chapter02 angles

Geometry

Angles

2

Whether you are playing snooker or soccer, building houses or bridges, or designing logos or computer graphics, you need to know all about angles. Look around you for a momentyou will see angles everywhere.

name and classify different types of angles and lines revise protractor skills for measuring and drawing angles recognise and use notations for angles, parallel lines and perpendicular

lines discover different angle facts and use them to solve geometry problems identify and measure pairs of alternate, corresponding and co-interior

angles for two lines cut by a transversal discover and use the properties of alternate, corresponding and

co-interior angles for two parallel lines cut by a transversal.

vertex The corner or point of an angle. protractor An instrument for measuring the size of an angle. complementary angles Two angles that add to 90. supplementary angles Two angles that add to 180. parallel lines Lines that point in the same direction and do not intersect. perpendicular lines Lines that intersect at right angles. transversal A line that cuts across two or more other lines.

The Earth takes one year to make a complete revolution around the Sun. How long does it take to travel an angle of 30?

In this chapter you will:

Wordbank

Think!

ANGLES 31

CHAPTER 2

32

NEW CENTURY MATHS 7

1 How many degrees are there in:a a quarter turn (right angle)?b a half turn (straight angle)?c a three-quarter turn?

2 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?

3 In the diagram in Question 2, nd one pair of labelled lines which have a 19 angle between them.

4 In the diagram in Question 2, nd two pairs of labelled lines which have a 90 angle between them.

5 In the diagram in Question 2, nd the pairs of labelled lines which have the following angles between them:a 7 b 8 c 13 d 28e 50 f 89 g 95 h 114

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

A

B

C

D

E

F

GHI

J

Start up

Worksheet 2-01

Brainstarters 2

ANGLES

33

CHAPTER 2

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

7 Decide whether each of these angles is:i acute ii obtuse iii reexa b c

d e f

g h i

j k l

m n o

Skillsheet 2-03

Types of angles

G

P

H

vertex arm

arc

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

{

G

G G

34 NEW CENTURY MATHS 7

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.

X

Y

Z

P

Q SR

a b

1 Name each of these angles in two different ways:

2 Name the angle marked with in each of these diagrams:

a b cP

Q K O

R

C G

V E

A

G

T PQ

D R C

D

d e f

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

Exercise 2-01Example 1

ANGLES 35 CHAPTER 2

Comparing angle sizeCan you tell which of these is larger?

Angle A Angle BIf you can tell these apart, your eyes can detect a difference of two degrees. Angle B is 2 larger than Angle A.

3 Draw each of these angles, labelling them correctly:a POT b TAF c AFE d H

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

b What type of angle is NCY?

5 Name the angles marked and in each of the following diagrams:

6 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 5.

N

A

Y

DC

xa 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


1 List the angles in each of the given sets in order, from smallest to largest, without usinga protractor.a These angles vary by 5.

b These angles vary by 3.

c These angles vary by 2.

d These angles vary by 1.

e These angles vary by .

f These 10 (i to x) angles will provide a harder challenge:

i ii iii

i ii iii

i ii iii

i ii iii

12---i ii iii

i ii iii

iv v vi

Exercise 2-02SkillBuilder 24-01

Measuring angles

ANGLES 37 CHAPTER 2

The protractorMeasuring anglesA protractor is an instrument used to measure angles.

vii viii ix x

2 List the following angles in order, from smallest to largest, without using a protractor.

a cbf

e

d

Worksheet2-03

Make your own protractor

Worksheet2-02

360 scale

Worksheet2-04

A page of protractorsCentre mark

90 100 110 120 130

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130

140

150

160

170

180 Base line

Outside scale

Inside scale


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

807060

5040

3020

100

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13014

015

016

017

018

0 B

A

O

MQ

P

90 100 110 120 130

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50

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140

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180

MQ

P

ANGLES 39 CHAPTER 2

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

Measure the reex angle GHK.

Solution Actually measure the obtuse

angle GHK rst (140). Subtract 140 from 360.

360 140 = 220Reex GHK = 220

X

E T

90 100 110 120 130

140150

160170

180

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50

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X

E T

Example 3

GH

K

90

10011012

0130

140

150

160

170

180

8070

6050

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2010

090

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100110

120

130

140150

160170

180 GH

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.

90 100 110 120 130 140150

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E

T

90 100 110 120 130 140150

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0

B

AO O

a b

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180D

N

O P O

Mc d

90 100 110 120 130 140150

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180G

U

Y 90 100 110 120 130 140150

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A F

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180I

U

R

90

10011012

0130

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80 7060

50

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90

807060

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100110

120

130

140150

160170

180

H

BK

e f

g h

A

B

O

P

Q

a b

D

Exercise 2-03

Worksheet 2-05

A page of angles

Example 2

SkillBuilders 24-02 & 24-03

Measuring angles

ANGLES 41 CHAPTER 2

N

M

A

Y

XP

S

Z

X

Y

T

c d

e

f g M

NL

i

j

k l

h

G

D

A

M

G

E

C

A

B

Z

Q

F D

P

BF


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

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

a b

c d

C

BA

N

ML

X

Z

Y

G

K

H

x

a b

c d

x

x

x

x

Example 3

ANGLES 43 CHAPTER 2

e f

x

x

Skillsheet 2-01

Starting GeometersSketchpad

Skillsheet 2-02

Starting Cabri Geometry

Estimating angles1 Using a drawing package and estimation skills, draw and label each of the following

angles.a ABC = 45 b DEF = 30 c GHI = 60d JKL = 90 e MNO = 120 f PQR = 155

2 Print out the six angles that you have drawn. Measure the angles with a protractor.3 What was the error each time between the angle you drew and the actual angle

requested?

Using technology Geometry

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.

Just for the record


Drawing anglesYou can also use your protractor to draw angles.

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

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180

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50

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90 80 70 6050

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0

100110120

130

140

150

160

170

180

PM

choose scale with 0 near M

mark 76

P M

K

line ruledfrom P throughmark at 76

1 Accurately draw these angles, using your protractor:a 35 b 115 c 150 d 40e 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 = 32d ABC = 45 e SAQ = 110 f NMH = 265g KLY = 28 h LMN = 180 i LKY = 90

3 You can use a geometry program, such as Cabri Geometry or Geometers Sketchpad, to draw accurate angles. The accompanying activity shows you how to make a protractor.


Geometry 2-01

Making a protractor

ANGLES 45 CHAPTER 2

Angle geometryClassifying anglesAngles may be classied according to their size:

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)

reex greater than 180 but less than 360

revolution 360 (complete turn)

Worksheet2-06

Angle cards

Skillsheet 2-03

Types of angles

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

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

Exercise 2-05


3 Decide whether each of these angles is acute, obtuse or reex.

4 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.)

5 Look up complement in a dictionary. Write one non-mathematical meaning you nd.6 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?

A

B C Z

YXD

Complementary angles add to 90.

a b

c d

SkillBuilder 24-05

Complementaryangles

ANGLES 47 CHAPTER 2

7 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.)

8 Look up supplement in your dictionary. Write a non-mathematical meaning for it.9 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?

10 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 =

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

11 Use Cabri Geometry or Geometers Sketchpad to illustrate the meaning of as many angle words as you can.

D

A B CP

S

Q

R

Supplementary angles add to 180.

AD

C

B

D

E

A B

C

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

SkillBuilder 24-04

Supplementary angles

Geometry 2-02Angle

vocabulary


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

12 Use the given information to nd the size of the angle shown by the letter each time.

13 You can use a geometry program, such as Cabri Geometry or Geometers Sketchpad, to demonstrate the rules for supplementary angles and angles at a point. Use this link to go to the accompanying activity.

m

160

q150

17070

6287

x

120y

95

25102

a

135

116

22d

7155

110105w

132 123

f48

ba c d

fe g h

ji k l

30

220

n 152k

118t 47

15

h

303

Geometry 2-03

Revolutions and straight angles

a

bc

d

Example 5

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

SolutionWKX is vertically opposite ZKY.Note: Angles that are equal in size are marked on diagrams with the same type of arc or symbol.

W

Z Y

X

K

ANGLES 49 CHAPTER 2

Vertically opposite angles are equal.

Example 6

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

Solutionk = 130 m = 50(since vertically opposite angles are equal).

13050

km

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, nd the size of the angle shown by the letter each time.

b

a

dc

uw

vu

da

c

b

f eh g kd

ih

pn

ml

70

a

110m 85

k

m

90 135 x

25 f

a b c

d e f


Example 6


Angle facts

Remembering these facts will help you complete the next exercise.

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)

w 133

29

n

62q

163t

hg

16020

r

sq90

g h i

j k l

B D

C

A

x

ab

m n

cb

ad

a

cb

ANGLES 51 CHAPTER 2

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 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 Refer to the diagram 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?

4 Calculate the size of the angle shown by the letter each time.

D

P

Q

N

M

Q P

R

S

6723

a 120 70

a

100

y

45

m

150p

19

m 41

xf

15

100100 40

a

cba

fed

ihg

Exercise 2-07

SkillBuilder 24-07

Combination figures


Naming linesA 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 and form four angles between them.For example, in this diagram, line DE intersects with line FG at point H. One of the angles formed is FHE.

a

32b

82135

y

dd

fee 112

48lk

j

11875yx 15585

p

yx

20

p

rq

s t

p

onm

lkj

170h

a

a

t t

e

u

e

e

BA

D

E

G

F

H

ANGLES 53 CHAPTER 2

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 symbol stands for is perpendicular to.

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 stands for is parallel to.

X

P

Q

Y

M

R

S

N

indicates these linesare parallel

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 your diagram for Question 4b, name two angles that are:a adjacent b vertically opposite c supplementary.

A B

CD

G

F

E D

C

B

AH

Exercise 2-08 SkillBuilder 24-08Parallel lines


Angles and parallel linesA line that crosses two or more other lines is called a transversal. Transverse means across.

If a pair of parallel lines are crossed by a transversal, then special pairs of angles are formed: alternate angles, corresponding angles and co-interior angles. We shall identify these angles and discover their properties.

6 State all the examples of parallel lines, perpendicular lines and intersecting lines you can nd in this photograph.

transversaltransversal

transversal

ANGLES 55 CHAPTER 2

Alternate anglesAlternate angles are on opposite sides of the transversal but between the parallel lines. They are marked with red dots on the diagram. Alternate means going back and forth.Draw a pair of parallel lines and mark the alternate angles as shown. Draw in the broken line and cut along it.Rotate the two alternate angles and place them on top of each other. You should see they are the same.

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.)

Alternate angles on parallel lines are equal.

Pairs of alternate angles

x

x

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 each time.a b c

a

b cg f

ed

f ge

c

bd

a

gf e

c

ba

d

a b c

Exercise 2-09 SkillBuilder 24-09

Z in parallel lines


Corresponding anglesCorresponding angles are on the same side of the transversal and are both either above or below the parallel lines. Corresponding means matching.

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

4 Without the use of instruments, calculate the size of each angle shown by a letter:a b c

d e f

g h i

m

11050

a n

80

122

b h

20n

m

p50

ba

40

130a

bb

a

c 44

Corresponding angles on parallel lines are equal.

Pairs of corresponding angles

x

x

ANGLES 57 CHAPTER 2

We can prove that corresponding angles are equal using the following method:

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

So a = c.

a

b

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 each time.a b c

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

4 Calculate the size of each angle shown by a letter:

cb a

gf

ed

f eg

abc

ab

d

dc

g

e f

d e f

a

y

m

c

t

a b

120

28

10874

63

60

50

a b c

Exercise 2-10


Co-interior anglesCo-interior angles are on the same side of the transversal but between the parallel lines. Co-interior means together inside.

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

We can also show that co-interior angles 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

5 Without measuring, nd the size of the other seven angles in this diagram.

g h im

110105

n

cy140

y a

dx

105

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

Pairs of co-interior angles

x

x

a b

c

ANGLES 59 CHAPTER 2

Example 7

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

Solutiona + 80 = 180 Co-interior angles are supplementary.

a = 180 80So a = 100

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

m + 55 = 180 Co-interior angles are supplementary.m = 180 55

So m = 125

a

80

55

m

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

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

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

adb

c

gfe

ab

c

gd

ef ca

bd

ef g

a b c

Exercise 2-11


4 Without the use of instruments, calculate the size of the angles shown by letters:

d e

a

50 m

90 75

b

112d

68

m

98 ab

f

f g130 k

j55

c

b

a

51

g h i

a b cExample 7

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 metres above the ground.

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

Just for the record 4.1 m

55 m

ANGLES 61 CHAPTER 2

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

2-07Matching angles

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.

2 Without the use of instruments, nd the size of each angle shown by a letter:

d e f

g h i

j k l

Q

AX

Z

W

VP

S

p

11571

t

105

k

a b c

120m

70 132n

a

x

28 72s

k

85

p

93

81y 150 w

Exercise 2-12


m n o

3 Without measuring, nd the size of all angles labelled with letters in these diagrams:

4 You can use Geometers Sketchpad or Cabri Geometry to show that the rules for parallel lines and traversals are always true. The instructions for this can be found in the accompanying activity.

128

dj 66

q

109

j k l

g h i

d e f

m n o

a b c

b67

a

133j

kl

m

n

p

52

y

z

42 95

l

m

b c

45 30

qp

75

85

m

k p

w 63

k

130

x

y

5562

a

72

b

n p

m

83 132

g

27

ab

c

SkillBuilders 24-10 to 24-12Combination

figures

Geometry 2-04

Angles and parallel lines

ANGLES 63 CHAPTER 2

Finding parallel linesWe can use what we know about angles and parallel lines to show that two lines are parallel.

Example 8

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

SolutionAXY 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?

SolutionMXY is co-interior with PYX.MXY + PYX = 110 + 80 = 190

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

75Y

75X B

DC

A

80

110M

PY Q

NX

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

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

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

64

64

AB

DC

100

A C

DB

100A

C

DB

3235

E

FG H E F

C

A

B

D79

82A C

B

D

63 63 C

D

A117

110

B

G

E

FE F

GE

F

G

A

B

D

CA

B

CD

120

60

100

85

A C

B D

90 90

E

F

E

F

E F

Exercise 2-13


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

P

AM

C

D

Q

NB

99

81

N Q

YX

PM

E G IK

M

P

F H JL

Q

N

87

8778

102

a b

c

7878

P

NM

X

Q105

f

K

D

M

PC Q

L

A

65

120d

P

AM

E DQ

NB8095

e

N

65

B

80C

9585F

75 75

Example 8

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 60c Right 200, left 70, right 40, right 10 d Left 240, right 190, right 100, left 50

Power plus

ANGLES 65 CHAPTER 2

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

250c

80

145

k

50

x35

120m

45 2095

k

d e f

g h i

Language of mathsacute adjacent alternate arcarm co-interior complementary correspondingdegree intersecting line obtuserevolution right angle scale straight anglesupplementary transversal vertex vertically opposite

1 Name the two parts of an angle.2 Find out what the word acute means 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 perpendicular6 Mr Transversal visits his parents on alternate days. What does this mean? How is

it similar to the mathematical meaning of alternate?

Worksheet2-07

Matchingangles

Worksheet 2-08Angles

crossword


Topic overview Give three examples of where angles are used. Do you think this chapter is very useful to you? Why? How condent do you feel in working with angles? Is there anything you did not understand about angles? Ask a friend or your teacher for help. The diagram below provides a summary of this section of work. Copy it into your workbook

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.

90 100 110120

130140

150160

170180

807060504030

20

100

9080

7060

5040

3020

100

100110120130140

150

160

170

180

Protractor

Co-interior

ANGLES

Acute

Adjacent

Revolution

Vertically opposite

Supplementary

Complementary

Transversal

x

Perpendicular

B

D

A

C

E

H

F

G

CorrespondingAlternate Parallel

LINES

ANGLES 67 CHAPTER 2

1 Draw labelled diagrams 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 = 41d DGE = 150 e SAR = 96 f XDW = 210g 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, reex or straight.

5 Without measuring, nd the size of each angle shown by a letter:

Chapter 2 Review Topic testChapter 2

Ex 2-01

Ex 2-03

Ex 2-04

Ex 2-05

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

Ex 2-07

a b cm

28k

47 xy 122

d e f140

75p

x

48

110 f


6 Label the marked angles as alternate, co-interior or corresponding:

7 Find the size of each angle shown with a letter:

g h i

82t

105 25pqr

x

xx

Ex 2-12

a b c

d e f

x

x

x

x

Ex 2-12

y

g h i

x

37

z62 pm

112

t

d a

a b ca

115m

35

k

65

130q

62

x

d

125

d e f

ANGLES 69 CHAPTER 2

8 Without measuring, nd the size of each angle shown with a letter:

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

10 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 reex anglei complementary angles j co-interior angles

Ex 2-12

x130

y

x64

m 70

az

c

38

57x y

145z

a

38

c

a b c

d e

Ex 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

Student textImprint pageTable of contentsPrefaceHow to use this bookHow to use the CD-ROMAcknowledgementsSyllabus reference grid1 The history of numbersDifferent number systemsThe HinduArabic number systemPlace valueExpanded notationThe four operationsArithmagonsDividing by a two-digit numberOrder of operationsThe symbols of mathematicsTopic overviewChapter review

2 AnglesNaming anglesComparing angle sizeThe protractorDrawing anglesAngle geometryNaming linesAngles and parallel linesFinding parallel linesTopic overviewChapter review

3 Exploring numbersSpecial number patternsTests for divisibilityFactorsPrime and composite numbersPrime factorsIndex notationSquares, cubes and rootsTopic overviewChapter review

Mixed revision 14 SolidsNaming solidsConvex and non-convex solidsPolyhedraPrisms and pyramidsCylinders, cones and spheresClassifying solidsEulers ruleEdges of a solidThe Platonic solidsDrawing and building solidsDifferent views of solidsTopic overviewChapter review

5 IntegersNumber linesNumbers above and below zeroDirected numbersOrdering directed numbersAdding and subtracting integersMultiplying integersDividing integersThe four operations with integersReading a map gridThe number planeThe number plane with negative numbersTopic overviewChapter review

6 Patterns and rulesNumber rules from geometric patternsUsing pattern rulesThe language of algebraTables of valuesFinding the ruleFinding harder rulesFinding rules for geometric patternsAlgebraic abbreviationsSubstitutionSubstitution with negative numbersTopic overviewChapter review

Mixed revision 27 DecimalsPlace valueUnderstanding the pointOrdering decimalsDecimals are special fractionsAdding and subtracting decimalsMultiplying and dividing by powers of 10Multiplying decimalsCalculating changeDividing decimalsDecimals at workConverting common fractions to decimalsRecurring decimalsRounding decimalsMore decimals at workTopic overviewChapter review

8 Length and areaThe history of measurementThe metric systemConverting units of lengthReading measurement scalesThe accuracy of measuring instrumentsEstimating and measuring lengthPerimeterAreaConverting units of areaArea of squares, rectangles and trianglesAreas of composite shapesMeasuring large areasTopic overviewChapter review

9 Geometric figuresPolygonsClassifying trianglesNaming geometric figuresConstructing trianglesClassifying quadrilateralsConstructing perpendicular and parallel linesConstructing quadrilateralsTopic overviewChapter review

Mixed revision 310 FractionsHighest common factor and lowest common multipleNaming fractionsEquivalent fractionsOrdering fractionsAdding and subtracting fractionsAdding and subtracting mixed numeralsFractions of quantitiesMultiplying fractionsDividing fractionsTopic overviewChapter review

11 Volume, mass and timeVolumeVolume of a rectangular prismCapacity and liquid measureMassTimelinesConverting units of timeTime calculationsWorld standard timesTimetablesTopic overviewChapter review

12 AlgebraAlgebraic expressionsAlgebraic abbreviationsFrom words to algebraic expressionsLike termsMultiplying algebraic termsExpanding an expressionExpanding and simplifyingAlgebraic substitutionTopic overviewChapter review

13 Interpreting graphs and tablesPicture graphsColumn graphs and divided bar graphsSector graphsLine graphsTravel graphs and conversion graphsStep graphsReading tablesTopic overviewChapter review

Mixed revision 4General revisionAnswersIndex

GlossaryABCDEFG HI JK LMNOPQRSTU VW X Y Z

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Using technology menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 13

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