Chapter 2

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SPACE AND GEOMETRY

Look around you for a moment—you will see that there are angles everywhere. The knowledge of angles is important in architecture, landing planes, graphic designing, and even in playing sports such as football or snooker.

02 NCM7 2nd ed SB TXT.fm Page 32 Saturday, June 7, 2008 2:53 PM

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In this chapter you will: Wordbank

• label and name images • estimate, measure and construct angles• classify angles as right, acute, obtuse, reflex,

straight or a revolution• identify and name adjacent angles, vertically

opposite angles, straight angles and angles of complete revolution

• use angle relationships to find unknown angles in diagrams

• use the words ‘complementary’ and ‘supplementary’ for angles

• use the common symbols for ‘is parallel to’ (

II

) and ‘is perpendicular to’ (

�

)• use the common conventions to indicate right

angles, equal angles and parallel lines• identify, name and measure alternate angle pairs,

corresponding angle pairs and co-interior angles for two lines cut by a transversal

• recognise the equal and supplementary angles formed when two parallel lines are cut by a transversal

• use angle properties to identify parallel lines.

•

complementary angles

Two angles that add to 90°.

•

parallel lines

Lines that point in the same direction and do not intersect.

•

perpendicular lines

Lines that intersect at right angles.

•

protractor

An instrument for measuring the size of an angle.

•

supplementary angles

Two angles that add to 180°.

•

transversal

A line that cuts across two or more other lines.

•

vertex

The corner or point of an angle.


34

NEW CENTURY MATHS 7

Start up

1

In this diagram, each gap represents 1° of angle size.

What is the angle, in degrees, between the lines labelled:

a

A

and

C

?

b

A

and

D

?

c

B

and

C

?

d

C

and

F

?

e

A

and

F

?

f

B

and

G

?

g

D

and

G

?

h

E

and

H

?

i

D

and

I

?

j

C

and

J

?

k

B

and

E

?

l

E

and

J

?

2

In the diagram in Question

1

, find one pair of labelled lines which have a 19° angle between them.

3


1

, find two pairs of labelled lines which have a 90° angle between them.

4


1

, find the pairs of labelled lines which have the following angles between them.

a

7°

b

8°

c

13°

d

28°

e

50°

f

89°

g

95°

h

114°

5

The word ‘degree’ has many meanings. Find four non-mathematical meanings for the word.

6

Decide whether each of these angles is:

i

acute

ii

obtuse

iii

reflex

Worksheet2-01

Brainstarters 2

A

B

C

D

E

F

GHI

J

Skillsheet2-01

Types of angles

a b c d


35

CHAPTER 2

ANGLES

2-01 Naming angles

An angle is a description of the size of a turn or rotation.It is drawn with two arms which meet at a

vertex

. Angles are normally marked with a curved line called an

arc

. This shows the size of the turn. The angle marked in this diagram can be written as:

•

∠

G or

•

∠

PGH

or

∠

HGP

•

P H

or

H P

e f g h

i j k l

m n o

G

P

H

vertex arm

arc

The middle letter is always the letter that labels the vertex of the angle.

{G

G G

Example 1

Name the angle marked with in each of these diagrams.

Solutiona ∠Y or ∠XYZ or ∠ZYXb ∠PQS or ∠SQP

Note: We cannot name this ∠Q because it is not clear which angle that means. There are three different angles whose vertex is ∠Q. They are ∠PQS, ∠SQR and ∠PQR.

a b

X

Y

Z

P

Q S

R


36

NEW CENTURY MATHS 7

1

Name each of these angles in two different ways.

2

The name of the angle marked is which of the following? Select

A

,

B

,

C

or

D

.

A

∠

ABD

B

∠

CBD

C

∠

ABC

D

∠

BCA

3

Name the angle marked with in each of these diagrams.

4

Draw each of these angles, labelling them correctly.

a

∠

POT

b

∠

TAF

c

∠

AFE

d

∠

H

Exercise 2-01

Ex 1

a b cP

Q K O

R

C G

V E

A

G

T PQ

Dd e f R C

D

A

B

C

D

D

B

C

A N

M

QP

P

T

S

R

Q

F

E

H

C

BA

DZ

W

Y

a b c

d e f

XE

G


37

CHAPTER 2

ANGLES

5 a

There are 13 different angles inside this diagram. Name them all.

b

What type of angle is

∠

NCY

?

6

Name the angles marked and in each of the following diagrams.

7

Angles

AMP

and

PMN

share a common arm,

PM

. They also share a common vertex,

M

. Angles that are next to each other in this way are called

adjacent angles

.Name a pair of adjacent angles for each diagram in Question

6

.

2-02 Measuring angles

A protractor is an instrument used to measure angles.

N

A

Y

DC

x

a b c

d e f

A

D

CB

R

S

P

Q

Q

R

P

M N

ZYX

W

H

F

DE

G

A

B

C

GH

I

E

F

x

x

x

x

x

x

NM

AP

arm Worksheet2-03

360° scale

Worksheet2-04

Make your own protractors

Skillsheet2-03

Starting Cabri Geometry

Geometry2-01

Making a protractor

Worksheet2-05

A page of protractors

Worksheet2-02

Comparing angle size

Centre mark

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

Outside scale

Base line

Inside scale

Skillsheet2-02

Starting The Geometer’s Sketchpad


38 NEW CENTURY MATHS 7

Example 2

1 Measure angle AOB.

Solution• Line up OB with the base line of the protractor.• Place the centre mark over the vertex, O.• The angle is smaller than 90°.• Use the inside scale,

counting from 0°.Angle AOB = 54°

2 Measure ∠PMQ.

Solution• Line up QM with the base line of the protractor.• Place the centre mark over the vertex, M.• The angle is greater than 90°.• Use the outside scale,

counting from 0°.∠PMQ = 155°

B

A

O

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

13014

0

150

160

170

180 B

A

O

MQ

P

90 100 110 120 130

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160170

180

8070

6050

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2010

0

90 80 70 6050

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2010

0

100110

120

130

140

150

160

170

180

MQ

P


39CHAPTER 2 ANGLES

3 Measure ∠TEX.

Solution• Line up TE with the base line of the protractor.• Place the centre mark over the vertex E.• ∠TEX is bigger than 90°.• Use the inside scale.

∠TEX = 134°

X

ET

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

13014

0

150

160

170

180

X

ET

Example 3

Measure the reflex angle ∠GHK.

Solution• Actually measure the

obtuse angle ∠GHK first (140°).

• Subtract 140° from 360°.360 − 140 = 220

Reflex ∠GHK = 220°

GH

K

90100110120130

140

150

160

170

180

8070

6050

4030

2010

0

9080706050

4030

2010

0

100110

120

130

140

150160

170180 G

H

K



1 Find the size of each of these angles.

2 Estimate the size of each of these angles. Name each angle and use a protractor to measure the angles accurately.

Exercise 2-02

Worksheet2-06

A page of angles

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

90 100 110 120 130

140150

160170

180

8070

6050

4030

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0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

90 100 110 120 130

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160170

180

8070

6050

4030

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0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

90 100 110 120 130

140150

160170

180

8070

6050

4030

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0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

E

T

90 100 110 120 130

140150

160170

180

8070

6050

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2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

B

AO O

a b

D

N

OP

O

Mc d

GU

Y

LA

F

I

U

R

90100110120130

140

150

160

170

180

8070

6050

4030

2010

0

9080706050

4030

2010

0

100110

120

130

140

150160

170180

H

BK

e f

g h

Ex 2

A

B

O

P

Q

a b

D


41CHAPTER 2 ANGLES

N

M

A

Y

XP

S

Z

X

Y

T

c d

e

f gM

N

L

i

k

l

h

G

D

A

M

G

E

C

A

B

Z

Q

FD

P

B

F

j



3 Estimate the size of each of these angles. Name each angle and use a protractor to measure the angles accurately.

4 The diagram shows Daniel shooting for goal in a game of football. His shooting angle is shown on the diagram. Estimate the size of this angle. Select A, B, C or D. A 60° B 120°C 150° D 240°

5 Measure the angles marked with and on each of these diagrams.

Ex 3

a b

c

d

C

BA

N

ML

X

Z

Y

G

K

H

x

a b

x

x


43CHAPTER 2 ANGLES

e f

x

x

c d

x

x

Just for the record

Why 360 degrees?Why are there 90° in a right angle and 360° in a revolution? Why do we use such strange numbers instead of more conventional numbers like 10 and 100?

The reason is that, in 2000 BC, the ancient Babylonians used a base 60 system of numbers. They used a base 60 number system because:• 60 is a rounder, more convenient number which has more factors than 10. You can

divide 60 by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.• 6 × 60 = 360, which was the Babylonian approximation of the number of days in a

year. They defined a revolution as being 360° so that, each day, the Earth would travel 1° around the Sun. A right angle, being a quarter-revolution, thus became 360° ÷ 4 = 90°.

Some people who prefer a base 10 system of measurement use grads instead of degrees to measure angles. With this system, a right angle is 100 grads and a revolution is 400 grads.

Find out more information about grads, including the exact relationship between degrees and grads.



2-03 Drawing anglesYou can also use your protractor to draw angles.

1 Accurately draw these angles, using your protractor.a 35° b 115° c 150° d 40°e 15° f 170° g 117° h 200°

2 Use your protractor to accurately draw and label these angles.a ∠DRE = 65° b ∠BGH = 145° c ∠GRT = 32°d ∠ABC = 45° e ∠SAQ = 110° f ∠NMH = 265°g ∠KLY = 28° h ∠LMN = 180° i ∠LKY = 90°

Exercise 2-03

Example 4

Use a protractor to draw angle KPM which measures 76°.

Solution• Draw a line with endpoints P and M.• Line up the base line of the protractor over PM. Place the centre mark over P. Follow

the inside scale around on the protractor, from 0° to 76°. Mark this point.

• Draw a line from P through this mark. Label the end of this line K.You have now drawn angle KPM, measuring 76°.

MP

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

PM

choose scale with 0° near M

mark 76°

P M

K

line ruledfrom P throughmark at 76°

Ex 4


45CHAPTER 2 ANGLES

2-04 Classifying anglesAngles may be classified according to their size as shown below.

1 Draw two different examples of:a an acute angle b an obtuse angle c a right angled a reflex angle e a straight angle f a revolution

2 Classify each of the following angles.a 37° b 107° c 252°d 195° e 79° f 180°g 163° h 179° i 360°j 5° k 345° l 91°m 14° n 299° o 90°p 205° q 126° r 44°

Angle Type Description

acute less than 90°

right 90° (quarter turn)Note that a right angle is marked with a box symbol.

obtuse greater than 90° but less than 180°

straight 180° (half turn)

reflex greater than 180° but less than 360°

revolution 360° (complete turn)

Exercise 2-04

Worksheet2-07

Angle cards

Skillsheet2-01

Types of angles



3 List the following angles from smallest to largest.

4 Decide whether each of these angles is acute, obtuse or reflex.a b

c d

5 Select A, B, C or D. Angles m° and n° are respectively:a obtuse and reflexb obtuse and a revolutionc acute and a revolutiond acute and reflex

a

b

c

d

e

f g

h

n° m°


47CHAPTER 2 ANGLES

2-05 Angle relationshipsIn the previous exercise, we described angles according to their sizes. Angles can also be described by how they relate to each other. In the following exercise we will discover some of these relationships.

1 Copy and complete the information below each of these diagrams. Use your protractor to measure the angles.a b

∠ABD = ∠XYZ = ∠CBD = ∠XZY = ∠ABD + ∠CBD = ∠XYZ + ∠XZY = (The angles you measured are called complementary angles. They complement each other to form 90°.)

2 Look up ‘complement’ in a dictionary. Write one non-mathematical meaning you find.

3 What is the complement of:a 30°? b 70°? c 25°? d 38°? e 89°? f 57°?g 42°? h 66°? i 11°? j 74°? k 1°? l 12°?

4 Copy and complete the information below each of these diagrams. Use your protractor to measure the angles.a b

∠ABD = ∠PQR = ∠CBD = ∠SRQ = ∠ABD + ∠CBD = ∠PQR + ∠SRQ = (These pairs of angles are said to be supplementary. They supplement each other, together forming 180°.)

Exercise 2-05

Geometry2-02

Angle vocabulary

Geometry2-03

Revolutions and straight angles

A

B C Z

YXD

Complementary angles add to 90°. !

D

AB

CP

S

Q

R



5 Look up ‘supplement’ in your dictionary. Write a non-mathematical meaning for it.

6 What is the supplement of:a 18°? b 150°? c 35°? d 125°? e 62°? f 87°?g 111°? h 173°? i 54°? j 132°? k 8°? l 91°?

7 a How many degrees are there in a complete turn or revolution?b Copy and complete the statements below each of these diagrams.

i ii

∠ADB = ∠AEB = ∠ADC = ∠BEC = ∠BDC = ∠CED = ∠ADB + ∠ADC + ∠BDC = ∠DEA = (These angles all meet at a point.) ∠AEB + ∠BEC + ∠CED + ∠DEA =

8 Use Cabri Geometry or The Geometer’s Sketchpad to illustrate the meaning of as many angle words as you can.

9 Use the given information to find the size of the angle shown by the letter each time.

Supplementary angles add to 180°.!

AD

C

B

D

E

A B

C

Angles at a point (in a revolution) add to 360°.!

Geometry2-02

Angle vocabulary

m°160°

q°150°

170°70°

62°87°

x°120°

y°95°

25°102°

a°135°

116°

22°d°

71°55°

110°105°

w°

132° 123°

f°48°

ba c d

fe g h


49CHAPTER 2 ANGLES

10 Find the value of d . Select A, B, C or D.A 122 B 61C 142 D 81

2-06 Vertically opposite anglesWhen two lines cross, four angles are created.• Which of these angles are equal?• Can you prove it using supplementary angles?

ji k l

30°

220°

n° 152°k°

118°t° 47°

15°

h°

303°

38°

160°d° d°

a°b°

c°d°

Example 5

∠WKZ is vertically opposite and equal to ∠XKY.What angle is vertically opposite ∠ZKY?

Solution∠WKX is vertically opposite ∠ZKY.Note: Angles that are equal in size are marked ondiagrams with the same type of arc or symbol.

W

Z Y

X

K

Vertically opposite angles are equal. !

Example 6

Find the size of the angles shown by the letters in this diagram.

Solutionk = 130 (vertically opposite angles)m = 50 (vertically opposite angles)

130°50°

k°m°



1 What angle is vertically opposite to:a the angle marked a°? b the angle marked w°? c the angle marked c°?

d the angle marked h°? e the angle marked k°? f the angle marked m°?

2 Without measuring, find the size of the angle shown by the letter each time.

Exercise 2-06

Ex 5

b°

a°

d°

c°

u°w°

v°u°

d°a°

c°b°

f °e°

h°g° k°

d°

i°h°

p°n°

m°l°

Ex 6

70°

a°

110°

m° 85°

k°

m°90°

135° x°

25° f °

a b c

d e f

w° 133°

29°

n°

62°q°

163°t°

h°g°

160°

20°r°

s°q°90°

g h i

j k l


51CHAPTER 2 ANGLES

2-07 Angle geometry

Types of angles Meaning Diagram

Adjacent angles Angles that share a common arm and a common vertex.(∠ABD and ∠DBC are adjacent angles.)

Complementary angles Two angles that add to 90°.(a + b = 90)

Supplementary angles Two angles that add to 180°.(m + n = 180)

Vertically opposite angles

Formed when two straight lines cross. Vertically opposite angles are equal.(a = c, b = d)

Angles at a point Form a revolution and add to 360°.(a + b + c = 360)

B D

C

A

x

a°b°

m° n°

c°b°

a°d°

a°c°b°

Example 7

Calculate the size of the angle shown by the letters in these diagrams.a b

Solutiona x + 130 = 180 (angles in a straight line) b y + 60 + 90 = 360 (angles at a point)

x = 180 − 130 y = 360 − 60 − 90 = 50 = 50

x°130°

60°

y°



1 a If ∠TAF = 42°, what is the size of its complementary angle?b If ∠ZAB = 127°, what is the size of its supplementary angle?

2 Refer to the diagram shown on the right.a Which angle is vertically opposite to ∠NDP?b Which angle is equal to ∠MDQ?c Name two straight angles in the diagram.d Name two different pairs of supplementary angles

in the diagram.

3 Which of the following is an angle adjacent to ∠AXB? Select A, B, C or D.A ∠BXC B ∠DXEC ∠DXC D ∠CXE

4 Refer to the diagram shown on the right.a Name a pair of adjacent angles.b Name a pair of complementary angles.c How do you know that the angles you named are

complementary?

5 Calculate the size of the angle shown by the letter. State which type of angles you used.

Exercise 2-07

D

P

Q

N

M

AB

C

DE

X

Q P

R

S

67°23°

Ex 7

a° 120°70°

a°

100°

y°

cba

45°

m°

150°p°

19°

m°41°

x°

f°

15°

100°100° 40°

a°

fed

ihg


53CHAPTER 2 ANGLES

2-08 Lines in geometryA line is named using two points on the line. For example, this is the line AB.

When two lines cross, we say that they intersect.Two lines intersect at a point.For example, in this diagram, line DE intersects withline FG at point H.

Perpendicular linesLines that intersect at right angles are called perpendicular lines.For example, in this diagram, PQ is perpendicular to XY.This is written as ‘PQ � XY’, where the � symbolstands for ‘is perpendicular to’.

a°

32°b°

82°

135°

y°

d°

d°

onm

lkj

170°h°

a°a°

t° t°

f°e°e° 112° 48°

l°k°

j°

y°x°

20°

rqp

118°

75°y°x°

155°85°p°

p°

s t

e°

u

e°e°

BA

D

E

G

F

H

X

P

Q

Y



Parallel linesLines that point in the same direction and never intersect are called parallel lines. Parallel lines are marked with identical arrowheads and are always the same distance apart. For example, in this diagram, MN is parallel to RS.This is written as ‘MN II RS’, where the symbol II standsfor ‘is parallel to’.

TransversalA line that crosses two or more other lines is called a transversal. Transverse means ‘crossing’.

1 Name the six different lines in this diagram.

2 In this diagram, name two lines that:a are perpendicularb are parallelc intersect.

3 Rewrite your answers to Question 2 parts a and b using the symbols for ‘is perpendicular to’ and ‘is parallel to’.

4 Draw and label correctly:a line FG b line AB intersecting line CD at point Ec line PQ parallel to line YZ d line JK perpendicular to line LM.

5 In the diagram on the right, name two angles that are:a adjacentb vertically oppositec supplementary.

Exercise 2-08

M

R

S

N

indicates these linesare parallel

transversal

transversal

A B

CD

G

F

E D

C

B

AH

A

B

C

D

E


55CHAPTER 2 ANGLES

6 Which line is parallel to line FG? Select A, B, C or D.A CD B LM C AB D PQ

7 In the diagram on the right, Frank Road is perpendicular to which of the following? Select A, B, C or D.A Emilia ParadeB Rosalia RoadC Daniel StreetD Christina Road

8 State all the examples of parallel lines, perpendicular lines and intersecting lines you can find in the photograph below.

D

C

F

G

A

B

L

M

P

Q

Christina Road

Emilia Parade D

anie

l S

tree

t

Frank Road

Ros

alia

R

oad



2-09 Alternate angles on parallel linesAlternate angles are between two lines and on opposite sides of a transversal crossing the lines. Alternate angles on parallel lines are equal.On this diagram the alternate angles are marked with dots. ‘Alternate’ means ‘going back and forth’.Draw a pair of parallel lines and mark the alternateangles as shown. Draw in the broken line and cutalong it.

Rotate the two alternate angles and place them on top of each other. You should see they are the same.

Mental skills 2

Changing the orderHave you noticed that 4 + 7 = 7 + 4? Have you also noticed that 3 × 5 = 5 × 3? Numbers can be added or multiplied in any order. We can use this property to make our calculations simpler.

1 Examine these examples.

a 19 + 5 + 5 + 1 = (19 + 1) + (5 + 5)= 20 + 10= 30

b 13 + 8 + 20 + 27 + 80 = (13 + 27) + (20 + 80) + 8= 40 + 100 + 8= 148

c 2 × 36 × 5 = (2 × 5) × 36= 10 × 36= 360

d 25 × 11 × 4 × 7 = (25 × 4) × (11 × 7)= 100 × 77= 7700

2 Now simplify these examples.a 45 + 16 + 45 + 4 + 7 b 38 + 600 + 50 + 12 + 40c 18 + 91 + 9 + 20 d 75 + 33 + 7 + 25e 24 + 16 + 80 + 44 + 10 f 56 + 5 + 20 + 15 + 4g 100 + 36 + 200 + 10 + 90 h 54 + 27 + 9 + 16 + 3

Maths without calculators

transversal

Alternate angles on parallel lines are equal.!


57CHAPTER 2 ANGLES

The marked pairs of angles are alternate. Measure them and check that alternate angles are equal. (Remember: Equal angles are marked by the same symbol.)

1 Which angle is alternate to the marked angle each time?

2 Copy each of these diagrams and mark in the alternate angle to the one shown.a b c

3 Which angle is alternate to the marked angle? Select A, B, C or D.A d° B e°C b° D a°

Exercise 2-09

Alternate angles on parallel lines

x

x

a°

b° c°g° f °

e°d°

f ° g°e°

c°b°

d°a°

g°f °

e°

c°

b°a°

d°

a b c

c°

d°e°

f°g°

a°

b°



4 Copy these diagrams and mark in a pair of alternate angles on each one.a b c

5 Write the size of each angle shown by a letter.

2-10 Corresponding angles on parallel linesCorresponding angles are on the same side of the transversal and are both either above or below the other two lines. ‘Corresponding’ means ‘matching’.

a b c

d e f

g h i

m°

110°50°

a° n°

80°

122°

b°h°

20°

n°m°

p°50°

b°a°

40°

130°

a°

b°

b°a°

c° 44°

Corresponding angles on parallel lines are equal.!Corresponding angles on parallel lines

x

x


59CHAPTER 2 ANGLES

We can prove that corresponding angles on parallel linesare equal.

a = b They are vertically opposite angles.b = c They are alternate angles.

So a = c.

1 Which angle is corresponding to the marked angle each time?a b c

2 Copy each diagram and mark the corresponding angle to the one shown.a b c

3 Copy each of these diagrams and mark in a pair of corresponding angles on each one.a b c

4 Which angle is corresponding to the marked angle? Select A, B, C or D.

5 Write the size of each angle shown by a letter.

Exercise 2-10

a°b°

c°

c°b° a°

g°f °e°

d°

f ° e°

g°a° b°

c°

a°b°

d°

d°c°

g°

e°f °

a°b°

c°

A D

BC

a°y°

m°120°

28°

63°

a b c



6 Without measuring, find the size of theother seven angles in this diagram.

2-11 Co-interior angles on parallel linesCo-interior angles are on the same side of the transversal but between the other two lines. ‘Co-interior’ means ‘together inside’.

Measure the following pairs of angles and see if they really are supplementary.

g h im°

110° 105°

n°

c°y° 140°

y° a°

d°

x°

d e f

c°

t°

a°b°

108°74°

60°

50°

105°

d°

f ° g°e°

c°b°a°

Co-interior angles on parallel lines are supplementary. They add to 180°.!Co-interior angles on parallel lines

x

x


61CHAPTER 2 ANGLES

We can also show that co-interior angles on parallel lines add to 180° using the following method.

a + b = 180 They are angles on a straight line.a = c They are alternate angles.

So c + b = 180

1 Which angle is co-interior with the marked angle each time?a b c

Exercise 2-11

a° b°

c°

Example 8

1 Find the size of the angle marked a° in this diagram.

Solutiona + 80 = 180 (co-interior angles on parallel lines)

a = 180 − 80= 100

2 Find the size of the angle marked m° in this diagram.

Solutionm + 55 = 180 (co-interior angles on parallel lines)

m = 180 − 55= 125

a°

80°

55°

m°

a°d°b°

c°g°

f°e°

a°b°

c°

g°d°

e°f° c°a° b°

d°

e° f °g°



2 Copy each of these diagrams and mark the angle that is co-interior with the marked angle.a b c

3 Copy each of these diagrams and mark pairs of co-interior angles.

4 Which angle is co-interior with the marked angle? Select A, B, C or D.A d° B b°C e° C g°

5 Without the use of instruments, find the size of the angles shown by letters.

a b c

a°

b°c°

d° e°f°g°

Ex 8

d e

a°

50° m°

90° 75°

b°

112°d°

68°

m°98° a°

b°

f

f ° g°130° k°

j°

55°

c°b°

a°51°

g h i

a b c


63CHAPTER 2 ANGLES

2-12 Angles on parallel linesBelow is a summary of all we have found out about the angles in parallel lines.

1 In the diagram on the right, name the angle that is:a corresponding to ∠VWAb alternate to ∠QXWc co-interior with ∠PWXd supplementary with ∠AWXe alternate to ∠SXVf corresponding to ∠ZXS.

Exercise 2-12

Just for the record

The Leaning Tower of PisaThe Leaning Tower of Pisa, Italy, began leaning shortly after its construction commenced in 1173. In 1350, it was leaning at 2.5°, or 4 m, from the vertical. By 1990, its lean had grown to 5.5°, or 4.5 m, and was increasing at 1.2 mm per year. Architects estimated that the tower would have toppled over by the year 2020 so it was closed for 12 years to allow $25 million worth of engineering work to take place. When it reopened in 2001, its lean had been pushed back to 5° or 4.1 m, and it is now guaranteed to stay up for at least another 300 years.

1 Draw a scale diagram of the Leaning Tower of Pisa given that its top is 55 m above the ground.

2 Research how engineers prevented the tower from leaning further. Use the library or the Internet to conduct your research.

4.1 m

55 m

Worksheet2-08

Matching angle

Worksheet2-09

Find the missing angle

When parallel lines are crossed by a transversal:• alternate angles are equal• corresponding angles are equal• co-interior angles are supplementary (add to 180°).

!

Q

AX

Z

W

VP

S



2 Without the use of instruments, find the size of each angle shown by a letter.

3 Without measuring, find the size of all angles labelled with letters in these diagrams.

p°

115°71°

t°

105°

k°

a b c

120°

m° 70° 132°n°

a°

d e f

g h i

x°

28° 72°s°

k°

85°

j k l

p°

93°

81°y° 150° w°

m n o128°

d°j°

66°q°

109°

a b c

b°67°

a°

133°

j°

k°

l°

m°n°p°

52°


65CHAPTER 2 ANGLES

4 Which of the following does y equal? Select A, B, C or D.A 28 B 47C 77 D 152

j k l

g h i

d e f

m n o

y°

z°

42° 95°

l°

m°b° c°

45° 30°

q°p°

75°

85°

m°

k° p°

w° 63°

k°

130°

x°y°

55°62°

a°

72°

b°

n° p°

m°83°

132°g°

27°

a°b°

c°

28°

105°

y°

Using technology

Constructing angles using geometry softwareNote: The activities have been demonstrated using The Geometer’s Sketchpad.

1 a Construct each of the following angles using the straightedge tool. i acute ii right iii obtuse iv reflex b Now label each of the four angles you have drawn using the text tool.

Starting The Geometer’s Sketchpad

Skillsheet2-02

Starting Cabri Geometry

Skillsheet2-03



2-13 Proving lines are parallelWe can use what we know about angles and parallel lines to show that two lines are parallel.Two lines are parallel if:• alternate angles are equal, or• corresponding angles are equal, or• co-interior angles are supplementary (add up to 180°).

c Measure the size of each angle you have drawn, correct to the nearest degree.ExampleThe diagram on the right shows acute angle ∠ABC = 52°.

2 a Start a new sketch and accurately construct separate angles of the following sizes.i 72° ii 310° iii 165° iv 98° v 236° vi 90°b Using the text tool, label each angle according to its classification, i.e. ‘acute’,

‘reflex’, etc.Example

3 For each of the following, sketch three different angles that can be classified as:a acute b reflex c obtuse

4 Using geometry software, construct the following. a b

c A pair of: i complementary angles

ii supplementary anglesiii corresponding angles of 28°, on parallel lines iv alternate angles of 65°, on parallel linesv co-interior angles on parallel lines, as shown on the

right, where one of the supplementary angles is 130°

m∠ABC = 52°A

B

C

Acute angleA

B

C

m∠ABC = 52°

B

23°

102°

DC

A

27°

27°

130°


67CHAPTER 2 ANGLES

1 In each diagram below, name a pair of alternate angles and use them to decide if ABis parallel to CD.a b c

2 In each diagram below, name a pair of corresponding angles and use them to decideif AB is parallel to CD.a b c

Exercise 2-13

Example 9

1 Is AB parallel to CD in the diagram on the right?

Solution∠AXY is alternate to ∠DYX.∠AXY = ∠DYX = 75°∴ AB II CD since a pair of alternate angles are equal.(∴ means ‘therefore’)

2 Is MN parallel to PQ in the diagram on the right?

Solution∠MXY is co-interior with ∠PYX.∠MXY + ∠PYX = 110° + 80° = 190°

≠ 180°Since co-interior angles do not add to 180°, MN is not parallel to PQ.

75°

Y

75°

X B

DC

A

80°

110°M

PY Q

NX

Ex 9

64°

64°

AB

DC

100°

AC

DB

100°

A C

DB

32°35°

E

FG H E F

C

A

B

D79°

82°

A C

B

D

63° 63°C

D

A117°

110°

B

G

E

F

EF

GE

F

G



3 In each diagram below, name a pair of co-interior angles and use them to decide if AB is parallel to CD.a b c

4 For each diagram below, determine if line PQ is parallel to line MN. Explain your reason.

5 What reason can be used to prove GC II HE? Select A, B, C or D.A ∠ABC = ∠HDF (alternate angles)B ∠CBD = ∠BDH (alternate angles)C ∠ADE = 91° (corresponding angles)D ∠BDE = ∠FDH (vertically opposite angles)

A

B

D

C

A

B

C

D

120°

60°

100°

85°

A C

B D

90° 90°

E

F

E

F

E F

P

AM

C

D

Q

NB

99°

81°

N Q

YX

PM

E G I K

M

P

F H J L

Q

N

87°

87°78°

102°

a b

c

78°78°

P

NM

X

Q105°

f

K

D

M

PC Q

L

A

65°

120°d

P

AM

E DQ

NB80°95°

e

N

65°

B

80°

C

85°85°

F

75° 75°

H

E

G

C

AB D

F89° 91°

91°91° 89°


69CHAPTER 2 ANGLES

Power plus

1 a Draw any triangle with angles of 70° and 55°.b Draw any parallelogram with angles of 50° and 130°.c Draw any four-sided shape with angles of 45°, 160°, 70° and 85°.

2 a Draw any triangle and measure the sizes of all three angles.b What is the sum of the angles in any triangle?c Draw any quadrilateral and measure the sizes of all four angles.d What is the sum of the angles in any quadrilateral?

3 How many degrees does the Earth spin on its axis in:a one day? b one hour? c 8 hours? d 10 minutes?

4 Work out which direction (left, right, front or behind) you would be facing after making each of these series of turns.a Right 80°, right 240°, left 90°, right 40°b Left 140°, left 140°, left 140°, right 60°c Right 200°, left 70°, right 40°, right 10°d Left 240°, right 190°, right 100°, left 50°

5 Find the size of each angle shown with a letter. Give reasons for your answers.

a b c

51°

m° 62°

x°

y°

125°

a°

82°

40°

m°

y°

35°

250° c°

80°

145°

k°

50°

x°35°

120°m°

45° 20°

95°

k°

d e f

g h i



Chapter 2 reviewChapter 2 review

Language of mathsacute adjacent alternate arc armco-interior complementary corresponding degree intersectingline obtuse revolution right angle scalestraight angle supplementary transversal vertex vertically opposite

1 How many degrees are there in a half turn (straight angle)?

2 Find the meaning of ‘acute’ when referring to a disease, for example acute appendicitis.

3 What is the difference between ‘complementary’ and ‘complimentary’?

4 When something happens that dramatically changes the way we think or do things, it is called ‘revolutionary’. Why do you think this is so?

5 Write the mathematical symbol for: a parallel b perpendicular.

6 Mr Transversal visits his parents on alternate days. What does this mean? How is it similar to the mathematical meaning of ‘alternate’?

Topic overview• Give three examples of where angles are used.• How confident do you feel in working with angles?• Is there anything you did not understand? Ask a friend or your teacher for help.• The diagram below provides a summary of this chapter. Copy and complete it, using

colour, pictures and key words to make your overview easy to read and remember. Check your completed overview with your teacher.

Worksheet2-08

Matching angles

Worksheet2-10

Angles crossword

90100

110120

130140

150160

170180

8070605040

3020

100

9080

7060

5040

3020

100

100110120130

140150

160

170

180

Protractor

Co-interior

ANGLES

Acute

RevolutionVertically opposite

Transversal

x

Perpendicular

BD

AC

E

H

F

G

CorrespondingAlternate

Parallel

LINES


71CHAPTER 2 ANGLES

Chapter revision1 Draw labelled diagrams of each of these angles.

a ∠BKT b ∠FPR c angle MZQ

2 Use a protractor to measure each angle you drew in Question 1. Name the smallest angle and the largest angle.

3 Use a protractor to draw these angles.a ∠JUG = 84° b ∠QRA = 117° c ∠POT = 41°d ∠DGE = 150° e ∠SAR = 96° f ∠XDW = 210°g ∠MNB = 195° h ∠PLO = 270° I ∠AMP = 300°

4 Write the name of each of these angles. Then label each one as acute, obtuse, right, reflex or straight.

5 a What is the complement of each of these angles?i 35° ii 78° iii 4°

b What is the supplement of each of these angles?i 45° ii 100° iii 178°

c Without measuring, find the size of the angle shown by each letter.i ii iii

Exercise 2-01

Exercise 2-02

Exercise 2-03

Exercise 2-04

W I

H

A R

D

GL

UV

RP

P

NE

S

M

M

V

Z M Q

P

A

T

X

Y

a b c

d ef

g h i

Exercise 2-05

70°25°

m°70°a°

35°

y°

Topic test 2



6 Find the size of each angle shown by a letter. Do not use a protractor to measure the angle.a b c

7 Without measuring, find the size of each angle shown by a letter.

8 In this diagram, name two lines that:a are parallelb are perpendicularc intersect.

9 a Copy each diagram and mark in the alternate angle to the one shown.i ii

Exercise 2-06

m°

a°

100°

44°a°

b°95°

Exercise 2-08

a b cm°

28°k°

47° x°y° 122°

d e f140°

75°p°

x°48°

110°f°

g h i

82°t°

105°25°p°

q°r°

x°x°

x°

Exercise 2-08

A

B

C

D

E

F

G

H

Exercise 2-09


73CHAPTER 2 ANGLES

b Without the use of instruments, find the size of each angle shown by a letter.i ii iii

10 a Copy each diagram and mark in the corresponding angle to the one shown.i ii

b Without the use of instruments, find the size of each angle shown by a letter.i ii iii

11 Copy each diagram and mark in the co-interior angle to the one shown.i ii

b Find the size of the angle shown by each letter.i ii iii

126°

a°b°

b°

120°

a°

38°

Exercise 2-10

p° n°

150°

117°p°

112°

a°

Exercise 2-11

72°

k°

x°y°

112° 82°

x°



12 Label the marked pairs of angles as alternate, co-interior or corresponding.

13 Find the size of each angle shown with a letter.

Exercise 2-12

a b c

d e f

x

x

x

x

Exercise 2-13

y°

g h i

x°

37°

z°62° p°

m°

112°

t°

d° a°

a b ca°

115° m°

35°

k°

65°

130°q°

62°

x°

d°

125°

d e f


75CHAPTER 2 ANGLES

14 Find the size of each angle shown with a letter.

15 Draw a neat diagram to illustrate each of the following.a an acute angle b supplementary anglesc a straight angle d vertically opposite anglese alternate angles f an obtuse angleg corresponding angles h a reflex anglei complementary angles j co-interior angles

16 In each diagram below, is AB parallel to CD? Give a reason for your answer each time.

Exercise 2-13

x°130°

y°

x°64°

m° 70°

a°z°

c°

38°

57°x° y°

145°

z°a°

38°

c°

a b c

d e

Exercise 2-13

Exercise 2-13

a b c

A

C D

B45°

135°

110°

112°

B

D

C

A

A

C

D

B

74°

74°

E

F

G

H

E

F

G

H

E

F

G

H


Date post:	27-Nov-2014
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Chapter 2

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