and R�S�, RT and R�T�, ST and S�T�, �R and �R�, �S and �S�, �T and �T�

6. �RST is the image of �DEF flipped in both the x- and y-axes.

y

x420

–2

–4

2

4

6

8

–2–4

T

R

S

T‘

R‘

S‘

T‘‘

R‘‘S‘‘

F

ED


3.1 Unique Triangle

W A R M - U P

Multiply using the distributive property.

1. 9 � 1 <12> 2. 14 � 2 <35>

3. 2 � 4 <9> 4. 5 � 3 <16>

5. 8 � 1 <10> 6. 12 � 2 <26>

7. 7 � 3 <25> 8. 4 � 7 <31>

9. 10 � 8 <88> 10. 12 � 1 <22>

11. 3 � 3 <11> 12. 2 � 15 <39>

13. 5 � 4 <23> 14. 6 � 6 <40>

15. 16 � 2 <34>

M u l t i p l y i n g a M i xe d N u m b e r by a Wh o l e N u m b e r

You can use the distributive property when multiplying a mixed number by a whole

number. When multiplying 6 � 7 for instance, you can break the into parts,

7 � . Then you can multiply each part by 6:

6 � 7 � (6 � 7) � a6 � b� 42 � 2

� 44

Examples:

5 � 2 � (5 � 2) � a5 � b 8 � 3 � (8 � 3) � a8 � b� 10 � � 24 � 6

� 10 � 30


Construct three quadrilaterals using three sets of four R2 Geostrips® and have them

ready.

Since this section deals with uniqueness of triangles, the idea of being unique

can serve as an introduction. For example, ask students to draw a square with side

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Materials• Geostrips®• protractors• Bullseye compasses• pattern block triangle

paper

Related Resources• BLM 3.1 Assessment

Question• BLM 3.1 Extra Practice

Specific CurriculumOutcomesE1 make and apply informal

deductions about theminimum sufficientconditions to guaranteethe uniqueness of atriangle and thecongruence of twotriangles


Link to Get ReadyStudents should havedemonstrated understandingof Congruent Polygons inthe Get Ready prior tobeginning this section.


lengths of 4 cm. Does everyone draw the same square? This square is unique because

everyone’s square is a replica of the same square: all the squares in the room are

congruent to one another.

Ask three students to come forward and give them each one of the Geostrip®

quadrilaterals. Ask them to show the class a square, and note they all showed the

same square. Ask them to make a rhombus without looking at one another and show

the class. Note they did not show the same rhombus. Therefore, given four equal

Geostrips®, the square that can be made is unique but the rhombus is not.

Ask students to write two factors that have a product of 12 and then two

factors that have a product of 13. Ask them to state their factors for 12. Do they all

state the same two factors? Then ask several students for their factors for 13. Do they

all state the same two factors? Which product, 12 or 13, is unique? Connect this

uniqueness to prime numbers.

D i s cove r t h e M at h

In this activity, students will be presented with various combinations of measures of

three parts of a triangle, some of which will produce a unique triangle and some of

which will not. The emphasis is on the measures of the parts and for students to be

convinced that certain combinations will or will not produce unique triangles. These

ideas are presented in a more abstract form in high school with SSS, SAS, ASA, and

SAA for proving triangles congruent; however, in grade 8, be sure that students are

confident working with the measurements before introducing these abbreviations.

Prepare sets of the materials students will need: Geostrips®, Bullseye compasses,

and protractor. If possible, have large pieces of paper for partners to draw on.

It is recommended that you have students work in pairs with their books closed

while you read the instructions and guide them through the activity. After each

combination is investigated, make lists under the headings: Will Produce Unique

Triangle (three side lengths, two side lengths and the included angle, one side length

and two specified angles) and Will Not Produce Unique Triangle (two angles, three

angles, two sides and one not-included angle). It will be helpful if you periodically

review what they have discovered, and help them connect the investigations to the

principal goal of the section.

After question 2, part c), and question 4, part b), have students cut out their

triangles and post them on chart paper or on the board so the class can see all the

triangles produced. This will make a very visual statement about unique and not

unique triangles.

For question 8, ask students to tell you the information Fred and Gurda have

about the triangle. (The lengths of two sides and the measure of one not-included

angle.) Ask, “Do these three specified measurements produce a unique triangle?”

Help students see why there are two possible triangles by having them draw the

triangles from the given measures. Once they have drawn �D and side DE, they

should see that a 3-cm line segment from E will intersect the extended arm of �D in

two places. This is how two different triangles were formed: Fred used one of the

points of intersection to construct his triangle and Gurda used the other.


Journal

Students could use these prompts for question 7.

• I think congruent means ….

• In the real world unique means �but in mathematical terms it means ….

D i s cove r t h e M at h An s we r s

1. a) Drawings may vary. b) Yes; No, because there is only one way to attach a third

side given two sides. c) same results

2. b) AB � 7 cm, AC � 5 cm, BC � 6 cm, �A � 57°, �B � 44.5°, �C � 78.5°

c) Yes; Yes

3. Yes, there is only one place where the other two sides can meet given the first side.

4. a) 80°; The sum of the angles of a triangle is 180°.

b) No, the triangles have different side lengths.

5. a) Subtract the measures of the two angles from 180°.

b) No, there are other possible triangles with different side lengths.

6. a) Y1 b) Yes; Y1 c) Same result with R3 as the third side.

7. a) Congruent: exactly the same side lengths and angle measures; unique: the

only possible triangle that can be made using certain measurements.

b) Everyone would draw congruent triangles because the triangle is unique.

Once you have two given side lengths with an included angle, the only way to

make a triangle is to draw a line segment from the end of one side to the other.

8. The triangles are not congruent even though they have two side lengths the

same and another angle is the same, because the third side and other two angles

are different in the two triangles.

9. a) Yes, there is only one way to finish constructing that triangle.

b) No, there might be more than one triangle with those measurements.

10. b) Yes; All the corresponding sides and angles are congruent.

11. Yes, because there is only one way to finish constructing that triangle.

12. Unique triangle: knowing all three sides, two sides and the included angle, or

two angles and any side. Not unique triangle: knowing three angles or two sides

and one not-included angle.

Example 1 asks students to determine if the triangles with only the indicated meas-

urements known would be the only ones possible to construct. In part a), you could

review two mental math strategies for finding the third angle: use quick addition to

add 53 and 43 [(50 � 40) � (3 � 3) � 96], and up through 100 to get the third angle

(from 96 to 100 is 4 and from 100 to 180 is 80, so the answer is 4 � 80 � 84). For partb), draw students’ attention to the word similar to describe the relationship between

all the possible triangles that could be constructed with the indicated angles. These

triangles would all have the same shape because they have the same angle measures,

but they would be different sizes (enlargements or reductions of one another).

Example 2 asks students to apply what they learned in the Discover the Math

by supplying the third piece of information that would guarantee the triangles are

unique. Methods 2 and 3 both involve one side and two angles but stress that the

angles and side need to be designated because the triangle would not be unique

otherwise. That is, if you just asked someone to draw a triangle with a 5-cm side and


two of its angles with measure 40° and 30°, more than one triangle is possible;

however, if you asked someone to draw a triangle ABC with AB � 5 cm, �A � 40°,

and �C � 30°, there is only one possible triangle.

Co m m u n i c ate t h e Key I d e a s

Have students work in groups to answer and discuss all of the Communicate the Key

Ideas questions. Use this opportunity to assess student readiness for the Check Your

Understanding questions.

Co m m u n i c ate t h e Key I d e a s An s we r s

1. Ravi; Justina created a congruent triangle.

2. a) Yes, two angles and one side length are known.

b) No, the known angle is not contained by the known sides.

c) No, none of the side lengths are known.

O n g o i n g A s s e s s m e nt

• Can students accurately construct a triangle?

• Can students describe the properties of two congruent triangles?

• Do students know what an included angle means?

• Can students measure angles accurately?

• Do students know that the sum of the angles in a triangle is 180°?

C h e c k Yo u r U n d e r s t a n d i n g

Q u e s t i o n P l a n n i n g C h a r t

For question 3, students should recognize that being given two sides and a

not-included angle does not guarantee a unique triangle. However, you might have

students try to draw the two possible triangles so they can see that in this particular

case it is not possible to draw two different triangles. (If PQ � 15 cm and QR � 12 cm,

you could draw two different triangles.) Students should appreciate that a general-

ization in mathematics has to apply to all situations and not just some situations.

For question 5, students should appreciate the difference between this situa-

tion and a situation in which the sides and angles are specified as in question 2. For

question 7, point out to students having difficulty that this is not a case of knowing

two sides and a not-included angle but, by using the Pythagorean relationship, a case

of knowing all three side lengths.

Level 1 Knowledge andUnderstanding

Level 2 Comprehension of

Concepts and Procedures

Level 3 Application and Problem Solving

11 1–6, 8, 10 7, 9, 12



• Students may believe that a triangle is not unique because other copies can

be made or drawn by others (as in Communicate the Key Ideas question 1).

Rx Remind students that unique does not mean that only one triangle can

exist, but that only one triangle is possible for a given set of measurements.

Something can be unique and still be copied many times, such as a CD

cover or a poster.

I nt e r ve nt i o n

• Students may need to frequently review the meaning of unique as it is used

in the context of the chapter.

A S S E S S M E N T

Q u e s t i o n 1 0 , p a g e 1 1 3 , An s we r s

a) unique; all three sides known

b) not unique; no sides known

c) unique; all three sides known because of Pythagorean relationship

d) unique; two angles and one side known

e) not unique; known angle is not the included angle

A D A P T A T I O N S

BLM 3.1 Assessment Question provides scaffolding for question 10.

BLM 3.1 Extra Practice provides additional reinforcement for those who need it.

V i s u a l / Pe rce p t u a l / S p at i a l / M o t o r

• Students with special needs should be paired with another student for peer

support.

• Instead of Geostrips®, some students may want to use strips of paper cut to

various lengths or Cuisenaire rods.

E x t e n s i o n

Assign question 12. You may wish to reduce the number of Check Your

Understanding questions to provide students with extra time to work on the Extend

questions. You could challenge students to create another question similar to ques-

tion 12 that uses different variable expressions for the two angles.

C h e c k Yo u r U n d e r s t a n d i n g An s we r s

1. Yes, all sides lengths are known.

2. Yes, two angles and a side are known.

3. Yes, the triangle is not unique since the angle given is not a contained angle.


4. �LMN is unique if LN is known (all three sides known), or if �M is known

(two sides and the contained angle known).

5. In one triangle, the 8.5-cm side could be contained between the two angles,

while in the other triangle it is not.

6. �DEF (two sides and the contained angle known) and �GHI (two angles and

one side known) are unique. �ABC is not unique since the angle given is not

contained.

7. Because �STU is a right triangle, she can use the Pythagorean relationship to

determine the measure of SU, so she knows the length of all three sides.

8. Yes, this is a unique triangle because all three sides are known.

9. no

11. 36 small triangles, 12 medium triangles (made up of 4 small triangles), 4 large

triangles (made up of 9 small triangles), and 4 extra-large triangles (each

touching one side of the square)

12. Andre; he can solve for x, so all three angles are known. Since he knows one side

and at least two angles, the triangle is unique.


3.2 Prove Triangles are Congruent

W A R M - U P

Write each fraction in simplest form.

1. < > 2. < >

Add, using the Make “1” strategy.

3. � <1 > 4. � <1 >

Add.

5. � < > 6. � < >

Subtract.

7. 5 � 1 <3 > 8. 7 � 3 <3 >

Multiply.

9. � 48 <8> 10. � 27 <21>

11. 8 � 2 <18> 12. 5 � 2 <13>

Divide.

13. 2 � <18> 14. � <4>

15. � 2 < >


This section involves the application of logical reasoning in mathematics. The

current interest in television programs such as CSI should help you convince

students of the role of logical reasoning. You could have students solve an equation

such as 3x � 12 � 30 and help them see that the steps they used to solve it are a chain

of logical reasoning. You could also review an application of the Pythagorean

relationship and make a similar connection to a chain of logical reasoning.


The purpose of Part A is to establish the important role logical reasoning plays in

convincing others that given relationships between triangles, angles, and sides guarantee

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Materials• rulers (millimetres)• protractors• Bullseye compasses

Related Resources• BLM 3.2 Extra Practice

Specific CurriculumOutcomesE1 make and apply informal

deductions about theminimum sufficientconditions to guaranteethe uniqueness of atriangle and thecongruence of twotriangles


Link to Get ReadyStudents should havedemonstrated understandingof Congruent Polygons inthe Get Ready prior tobeginning this section.


other relationships. These four questions cover a range of previously learned geometric

concepts. Keep in mind the purpose of the questions and do not get sidetracked into

teaching the concepts themselves; rather, select the questions, or others of your own

creation, that keep students focused on a chain of logical reasoning to convince others.

You could start by having students complete question 3 individually and discuss

how they knew which sides and angles corresponded. While many students may have

relied on the given diagram, they should realize that they would have known the

correspondences from the way the triangles were named. Emphasize that they were

able to logically find six measures by combining their knowledge of what it means

for triangles to be congruent and for triangle parts to correspond. Then, have

students work in pairs to complete questions 1, 2, and 4.

The purpose of Part B is for students to use logical reasoning to prove two

triangles are congruent, based on what they learned about situations that guarantee

unique triangles in section 3.1. They should understand that all replicas of unique

triangles will be congruent to one other. Have students work in pairs.

You could have students close their books and lead them through questions 1and 2. Have them discuss and summarize what they learned, and make the connections

to section 3.1. Make a list of comparisons that guarantee two triangles are

congruent, such as “Three pairs of corresponding sides congruent and two pairs of cor-

responding sides and included angles all congruent.” and add to the list throughout the

activity. Have students open their books and work through questions 3 and 4. Discuss

and check these questions, and add to the list: “One pair of corresponding sides and

two pairs of corresponding angles all congruent”.

Have students close their books again and lead them through question 5. Do

not have students trace the inside of their Geostrip® triangles because the thickness

of the strips makes it difficult to have those interior side lengths the same when the

angles vary. Instead, concentrate on the strips themselves. Students can make two

different triangles using the two given Geostrips® and a not-included angle of 30°.

You may wish to have students complete question 6 individually before they

share their work with their partners. The discussion of this question should focus on

the fact that if two pairs of corresponding angles are congruent then the third pair

must also be congruent. And, since one pair of corresponding sides are congruent,

the triangles are both replicas of the same unique triangle and thus are congruent.


P a r t A

1. a) All three sides are equal, so it must be an equilateral triangle, which has three

60° angles.

b) The third angle is a right angle because 180° � 67° � 23° � 90°, so the

Pythagorean relationship applies to the triangle.

2. a) The opposite angle is also 60° and the other two angles are 120° because each

is on a straight line with a 60° angle.

b) It must be a square. A rectangle has only two lines of symmetry through the

middle of pairs of opposite sides and a rhombus has only two lines of

symmetry through opposite angles.

3. All corresponding sides and angles are equal between the two congruent

triangles: DE � 4.8 cm, EF � 7.9 cm, DF � 10.3 cm, �D � 48°, �E � 105°,

�F � 27°.


4. a) false; it could have angles 90°, 45°, and 45°

b) true; if two sides are equal, then the triangle is isosceles and two angles must

also be equal

c) false; the third angle is 180° � 31° � 54° � 95°, which is obtuse

d) true; all three sides are known

P a r t B

1. a) Yes, all three sides are known.

b) Yes. They are congruent because they have the same angles and side measures;

No, this is the only possible triangle using these Geostrips® so it is unique.

c) �PQR and �LMN are unique because all three sides are known. The

triangles are congruent because they have matching angles and sides.

2. a) Yes, two sides and the contained angle are known.

b) Yes, they have matching sides and angles. No, because the triangle created

with those three side lengths is unique.

c) Both triangles are unique because two sides and the contained angle are known,

so �DEF and �JKL are congruent since they have matching angles and sides.

3. Linh; the side lengths could be different.

4. a) Yes, two angles and a side are known.

b) Yes, both have a pair of equal angles and the contained side in each triangle is

12 cm, so they are the same unique triangle.

5. b) Answers may vary. Yes, because B1 can be attached to two places along R3 so

there are two possible triangles.

c) No, because there are two possible lengths for the third side.

d) If the corresponding angles are not the contained angles between the two

known pairs of corresponding sides.

6. a) �C � 113°, �D � 30°, �P � 113°, �Q � 30°

b) CE � 2.5 cm, PR � 2.5 cm

c) Yes, because two angles are congruent and a corresponding pair of sides are

the same length.

d) �ABC and �DEF did not have any side lengths given so they were not

unique triangles. �CDE and �PQR have all matching angles and one pair of

matching sides so they are unique and congruent. Both pairs of triangles have

matching angles.

7. all three sides the same, two corresponding sides and the contained angle the

same, two angles and any corresponding side the same



Ideas questions. Be sure to have students share their responses to question 3 so that

all possibilities are considered. Use this opportunity to assess student readiness for

the Check Your Understanding questions.


1. yes

2. yes

3. Answers may vary. For example, measure all three sides of each triangle.


In Example 1, students continue to use a chain of logical reasoning to convince class-

mates of the truth of a statement. Suggest students draw diagrams to help them in their

thinking. Their prior work with reflective symmetry in triangles should lead them to

recall that isosceles triangles have one line of symmetry, so they can use the informa-

tion to convince someone that the triangle is isosceles. Point out that they are able to

find the measure of the third angle because they know that the sum of the three angles

is 180°. They should recall that an isosceles triangle has two equal sides and two equal

angles. The integration of these various understandings in a clear and logical sequence

makes a convincing argument for the truth of a statement.

Example 2 asks students to use the given information about two triangles to

convince a classmate that the two triangles are congruent. Point out the use of the

congruency symbol (�) as described in the Communicating Mathematically box.

This provides an opportunity for students to apply what have they learned. Point out

that a diagram will help them in their reasoning because only the measures of two

pairs of corresponding sides were given.


• Can students identify the properties of a triangle such as the sum of the

interior angles?

• Can students identify the properties of an isosceles triangle?

• Can students identify the properties of an equilateral triangle?

• Do students understand what is meant by minimum sufficient conditions?



For question 3, part d), refer students to section 3.1, Check Your Understanding,

question 7 for the special case of right triangles. In the follow-up to question 4, be

sure to get a variety of responses to show all the ways the triangle can be drawn. In

question 10, point out that AD and BE are transversals for the parallel lines; students

will have to recall the angle relationships for parallel lines.


• When students draw diagrams, some often fail to label the vertices of the

triangles and/or to include the units of measure.

Rx Post, review, and remind students to use a checklist when drawing:

– Does my diagram accurately show what I was given?

– Have I labelled the vertices of my triangle(s) correctly?

– Have I included the units for the side lengths and used the degree

symbol (°) for angles?





4 a)–d) 1–3, 4 e), 5, 11, 12 6–10, 13


• Students neglect to include a sketch of a triangle.

Rx Insist students make a sketch of all triangles even when it is not explicitly

stated in the question. Include a mark for a sketch when assessing their learning.

Make sure students are aware they are required to include the sketch.





support.

• Instead of Geostrips®, some students may want to use strips of paper cut to

various lengths or Cuisenaire rods

E x t e n s i o n

Assign question 13. You may wish to reduce the number of Check Your


question. If students do not start by drawing a diagram, remind them of this strategy.

Challenge students to create a similar problem with different algebraic expressions for

the angles.

Technology

The Geometer’s Sketchpad® can be used to explore congruence and similarity. There

are some activities available online. Go to www.mcgrawhill.ca/books/math8NS for

some interesting Web sites.


1. The third angle measure is 90°, so it is a right triangle with two equal angles.

2. WX � PR, WY � PT, XY � RT, �W � �P, �X � �R, �Y � �T

3. a) congruent; two pairs of corresponding sides and the contained angle are

congruent

b) congruent; two pairs of angles and a corresponding side are congruent

c) not definitely congruent; no side lengths are known, so they could be different

sizes

d) congruent; all corresponding pairs of sides are congruent because of the

Pythagorean relationship

4. Answers may vary.

a), c)

b) Measured sides AB and BC with a ruler, and �B with a protractor.

d) Two side lengths and the contained angle are enough to show congruency.

e) three measurements

Y

3.9 cm

3.7 cm32º

X

Z


5. Measure all three side lengths; At least one side length is needed to show

congruency so just measuring all the angles is not enough information.

6. Measure the length of the bottom side and the bottom angles.

7. �EDF � �GDF, so there is a congruent pair of angles between two congruent

pairs of sides in �DEF and �DGF.

8. Since there are two pairs of congruent angles and DF is a shared side in both

triangles, �DEF and �DGF are congruent so the corresponding sides DE and

DG are equal.

9. The third sides could have different lengths and the unknown angles could have

different measures.

10. Since the line segments are parallel, �B � �E and �A � �D, so there are two

pairs of congruent angles and the contained sides are congruent.

11. Measure all the side lengths and compare matching side lengths for congruency.

12. a) Yes; for example, measure all three sides of each triangle.

b) Yes; for example, measure all three sides of each triangle.

c) No; it has no line of symmetry, but it does have rotational symmetry.

13. Yes, �RST � �JKL. Since the sum of the angles in the triangle is 180°,

4x � 3x � 40 � 180, so 7x � 140 and x � 20°. By substituting x for the angle

measures I get all three pairs of angles congruent and a pair of corresponding

sides congruent.


3.3 Properties of Transformations

W A R M - U P

Evaluate.

1. 0.01 � 6000 <60> 2. � 70 <7>

3. 7200 � 1000 <7.2> 4. 0.1 � 85 <8.5>

5. � 90 <0.09> 6. 0.7 � 10 <0.07>

7. 0.001 � 6400 <6.4> 8. � 6600 <66>

9. 15 � 1000 <0.015> 10. 800 � 0.1 <80>

11. � 5.1 <0.051> 12. 82 � 100 <0.82>

13. 7.49 � 0.001 <0.007 49> 14. � 0.63 <0.063>

15. 572 � 10 <57.2>

M u l t i p l y by Te nt h s, H u n d re d t h s, a n dTh o u s a n d t h s

For this strategy, keep track of how the place values of the numbers change.

When multiplying by 0.1, the place values of a number change by one place:

0.1 � 560 � � 560 The 5 hundreds become 5 tens and the 6 tens

� 56 become 6 ones.

When multiplying by 0.01, the place values of a number change by two places:

0.01 � 8750 � � 8750 The 8 thousands become 8 tens, the 7 hundreds

� 87.5 become 7 ones, and the 5 tens become 5 tenths.

When multiplying by 0.001, the place values of a number change by three places:

0.001 � 314 � � 314 The 3 hundreds become 3 tenths, the 1 ten becomes

� 0.314 1 hundredth, and the 4 ones become 4 thousandths.

Notice that multiplying by tenths, hundredths, and thousandths is similar to

dividing by tens, hundreds; and thousands.

0.1 � 560 � 56 560 � 10 � 56

0.01 � 8750 � 87.5 8750 � 100 � 87.5

0.001 � 314 � 0.314 314 � 1000 � 0.314

1

1000

1

100

1

10

1

10

1

100

1

100

1

1000

1

10

Materials• centimetre dot paper• Bullseye compasses• protractors• rulers (millimetres)• transparent mirrors• tracing paper

Related Resources• BLM 3.3 Alternate DTM

Part A Activity• BLM 3.3 Assessment

Question• BLM 3.3 Extra Practice

Specific CurriculumOutcomesE2 make and apply

generalizations aboutthe properties ofrotations and dilatations,and use dilatations inperspective drawings ofvarious 2-D shapes

E4 perform various 2-Dconstructions and applythe properties oftransformations to theseconstructions


Link to Get ReadyStudents should havedemonstrated understandingof Transformations in the GetReady prior to beginning thissection.



Display objects or photos of objects that use transformations in their design, such as

wallpaper or a border, fabric, a tie or scarf, a tile, a plate, a stained glass window, an

Escher poster, etc. Ask students to find examples of translations, reflections, and

rotations in the designs. It is important that students appreciate that transformations

are used in real world of design, and that artists need to understand the properties of

these transformations to make their designs.

Transformations can be viewed as both operations on geometric shapes and as

relationships between geometric shapes, in the same way as addition, subtraction,

multiplication, and division are operations on numbers and also relationships

between numbers. For example, 18 is viewed as the result of the operation of multi-

plying 6 by 3, but 18 can also be viewed as a comparison to 6 (it is 3 times as much

as 6). Similarly, if you translate a triangle, you produce its image in a new position

and that image is viewed as a result of the translation. But the two triangles can also

be viewed as having a relationship: they are translation images of one another.


Have students work in pairs. It is important that students understand that the prop-

erties of the three transformations they observe in the activities apply to all similar

transformations. You could prepare overheads of other polygons under translations,

reflections, and rotations, and quickly display them after each activity to show that

the properties apply to other polygons as well. For each part of the Discover the

Math, have students list the properties of the transformation studied on chart paper

for future reference.

To do Part A using technology, see BLM 3.3 Alternate DTM Part A Activity for

detailed instructions on how to translate 2-D figures using The Geometer’s

Sketchpad®. The BLM also includes an exercise examining some of the properties of

translations. For question 9, parts c) to e) of the BLM, students should notice that

the translated pentagons have the same measurements as the original and that the

measurements change if they change the size or shape of the original pentagon.

After students finish question 1, have them discuss their results. Consolidate

their observations on a list titled Properties of Translations. Properties should

include:

• Corresponding sides have equal lengths (are congruent).

• Corresponding sides are parallel to each other.

• Corresponding angles have equal measures (are congruent).

• Corresponding shapes are congruent.

• All points change position and move the same distance.

• Line segments joining corresponding points have equal lengths.

• Line segments joining corresponding points are parallel to each other.

Refer students to the diagram on page 107 of the Get Ready. Ask them if these

properties apply to the red and blue triangles. This will also provide an opportunity

to discuss that line segments along the same straight line are parallel to each other.

Have students continue with questions 2 and 3, where they learn how to communi-

cate a translation using two different notations and how to apply the parallel prop-

erty of translations to constructions. Remind them that question 4 asks them to


name particular line segments, angles, and shapes; for example, DE � D�E�, DF � D�F�,

and EF � E�F� because corresponding segments have equal lengths. This question

could be used as a homework question for individuals to complete.

For Part B, have students open their books and follow along as you guide them

through question 1. Students should complete and discuss each part before you

consolidate the answers. After question 1, part g), direct students to the bottom

paragraph and the margin definition of orientation. You may want to add:

“Corresponding shapes have the same orientation.” to the list from Part A. For

question 1, part j), have students add their observations to a list titled Properties of

Reflections. Properties should include:




• Corresponding shapes have different orientations.

• All points, except those on the mirror line, change position.

• A point and its image are equidistant from the mirror line.

• Line segments joining corresponding points are parallel to each other.

• The mirror line is a perpendicular bisector of segments joining

corresponding points.

Have students complete questions 2 to 4 to check the properties of reflections

with quadrilaterals, and to see how the properties of reflections can be applied.

Question 5 could be used as a homework question for individuals to complete.

For Part C, have students open their books and follow along, doing the various

parts as you read them, and discuss the results. Have students add their observations

to a list titled Properties of Rotations. Properties should include:




• Corresponding shapes have the same orientation.

• All points, except the centre of rotation, change position.

• All points move through the same angle (the angle of rotation) with the

centre of rotation.

• Corresponding line segments joining the centre of rotation to correspon-

ding points have equal measures.

• The perpendicular bisectors of segments that join corresponding points

pass through the centre of rotation.

The three lists will help students answer question 5.


P a r t A

1. a) Corresponding angles look equal. They can be measured to check.

b) Corresponding sides look equal. They can be measured to check.

c) They are congruent since all corresponding angles and sides are congruent.


d) Answers may vary.

e) The line segments are all equal in length and parallel.

f) Yes; From the broken line segments you can see the vertices all move the same

distance, so all points on the sides between the vertices must also move the

same distance.

2. a) [1R, 3D] b) [3R, 4U]

3. a) For both line segments, the end point is 5 units to the right of the beginning

point and 7 units down.

b) Yes, because all points on AB get translated to their corresponding location

on A�B�.

c) Answers may vary. Translate X(1, 2) to T(4, 9): [3R, 7U].

d) Answers may vary. Y(7, 8) goes to V(10, 15). These lines are parallel because

each point on TV is the same distance and direction from a corresponding

point on XY.

4. �D�E�F� is also scalene so each triangle is made up of different side lengths and

angles. Both triangles are congruent to each other so all corresponding side

lengths and corresponding angles are congruent.

P a r t B

1. b) Corresponding angles look equal. They can be measured to check.

c) Corresponding sides look equal. They can be measured to check.

d) Yes; corresponding angles and sides are congruent.

e) No; B and B� are the same point. No; A and A� are farther away than C and C�.

f) Corresponding vertices are the same distance from the mirror line.

g) �ABC, �A�C�B�, �A�C�B�; The order is not the same for all triangles.

h) The distances from the vertices to the reflection line are different so the

broken lines are not congruent. The line segments are parallel.

i) 90°. The mirror line is a perpendicular bisector to each broken line.

j) Image is congruent to pre-image, orientation changes, mirror line is a

perpendicular bisector of line segments between corresponding points.

y

x20

–2

–4

–6

2

4

–2–4–6

A

BE

A‘

B‘


2. a), b) Image points are the same distance from

the line of reflection as the pre-image points.

c) Yes, reading clockwise, the quadrilaterals are

PQRS and P�S�R�Q�; The top and bottom have

been reversed.

d) Congruent images, different orientation, line

segments between corresponding points parallel

and bisected by line of reflection.

3. a) Reflected triangles have different orientations.

b) Draw the perpendicular bisector of LL�, MM�, or NN�.

4. a) CD is the mirror line reflecting A onto B.

b) F is along the mirror line reflecting D onto E, so the distances between D and

F and D and E are the same. Since DE and DF are the same length and are both

connected to DE and each other, the triangle formed must be isosceles.

5. The triangles are congruent, their orientations are different, the mirror line is a

perpendicular bisector of JJ�, KK�, and LL�.

P a r t C

1. a)–c) d) yes

e) Yes; corresponding angles and sides are

congruent.

f) Yes, corresponding vertices are in the

same order going clockwise.

g) Yes; line segments formed by joining the

centre of rotation to corresponding points

are congruent.

h) No; points farther from the centre of

rotation move farther when rotated.

i) All three points were rotated by the angle of rotation so all three angles are

equal; 60°

j) point C; yes

k) Image and pre-image are congruent and have the same orientation.

Corresponding points are the same distance from the centre of rotation. The

perpendicular bisectors of the line segments between corresponding points pass

through the centre of rotation. The angle the line segments formed with the

centre of rotation is the angle of rotation.

2. a) Answers may vary. Measure �APA�.

b) The perpendicular bisectors of the line segments between corresponding

points each pass through the centre of rotation.

LL’

M’

N’

C

M

N

y

x4 80

–4

4

8

12

16

–4

P

P‘

Q

Q‘

R

R‘

S

S‘


3. a), b) c) Yes; corresponding angles and sides are

congruent.

d) The centre of rotation; properties of

rotations.

e) 90° rotation clockwise about P.

f) Yes; they can be measured or, in this case,

calculated using the Pythagorean

relationship.

g) Yes; properties of rotations.

h) Every point around the centre of rotation moves around the centre of

rotation, staying the same distance from the centre of rotation.

i) Corresponding sides are parallel and congruent. The line segments between

corresponding points pass through the centre of rotation.

4. Triangles are congruent and have the same orientation. Corresponding points

are the same distance from the centre of rotation H. The perpendicular bisector

of the line segment between corresponding points passes through H. The angle

the line segments form with H as the vertex is 60°.

5. All three transformations have congruent images and pre-images. Translations

and reflections have line segments between corresponding points parallel.

Translations and rotations have the same orientation for image and pre-image.

Only reflections have a change in orientation from image to pre-image. Only

translations have corresponding points on the image and pre-image the same

distance apart.



Ideas questions. Use this opportunity to assess student readiness for the Check Your



1. Check if they are congruent and if all corresponding points are the same

distance apart.

2. Make a perpendicular bisector for the line segment between any pair of


3. Draw line segments from the vertices of the polygon to points on the mirror so

that the line segments are perpendicular to the mirror line. Extend these line

segments twice their length past the mirror line and join the endpoints to

construct the reflection.

4. Make perpendicular bisectors for the line segments between pairs of

corresponding points, and if they all meet at a point, the centre of rotation,

then it is a rotation.

Example 1 asks students to use properties of a translation to verify whether a given

diagram actually represents a translation. From Discover the Math, Part C, question

3, part i), students should understand that corresponding sides of a translation and

of a 180° rotation are parallel. This is why the unique property of translations is that

line segments joining corresponding vertices are congruent and parallel.

L‘

K‘

M‘J‘

J

L

K

M


Example 2 asks students to determine which transformation is shown in the

given diagram. Point out the thought bubble to the right. At first glance, the diagram

may appear to be a reflection but a check of its orientation shows this is not the case.

The corresponding sides are parallel so it might be a translation; however, segments

joining corresponding points are clearly not parallel, a requirement of a translation.

Therefore, this is likely a rotation. Point out that is the special case of a 180° rotation;

corresponding sides are parallel for this type of rotation only.


• Can students describe the properties of rotations, translations, and reflections?

• Can students used the notation shown in Example 2 to mark the original

vertex and the transformed vertex?

• Can students identify the properties of a scalene triangle and of a quadrilateral?

• Can students describe an angle bisector and mirror line?

• Do students understand the terms corresponding, pre-image, and map onto?



Following question 2, students could use a Venn Diagram with three circles labelled

Translations, Reflections, and Rotations to sort the properties. For example,

corresponding shapes are congruent would be placed in the intersection of the three

circles while corresponding sides are parallel would be placed in the Translations circle.

Question 3 provides an opportunity to discuss how one diagram can be

described in two ways because of the special case of the mirror line being a side of a

polygon: One quadrilateral is the reflected image of the other quadrilateral; The

hexagon has one line of reflective symmetry.

Following question 4, you could ask: “How would the result have been different

if the angle had been 70° rather than 72°? What is special about a 72° angle? What other

angles could have been used to guarantee a resultant polygon with rotational symmetry?”

This will lead into the idea of a central angle of a regular polygon in Section 3.4.

After question 7, ask if the same three possibilities would exist if the shapes

were rectangles or parallelograms. Students may be able to find a pair of rectangles

that are reflections, translations, and 180° rotations of each other, but not a pair of

parallelograms (unless they contain right angles). Question 8 might require some

discussion because the mirror line passes through the figure and the mirror line has

to be treated like a two-way mirror.


• Students often do not realize that all points on the figure are affected by a

transformation because they usually only find the images of the vertices of a

shape under a transformation.





1, 2, 6 a) 3–5, 6 b)–e), 7, 9–12 8


Rx When students are carrying out a transformation, sometimes have them

find the images of points other than the vertices. Periodically, when a trans-

formation is on the display, point to a variety of points (in the interior,

exterior, or along line segments) and discuss the locations of the images of

these points under the transformation.

• When carrying out a translation, some students mistakenly start counting

left or right at the vertex of the pre-image rather than 1 unit to the left or

right of that vertex.

Rx Have students make a scalene triangle on the top left of a 10 � 10 geoboard,

give them a translation using arrow notation for a right and down move-

ment (such as (3→, 4↓)), and ask them to use another elastic to show the

image. Watch to see if they count pegs or distances. If they count distances,

help them make the connection to grid paper by asking them to copy the

geoboard triangle onto grid paper. If they count pegs, ask them what (1→)

means and other questions to try to get them to self-correct before moving

on to grid paper.


• Some students may need to work with manipulates prior to being able to

respond to questions.

A S S E S S M E N T


a)

b) A reflection; corresponding vertices are joined by parallel line segments of

various lengths and the orientation is different.

c) i) A 180° rotation about E; the orientation is the same but line segments between

corresponding points are neither parallel nor congruent.

ii) A translation [5R, 4D]; the orientation is the same and line segments between

corresponding points are parallel and congruent.

d) They are congruent but do not have all the properties of any of the

transformations.




C’’

E

DC

BA A’B’

C’D’

E’ F’F

E’’’

D’’’C’’’

B’’’A’’’

F’’’

F’’

A’’B’’

D’’

E’’




support.

• Some students may benefit from having enlargements of square dot paper

or centimetre grid paper to record their work.

E x t e n s i o n

Assign questions 12. You may wish to reduce the number of Check Your


questions. Students should discover that both the x- and y-coordinates of any point

have opposite signs following the two reflections, and that the reflections can be

performed in any order. Challenge the students to find a single transformation that

would have the same result as these two reflections.

Te c h n o l o g y

Use Internet resources to explore transformations using interactive sites, real-world

applications, and The Geometer’s Sketchpad®. Go to www.mcgrawhill.ca/books/

math8NS for some interesting Web sites.


1. a) 180° rotations and translations

b) This piece could be rotated 180° about a point near its middle and

then the image and pre-image translated right repeatedly to create the

pattern.

2. a) translation, reflection, rotation b) translation

c) translation, reflection, rotation d) rotation e) reflection

f) translation, reflection g) translation

h) rotation, reflection (perpendicular bisectors intersect everywhere)

3. a), b) Draw line segments from the vertices perpendicular

to the right side and make the image point the same

distance from the right side as the pre-image point.

c) All points on the right side did not change because they are on the mirror line.

d) Hexagon; it has one line of symmetry because the right half is a mirror image

of the left half.

4. a), b) d) Q

P R


c) All points but P change position because only the centre of rotation stays in

the same position.

e) Decagon; it has rotational symmetry of order 5 because it is five rotations of

the original triangle.

5. b) Construct the perpendicular bisector to the line segment between


6. a), c) b) It is not a translation or a reflection

because line segments between

corresponding points are not parallel.

d) Construct perpendicular bisectors to line

segments between corresponding points

and find where they meet to locate P.

e) approximately 93° counter clockwise

about P

7. a) i) translated 5 units right

ii) reflected across the vertical line 1 unit right of the black square

iii) answers may vary; rotated 90° clockwise about the point 1 unit right and

1 unit below the black square

b) I would know which vertices were corresponding pairs and could check the

orientation of the image to determine the type of transformation.

8. a), b)

c) There were points to reflect on both sides of the mirror line.

9. a) sometimes true; points on the mirror line do not change position

b) never true; translations maintain orientation

c) sometimes true; only true for 180° rotations

d) always true; reflection images are congruent to the pre-image

e) never true; points farther from the centre of rotation move a greater distance

10. a) Each quarter has a mirror line between the two touching purple triangles. Each

quarter is a 90° rotation of the other quarters about the centre of the square.

b) The square has a horizontal mirror line, a vertical mirror line, and two

diagonal mirror lines all passing through the middle of the square. Each

V-shape is a 90° rotation of the others about the centre of the square.

12. b) i) All x-values stayed the same and all y-values were multiplied by �1.

ii) All x-values were multiplied by �1 and all y-values stayed the same.

iii) All x-values and all y-values of the pre-image were multiplied by �1.

iv) same as iii)

c) yes

d) All x-values and all y-values are multiplied by �1; It is also the same image as

a 180° rotation of the pre-image.

A

A’

D

D’

B

B’

C

C’


3.4 Regular Polygons

W A R M - U P

Solve, using the Make “1” strategy.

1. � <1 or 1 > 2. 1 � <2 or 2 >

Add.

3. � < > 4. � < >

Subtract.

5. 8 � 2 <5 > 6. 4 � 2 <1 >

Multiply.

7. � 36 <12> 8. � 48 <20>

9. 3 � 4 <14> 10. 8 � 3 <30>

Divide.

11. 5 � <15> 12. � <4>

Evaluate.

13. � 420 <42> 14. 0.001 � 60 <0.06>

15. 850 � 100 <8.5>


You could have students re-create the design in the section opener photo using

pattern blocks, (except for the outside shapes with rounded edges). Have students

suggest how the rhombuses are different from the other shapes they used. Review

what information about a polygon, such as an octagon, is conveyed when it is called

a regular polygon. Brainstorm examples of regular polygons used in the real world

and list them on chart paper. Challenge students to watch for other examples in the

days ahead, and add to the list when more examples are found.

1

10

2

9

8

9

1

3

3

4

2

3

5

12

1

3

2

5

3

5

1

10

9

10

7

9

1

9

2

3

11

12

5

12

1

2

1

2

3

6

5

6

2

3

1

3

4

12

5

12

11

12

Materials• Geostrips®• rulers (millimetres)• protractors• Bullseye compasses• tracing paper• transparent mirrorsOptional:• The Geometry Template®

Related Resources• BLM 3.4 Regular Polygons• BLM 3.4 Assessment

Question• BLM 3.4 Extra PracticeOptional:• BLM 3.4 DTM Polygon

Table• BLM 3.4 CYU Table• BLM 3.4 Alternate DTM

Activity

Specific CurriculumOutcomesE5 make and apply

generalizations aboutproperties of regularpolygons


Link to Get ReadyStudents should havedemonstrated understandingof Regular Polygons in theGet Ready prior to beginningthis section.



To do this activity using technology, see BLM 3.4 Alternate DTM Activity for detailed

instructions on how to construct regular polygons using circles and triangles using The

Geometer’s Sketchpad®.

The purpose of the activity is to move students’ thinking from recognizing,

naming, representing, and describing regular polygons to an understanding of the

properties of these regular polygons. They should notice the following patterns and

relationships:

• the pattern in the number and location of lines of reflective symmetry

• the pattern in the order of rotational symmetry

• the pattern in the number of isosceles triangles into which the regular

polygons can be partitioned

• the pattern in the measures of the central, interior, and exterior angles

• the relationship of the central, interior, and exterior angles to inscribed and

circumscribed circles

Throughout the activity, discuss how artists could make use of each property in their

work.

Question 1 should establish a need to know about the angle measures as

students probably eyeballed the angles to create a regular pentagon. Hand out BLM 3.4Regular Polygons for question 2 and have students work on their own, then discuss

their results. Lead students through questions 3 and 4. Supply BLM 3.4 DTM PolygonTable for question 4, part e), if needed. After completing the table, return to question 1

and have students discuss how they would now create the regular pentagon.

Have students complete question 5 on their own and discuss their findings.

Start a list titled Properties of a Regular n-gon on chart paper and ask students to

make suggestions. They should copy the list in their notebooks and use it to answer

question 6 individually. Properties could include:

• It has n equal sides and n equal angles.

• If n is even, it has (n � 2) pairs of opposite parallel sides. If n is odd, there

are no parallel sides.

• It has n lines of reflective symmetry. If n is even, half the lines of symmetry

go through midpoints of opposite sides and half go through pairs of

opposite vertices. If n is odd, the lines of symmetry are the perpendicular

bisectors of the sides.

• It has a centre that is the intersection of its lines of symmetry.

• It has two related circles that share its centre: an inscribed circle that

contains the midpoints of all its sides and a circumscribed circle that

contains all its vertices.

• It can be partitioned into n congruent isosceles triangles.

• It has rotational symmetry of order n.

• Its central angles have a measure of 360° � n.

• Each of its angles has a measure of 180° � (360° � n).

• Each of its exterior angles has the same measure as its central angles.



1. b) All sides are the same length; all angles change.

c) Some are concave and those with angles greater than 180° are convex.

d) All the angles are equal.

e) Concave. No, because all the angles must be equal and it would not be

possible to have all convex angles.

2. a) If a regular polygon has an even number of sides, it has parallel opposite sides.

b) 14-gon: 7 pairs of parallel opposite sides; 15-gon: no parallel opposite sides;

20-gon: 10 pairs of parallel opposite sides; 50-gon: 25 pairs of parallel opposite

sides

3. a) Each line of symmetry is the perpendicular bisector of one side, or it bisects

one of the angles. The number of lines of symmetry is the same as the number

of sides of a regular polygon.

b) The circle touches all five vertices. It is circumscribed because the vertices all

lie on the circumference.

c) The circle touches each side. It is inscribed because it is drawn inside the

pentagon.

d) Three are between opposite angles and three are between the midpoints of

opposite sides.

e)

f) For regular polygons with an odd number of sides, lines of symmetry are

between an angle and the midpoint of the side opposite the angle (bisects

both). For regular polygons with an even number of sides, half the lines of

symmetry are between opposite angles and the other half are between the

midpoints of opposite sides. A 19-gon has 19 lines of symmetry between the

angles and the midpoints of the sides opposite the angle. A 20-gon has 10 lines

of symmetry between opposite pairs of angles and 10 lines of symmetry

between the midpoints of opposite sides.

4. a) Isosceles triangles; each long side length is the radius of the circumscribed

circle, so the two sides are congruent. They are congruent triangles because they

all have congruent corresponding sides.

b) Divide 360° by the number of equal angles; 45°.

c) Their sum is 180° minus the measure of the central angle, so divide that by 2; 67.5°.

d) 135°


e)

f) Divide 360° by the number of sides and subtract that measure from 180°;

152.3°.

5. a) 108° b) Subtract the interior angle from 180°.

c) They are equal. Yes, because the central angle and the interior angle are

supplementary, and so are the interior angle and the exterior angle.

d) That is the angle you would cut out of the wood.

6. 15-gon: 15 lines of symmetry each bisecting one angle and the opposite side,

central angles and exterior angles measure 24°, interior angles measure 156°.

16-gon: 16 lines of symmetry, 8 through opposite pairs of angles and 8 through

the midpoints of opposite sides, central angles and exterior angles measure

22.5°, interior angles measure 157.5°.



Ideas questions. In this section, you may wish to have students work together as a

class. Use this opportunity to assess student readiness for the Check Your


Regular

Polygon

Number of

Congruent

Triangles

in Interior

Measure of

Each Central

Angle

Measure of

Each Base

Angle of Triangle

Measure of Each Interior

Angle of the Regular Polygon

square 4360° ÷ 4

= 90°(180° – 90°) ÷ 2

= 45°45° + 45°

= 90°

pentagon 5360° ÷ 5

= 72°(180° – 72°) ÷ 2

= 54°54° + 54°

= 108°

hexagon 6360° ÷ 6

= 60°(180° – 60°) ÷ 2

= 60°60° + 60°

= 120°

heptagon 7360° ÷ 7 � 51.4°

(180° – 51.4°) ÷ 2� 64.3°

64.3° + 64.3°� 128.6°

octagon 8360° ÷ 8

= 45°(180° – 45°) ÷ 2

= 67.5°67.5° + 67.5°

= 135°

nonagon 9360° ÷ 9

= 40°(180° – 40°) ÷ 2

= 70°70° + 70°

= 140°

decagon 10360° ÷ 10

= 36°(180° – 36°) ÷ 2

= 72°72° + 72°

= 144°

hendecagon 11360° ÷ 11 � 32.72°

(180° – 32.72°) ÷2 � 73.6°

73.6° + 73.6°� 147.2°

dodecagon 12360° ÷ 12

= 30°(180° – 30°) ÷ 2

= 75°75° + 75°

= 150°

20-gon 20360° ÷ 20

= 18°(180° – 18°) ÷ 2

= 81°81° + 81°

= 162°

60-gon 60360° ÷ 60

= 6°(180° – 6°) ÷ 2

= 87°87° + 87°

= 174°

n-gon n360° ÷ n

= 360°—

n

a180° – 360°—

nb ÷ 2

= 90° – 180°—

n

90° – 180°—

n+ 90° –

180°—

n

= 180° – 360°—

n



1. Reflective symmetry means an object can be folded so that one half perfectly

covers the other half. Rotational symmetry means a copy of an object can be

turned and still look exactly the same as the original.

2. The name tells you the number of sides, and you can tell where the lines of

symmetry are depending if the number of sides is even or odd.

3. There are 360° in a complete rotation, so the size of the central angle is 360° divided

by the number of central angles that can fit around the centre of a polygon.

4. Polygons with an odd number of sides cannot have pairs of parallel sides

because there would be one side without an opposite parallel side and that

would mean the angles are not congruent.

5. Bisect two of its angles, or draw perpendicular bisectors of two sides to find the

point where the line segments meet.

6. They are all equal.

Example 1 asks students to apply what they learned about the measures of the interior

and exterior angles of a regular polygon. Students could use mental math for the calcu-

lations. You could ask students to generalize about the sizes of the interior and exterior

angles as the number of sides increases. Example 2 asks students about the properties of

a hexagon as they relate to using a hexagon in design work. You could remind students

to refer to the list of general properties they made in the Discover the Math.


• Can students describe a polygon?

• Can students draw the line of symmetry in any polygon, if applicable?

• Can students describe a perpendicular bisector?



You could point out that question 3 is an alternative method for determining the

measure of the interior angles of regular polygons. This method also enables

students to determine the sum of the angles in any polygon. Supply BLM 3.4 CYUTable for part c) as needed. Following question 7, students could add to their list of

properties: “A regular n-gon will tessellate if the measure of its interior angles divides

360° with no remainder.”

Question 8 provides an opportunity to apply the Pythagorean relationship in

another context. Point out that segment TV is the perpendicular-bisector of the side

RS of the pentagon, and it is the length of TV that is the apothem. You could have

students measure the apothems of some other regular polygons on BLM 3.4 RegularPolygons and calculate their areas.





1, 2, 3 a)–c), 5, 12 a)3 d)–f ), 4, 6, 7, 9 a)–e), 10,

11 a), 12 b)–e)8, 9 f ), 11 b)


Examine and discuss the quilt block designs students make for question 11, part b).Students will be designing another quilt block in the Chapter Problem Wrap-Up and this

would be a good time to address any design problems and point out improvements.

Journal

Students could use these prompts for question 6.

• In my scale drawing of the “loonie”, the interior angles measured

�and the sides measured �.

• These are the steps I used to draw my scale drawing. Step 1….


• Some students describe a regular polygon only in terms of having equal

sides and forget that the angles must also be equal.

Rx Use Geostrips® to reinforce the equal-side and equal-angle nature of regular

polygons. If students join six of the same Geostrips®, they will soon realize

there are a number of possible hexagons with six equal sides, but only one

regular hexagon; and to make it they have to concentrate on making the

angles equal.

• Some students may have difficulty constructing the polygons.

Rx Provide students with a worksheet modelling the Check Your



• Some students may need help drawing the more complex polygons, such as

the one in question 9. Help students get started with the drawings in class

and then have them complete the questions on their own.

A S S E S S M E N T


a) heptagon

b) square, hexagon, octagon, decagon, dodecagon

c) equilateral triangle, pentagon, heptagon, nonagon, hendecagon

d) hexagon

e) decagon

f) dodecagon

g) equilateral triangle, square, hexagon

h) equilateral triangle, square, pentagon, hexagon, octagon, nonagon, decagon,

dodecagon






• Some students may benefit from having physical samples of regular

polygons. Supply Polydrons® or pattern blocks.

E x t e n s i o n

Assign question 12. You may wish to reduce the number of Check Your Understanding

questions to provide students with extra time to work on the Extend question. You

could also challenge students to find the measures of the angles in one of the star

polygons, or to use a star polygon in their quilt block design for question 11, part b).


1. a) it is a heptagon b) it is a hendecagon

c) it is an equilateral triangle d) it is a 20-gon

2. a)–d)

e) Regular decagon, because it has ten equal sides and ten equal angles.

3. a), b)

c)

d) The measure of an interior angle is the total number of degrees divided by the

number of angles.

e) Yes; 720°.

f) Multiply 180° by two less than the number of sides.

RegularPolygon

Numberof Sides

Number ofInterior Triangles

Total Numberof Degrees

Measure of EachInterior Angle

pentagon 5 3 3 � 180°= 540°

540° ÷ 5 = 108°

hexagon 6 4 4 � 180°= 720°

720° ÷ 6 = 120°

heptagon 7 5 5 � 180°= 900°

900° ÷ 7 � 128.6°

octagon 8 6 6 � 180°= 1080°

1080° ÷ 8 = 135°

nonagon 9 7 7 � 180°= 1260°

1260° ÷ 9 = 140°

20-gon 20 18 18 � 180°= 3240°

3240° ÷ 20 = 162°

n-gon n n – 2 (n – 2) � 180°= (180n)° – 360°

[(180n)° – 360°] ÷ n

= 180° – 360°—

n

P


4. b) Start by drawing a circle with radius 5 cm about point T. Draw a line segment

connecting the centre to a point on the circumference and continue drawing the

same lines at 40° intervals because the measure of the central angle of a

nonagon is � 40°.

5. 10 sides, 10 lines of symmetry (5 through opposite pairs of angles and 5

through the midpoints of opposite sides), central angles and exterior angles

measure 36°, interior angles measure 144°.

6. The measure of an interior angle is about 147°. The sides are about 3 cm long.

Start with a 5-cm line segment starting at the centre and make 5-cm line

segments at 33° intervals, then connect the endpoints to finish the hendecagon.

7. equilateral triangle, square, hexagon

8. a) about 12.65 cm

b) 57 cm2; 285 cm2; five triangles the size of �RST would fill the pentagon

perfectly.

c) 63.25 cm; the perimeter is five times the length of RS.

9. a)–c)

It touches all three vertices. It is the centre of a circumscribed circle.

d), e)

It touches each side exactly once. It is the centre of an inscribed circle.

f) The angle bisectors and perpendicular bisectors of the sides all meet at the

centre, so the point of intersection is both an incentre and a circumcentre.

11. a) There are seven of each polygon in the regular 7-gon, there are seven lines of

symmetry, and the order of rotational symmetry is 7.

b) Designs may vary. For example:

12. b) 5 lines of symmetry; degree of rotational symmetry 5

c) 7- and 9-pointed stars; 11-pointed star

d) 3 line segments

e) 6-pointed star

360

9


Chapter 3 Review

W A R M - U P

Solve, using the Make “1” strategy.

1. � <1 > 2. 2 � <3 >

Add.

3. � < > 4. � < >

Subtract.

5. 9 � 6 <2 > 6. 11 � 2 <8 >

Multiply.

7. � 42 <7> 8. � 108 <96>

9. 7 � 5 <40> 10. 9 � 3 <33>

Divide.

11. 6 � <24> 12. 1 � <10>

Evaluate.

13. 0.001 � 58 <0.058> 14. 8.2 � 10 <0.82>

15. � 490 <4.9>


Us i n g t h e C h a p t e r R ev i ew

The students might work independently to complete the Chapter Review, and then

compare solutions in pairs. Alternatively, the Chapter Review could be assigned for

reinforcing skills and concepts in preparation for the Practice Test. Provide an

opportunity for the students to discuss any questions, consider alternative strategies,

and ask about questions they find difficult.

For question 6, part c), point out to students that they should consider what

they know about the right triangles and the Pythagorean relationship. Part d)requires students to use their knowledge of angle relationships from grade 8 (alter-

nate interior angles).

1

100

1

8

1

4

1

4

2

3

5

7

8

9

1

6

5

8

3

8

7

11

4

11

13

15

2

5

7

15

9

10

7

10

1

5

1

8

1

4

7

8

2

15

4

15

13

15

Materials• Bullseye compasses• square dot paper• protractors• tracing paper

Related Resources• BLM 3R Extra Practice



After students complete the Chapter Review, encourage them to make a list of

questions they found difficult, and to include the related sections. They can use this

list as a guide on what to concentrate their efforts on when preparing for the final

chapter test.

A S S E S S M E N T

Chapter Review

This is an opportunity for the students to consolidate the chapter material by

completing selected questions and checking the answers. They can then revisit any

questions that they found difficult.

Upon completing the Chapter Review, students can also answer questions such

as the following:

• Did you work by yourself or with others?

• What questions did you find easy? difficult? Why?

• How often did you have to ask a classmate to help you with a question? For

which questions?


Have students use BLM 3R Extra Practice for more practice.

R ev i ew An s we r s

1. a) unique; two sides and the contained angle known

b) unique; two angles and a side known

c) unique; all three sides known

d) not unique; need to know one more side length or angle measure

2. �DEF � �LMN since two angles and a side are known. D and L are

corresponding angles and so are F and N, therefore DF � LN.

3. a) Make equal length arcs from corresponding vertices so that they intersect

twice and draw the mirror line through these two points of intersection.

b) Draw two equal length arcs from corresponding vertices until they intersect at

two points. Draw a line segment through those two points of intersection.

Repeat these steps for the other two corresponding vertices. The point of

intersection of these line segments is the centre of rotation.

4. a) dodecagon b) 150°

c) All sides are congruent and opposite sides are parallel.

d) 12 lines of symmetry, rotation symmetry of degree 12.

5. a) No; only the angles are known in each triangle.

b) Yes; two pairs of corresponding angles and one pair of corresponding sides

are congruent.

c) Yes; the hypotenuses and known legs are congruent in each right triangle.

d) Yes; using the properties of parallel lines, two corresponding angles and the

contained side are congruent.


6. a)

b) �PQR � �P�QR�, all corresponding sides and angles are congruent, the

triangles have the same orientations, perpendicular bisectors to line segments

between corresponding points will pass through Q, and any angle formed by

corresponding points with Q as the vertex will be 90°.

7. a)

b) �PQR � �PR�Q, all corresponding sides and angles are congruent, the

triangles have different orientations, corresponding points are an equal distance

from the mirror line, and the line segment between corresponding points is

bisected at a right angle by the mirror line.

8. There is only one way to arrange the angles of a triangle once all the side

lengths are known, but the angles in a quadrilateral can be adjusted to different

sizes.

9. Translation; all corresponding side lengths are congruent and parallel.

10. b) Draw a circle with a centre the same as the centre of the square and a radius

that is half the length of a side of the square.

c) Calculate the central angle, � 40°, and mark the vertices around the

circle at this angle.

d) No, because there are an odd number of triangles so symmetry is not possible

with using two colours.

11. a) Drawings may vary.

b) The image of L is itself because the centre of rotation does not move.

c) Yes, because it is the only point that has not moved, and the angle of rotation

is 180° so corresponding sides are parallel; The direction of rotation could be

clockwise or counter clockwise because 180° is half a full rotation in either

direction.

d) The image is congruent to the pre-image. They are parallel. When a line

segment is rotated 180°, it is upside-down, so it is parallel to its pre-image.

360°

9

R

P

R‘

Q

R

P

P‘

R‘

Q


Chapter 3 Practice Test


Us i n g t h e Pra c t i ce Te s t

This Practice Test can be assigned as an in-class or take-home assignment. If it is used

as an assessment, use the following guidelines to help you evaluate the students.

• Can students distinguish which three given parts of a triangle guarantee it is

a unique triangle?

• Can students distinguish which three corresponding parts of two triangles

can be compared to convince other people the two triangles are congruent?

• Can students list most of the properties of translations, reflections, and rotations?

• Can students recognize the common properties of the three transformations

and the properties that are unique to each transformation?

• Can students locate the mirror line in a given reflection without a transparent

mirror?

• Can students determine the centre and angle of rotation of a given rotation?

• Can students apply the properties of translations, reflections, and rotations?

• Can students list most of the properties of regular polygons, particularly as

they apply to a particular regular polygon?

• Can students determine the measures of the central, interior, and exterior

angles of any regular polygon?

• Can students draw the inscribed and circumscribed circles of any given regular

polygon?

St u d y G u i d e

Use the following study guide to direct students who have difficulty with specific

questions to appropriate areas to review.

A S S E S S M E N T

After students complete the Practice Test, you may wish to use BLM 3PT Chapter 3Test as a summative assessment.



• Allow the use of calculators.

• Let students give their answers verbally, either in an interview setting or recorded.

Question Refer to Section

1, 5 3.1

7, 9 3.2

2, 8, 11 3.3

3, 4, 6, 10, 14 3.4

Materials• tracing paper

Related Resources• BLM 3PT Chapter 3 Test



L a n g u a g e / M e m o r y

• Allow students to refer to personal math dictionaries, journals, index card

files, or notes.

Pra c t i ce Te s t An s we r s

1. C

2. B

3. D

4. Subtract 360° divided by the number of sides from 180°; 168°.

5. a) Two triangles of different sizes could have the same angle measures. b) a side length

6. No, because some lines of symmetry only bisect opposite side lengths and

others bisect a side length and an opposite vertex. For example, a square has two

lines of symmetry that do not touch the angles.

7. Mark a mirror line 0.5 m from the flowerbed. Use rope to locate corresponding

points an equal distance from the mirror line and along a line perpendicular to

the mirror line that goes through the original point.

8. a) reflection b) rotation

9. a) i) congruent; all three sides known ii) congruent; two angles and one side known

iii) congruent; two sides and the contained angle known

iv) not congruent; no sides known

b)

c) i) �X � �X�, �Y � �Y�, �Z � �Z� ii) �C � �C�, AB � A�B�, AC � A�C�

iii) �P � �P�, �R � �R�, PR � P�R� iv) �F � �F�

d) i) reflection across a vertical line ii) translation right

iii) 180° rotation about a point between the triangles

10. The octagon that is not regular could have sides and angles of different

measures while a regular octagon has sides and angles that are equal measures.

The octagon that is not regular may have fewer lines of symmetry and a lower

degree of rotational symmetry than the regular octagon.

11. Dodecagon, because a regular polygon with central angle 30° has � 12 sides.

12. Hendecagon, because only dodecagons and hendecagons have internal angles

between 140° and 150° and the lines of symmetry in a hendecagon are all

perpendicular bisectors of each side.

360

30

Y

X

Y’

X’

3.2 cm

8.5 cm

8.5 cm

7.0 cm

7.0 cm 3.2 cm

Z

Z’

B C

A

B’ C’

A’

2.5 cm116°

19°

2.5 cm116°

19°

Q R

R’

P

Q’

P’

3.6 cm

3.7 cm

3.7 cm

3.6 cm

67°

67°

E

D F

E’

D’ F’

110°

50°110°

50°


Chapter 3 Chapter Problem Wrap-Up

1. Introduce the problem.

2. Clarify the assessment criteria by reviewing BLM 3CP Chapter Problem Wrap-Up Rubric with students.

3. Remind individual students that they have worked on the chapter problem

during Chapter Problem revisits throughout the chapter and that these will help

them. Students can also be directed to section 3.1, question 11, section 3.2,

question 12, section 3.3, question 10, and section 3.4, question 11, at this point.

4. Brainstorm with students about different polygons and transformations that

can be used in the quilt block.

5. Allow students time to work on the design, either individually or in a group.

Students should prepare separate quilt blocks.

O ve r v i ew o f t h e Pro b l e m

The assignment should be accessible to all students. Weaker students may take some

time to create their design and may not include all the suggested aspects of the square.

As the design is to include congruent triangles, transformations, and regular polygons,

expect to see a reflection of each student’s ability in those topics in the final design.

A S S E S S M E N T

Use BLM 3CP Chapter Problem Wrap-Up Rubric to assess student achievement.

H i g h S co r i n g S a m p l e R e s p o n s e

Refer to the Exemplar at the end of this Teacher’s Resource chapter.

C r i t e r i a fo r a H i g h S co r i n g R e s p o n s e

• Student creates a design that meets all the requirements.

• Student makes accurate drawings of congruent triangles and regular polygons.

• Student includes several images of polygons after translations, rotations,

and reflections.

• Student draws the images of the transformed polygons accurately.

Wh at D i s t i n g u i s h e s Lowe r S co r i n g R e s p o n s e s

• Student may omit some aspects of the design.

• Student may not make accurate drawings of congruent triangles and regular

polygons.

• Student may not include all three types of transformations.

• Student may not draw images accurately, so transformation images are not

congruent to pre-images.

• Student basically understands the problem and can do some parts of it—

just cannot combine all the parts into one design.


C h a p te r Pro b l e m Wra p - Up, An s we r

Refer to student exemplar for a sample answer.

St u d e nt E xe m p l a r


Chapters 1–3 Review

W A R M - U P

Evaluate using compatible factors.

1. 5 � 13 � 2 <130> 2. 7 � 25 � 4 <700>

3. 15 � 17 � <85> 4. 3.5 � 12 � 2 < 84>

5. 24 � 25 <600> 6. 4.5 � 24 <108>

Evaluate.

7. � < > 8. 5 � 1 <3 >

Multiply.

9. � 54 <48> 10. 6 � 2 <14>

Divide.

11. 2 � <18> 12. 3 � <20>

Evaluate.

13. 0.01 � 785 <7.85> 14. � 450 <45>

15. 420 � 1000 <0.42>


Us i n g t h e Cu m u l at i ve R ev i ew

The students might work independently to complete the Chapters 1–3 Review, and

then compare solutions in pairs. Provide an opportunity for the students to discuss

any questions, consider alternate strategies, and ask about strategies or problems

they found difficult.

After students complete the Chapters 1–3 Review, encourage them to make a

list of questions that caused them difficulty, and include the related sections. They

can use this list to focus their studying for a final test on the book’s content.

A S S E S S M E N T

C h a p te r s 1 – 3 R ev i ew

This is an opportunity for the students to assess themselves by completing selected

questions and checking the answers. They can then revisit any questions that they

found difficult.

1

10

1

6

1

3

1

9

1

3

8

9

1

6

5

6

7

10

1

5

9

10

1

3

Materials• square dot paper



Upon completing the cumulative review, students can also answer questions

such as the following:

• Did you work by yourself or with others?

• What questions did you find easy? difficult? Why?

• How often did you have to ask a classmate to help you with a question? For

which questions?

St u d y G u i d e

Use the following study guide to direct students who have difficulty with specific

questions to appropriate areas to review.


• Allow the use of calculators for students having difficulty with the calcula-

tions, or as a method of checking answers.

• Question 13 requires students to use their knowledge of angle relationships

from grade 8 (corresponding angles).

C h a p te r s 1 – 3 R ev i ew An s we r s

1. a) 27 b) 12 c) 0.6 d) 99

2. approximately 9.2 cm

3. a) approximately 0.38 b) 1.2 c) 12 d) approximately 38

4. 2 cm

5. 12 congruent sides and 12 congruent angles all equal to 150°, the measure of

the central angle is 30°, rotational symmetry of degree 12, and 12 lines of

reflective symmetry: 6 bisecting opposite angles and 6 bisecting opposite sides.

6. a) b) 3 c) 3 d) 3

7. 6 cups; cup more9

10

1

10

11

30

9

10

5

6

5

8

Question Refer to Section

1 1.1

3 1.2

9 1.3

2, 4 1.4

6, 7 2.1

8, 12 2.2

10, 14 2.3

17 2.4

16 3.1

13, 15 3.2

11, 18 3.3

5 3.4


8. a) b) c) 1 d) 2

9. y � 9.9 cm

10. a) 20 b) 6 c) d) 1

11. 90° clockwise rotation. All points but the centre of rotation have moved, and

line segments between corresponding points are not parallel.

12. a) 3 laps b) 5 laps

13. Using the properties of parallel lines, �ABC � �ECD, so the triangles are

congruent because two pairs of corresponding angles and a pair of

corresponding sides are congruent.

14. $60.67

15. a) not congruent; only one pair of congruent angles and no sides known

b) congruent; two pairs of corresponding angles and a pair of corresponding

sides are congruent

c) not congruent; DG and DE are not congruent

16. a) not unique; only two angles are known

b) not unique; the known angle is not between the known sides

c) unique; all sides are known using the Pythagorean relationship

d) unique; two angles and one side known

17. a) 1 b) 1 c) 2

18. b) Translation [1R, 4D]; all lines segments between corresponding points are

congruent and parallel.

c) i) 90° rotation clockwise about the top left vertex of the blue pentagon;

perpendicular bisectors of line segments between corresponding points all meet

at that point.

ii) Reflection across the vertical line 0.5 units right of the black pentagon; different

orientation, all lines segments between corresponding points are parallel.

iii) Translation [1L, 4U]; all lines segments between corresponding points are

congruent and parallel.

1

42

1

2

1

4

1

2

1

8

3

5

1

2

1

3

3

5

3

4


CHAPTER 3 Geometry I - cbv.ns.ca 8_TR/Teachers Resource...CHAPTER 3 Geometry I ... has a rotational...

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