Chapter02 Angles

2

Geometry

Angles

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 moment—you will see angles everywhere.

In this chapter you will: ■ ■ ■ ■ ■ ■

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.

Wordbank ■ ■ ■ ■ ■ ■ ■

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.

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

ANGLES

31

CHAPTER 2

Start up Worksheet 2-01 Brainstarters 2

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. I

H

G

J

F E

D C

B A

What is the angle, in degrees, between the lines labelled: a A and C? b A and D? d C and F? e A and F? g D and G? h E and H? j C and J? k B and E?

c f i l

B and C? B and G? D and I? E and J?

3 In the diagram in Question 2, find one pair of labelled lines which have a 19° angle between them. 4 In the diagram in Question 2, find two pairs of labelled lines which have a 90° angle between them. 5 In the diagram in Question 2, 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° 6 The word ‘degree’ has many meanings. Find four non-mathematical meanings for the word.

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NEW CENTURY MATHS 7

7 Decide whether each of these angles is: i acute ii obtuse

Skillsheet 2-03

iii reflex

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

Types of angles

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 Gˆ

{

• ∠PGH or ∠HGP • PGˆ H or HGˆ P

P vertex G

arm

arc

H

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

ANGLES

33

CHAPTER 2

Example 1 Name the angle marked with a Y

in each of these diagrams. b

P

Q

S

Z

X

R

Solution

a ∠Y or ∠XYZ or ∠ZYX b ∠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.

Exercise 2-01 Example 1

1 Name each of these angles in two different ways: P a b C

c G R

Q

d

K G

V

O D

e

f R

E C

A

Q D

P

T

2 Name the angle marked with in each of these diagrams: M B a b

c

P

S Q

C T

D A

N Y

d

R

Q P

e

B

f

F

A E X G W

34

D Z NEW CENTURY MATHS 7

E

C H

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

d ∠H N

4 a There are 13 different angles inside this diagram. Name them all. b What type of angle is ∠NCY?

A C

D

Y

and x in each of the following diagrams: P b c M

5 Name the angles marked A a

N

x R

x D

Q

S

x

B C E

d

Q

R D

e

P W

f

E I

x C

F G

F

x B

x H

H G

X

A

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.

Z

Y

A P arm

M

N

Comparing angle size Can you tell which of these is larger?

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

35

CHAPTER 2

SkillBuilder 24-01 Measuring angles

Exercise 2-02 1 List the angles in each of the given sets in order, from smallest to largest, without using a protractor. a These angles vary by 5°. i ii iii

b These angles vary by 3°. i

ii

iii

c These angles vary by 2°. i

ii

iii

d These angles vary by 1°. i

ii

iii

1 e These angles vary by --- °. 2 i

ii

iii

f

36

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

iii

iv

vi

v

NEW CENTURY MATHS 7

vii

viii

ix

x

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

d a

b

c

e

f

The protractor Measuring angles A protractor is an instrument used to measure angles.

Outside scale

60

50

100 90 80 70 110

1 14

0

Inside scale

15

0

30

Worksheet 2-03 Make your own protractor

0

50

20

70 180 60 1 0 1 15 10 0

180 170 1 60

13

30

10 2 0

0

40

0

60

12

0

40

0 12

90 100 1 10

14

30

80 70

Worksheet 2-02 360° scale

Base line Centre mark

Worksheet 2-04 A page of protractors ANGLES

37

CHAPTER 2

Example 2 A

1 Measure angle AOB.

B

O

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, A 80 90 100 11 counting from 0°. 0 1 70 2 0 0 6 Angle AOB = 54°. 13 100 90 80 70 0 110 50 60 20 1

50

0 14

30

180 170 1 60 15 0

20 10

10 2 0

30

70 180 60 1 0 1

15

40

B

0

0

0

13

14

0

40

• • • •

O

2 Measure ∠PMQ.

P

Q

M

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, 80 90 100 11 counting from 0°. 0 1 70 20 0 6 90 80 ∠PMQ = 155°. 0 13 0 1 70 0 0 1 0 1 5 60

0

0 14

30 10 2 0

180 170 1 60 15 0

0

0

10 20

70 180 60 1 0 1

30

NEW CENTURY MATHS 7

15

38

40

Q

50

0

40

12

14

0

13

M

P

3 Measure ∠TEX.

X

T

E

Solution

1

50

14 0

0

30

15

10

0

E

170 180

180 170 1 60

20

160

30

10 2 0

0 15

40

0

0

0

13

14

Line up TE with the base line of the protractor. Place the centre mark over the vertex E. ∠TEX is bigger than 90°. 80 90 100 11 X 0 1 70 Use the inside scale. 20 0 6 90 80 0 13 0 1 ∠TEX = 134° 7 0 10 0 1 50 60 20 40

• • • •

T

Example 3 Measure the reflex angle ∠GHK. H

G

K

0 30

10 20

10 2 0

40

0

0

50

60

12

00 90 80 70 10 1 60 0 1

70

80

90 100 1 10

50

1

13

40

40

30

14

0

12 0

60 0 1 15

G

0

13

K

0

H

180 170 1 60 15

• Actually measure the obtuse angle ∠GHK first (140°). • Subtract 140° from 360°. 360 − 140 = 220 Reflex ∠GHK = 220°

170 180

Solution

ANGLES

39

CHAPTER 2

Exercise 2-03 1 Find the size of each of these angles:

40 0

0

40 0 14

150

30

10 2 0 0

40 0 14

150

30

0 14

180 170 1 60

14

U

40

0 13

70 110

100 90

0

120

80 90 100 11 0 60

120

13

0

50

14

150

30 20

10

0

70 180 60 1 0 1

13

40

10 2 0

80 7 0

40

0

30

10

60

15

30

14

40

50

0

40 150

10 2 0 0

180 170 1 60

0

40 180 170 1 60 150

10 2 0 0

14

30

0 14

150

180 170 1 60

10 2 0 0

30 20

F

0 14

30

170 180

0

0

0

0

A

K

0

120

60

110

70

I

13

50

60 0 1

L

10 2 0

150

100 90

80 7 0

80 90 100

60

0 14

13

120

15

30

60

10

10 2 0

80 7 0

20

0

0

100 90

30

10

170 180

180 170 1 60

110

40

60 0 1

180 170 1 60

70 180 60 1 0 1

0

170 180

10

120

80 90 100 11 0

0

0

70

14

15 20

50

110 120

15

170 180

60 0 1 15

60

h

B

0

20

0 30

g

H

50

U

13

50

O

0

40

G

120

60

1

13

50

13

80 7 0

30

60

100 90

40

70

110

f

Y 14

0

100

30

120

0

70 180 60 1 0 1 15 10 0

20

80 90 100 11 0 1 70 20 60 90 80 110 120

0

0 30

e

P

T

14

50

80 90 100 11 0

70

60

50

0

60

O

50

O

13

40

D

10

120

60 0 1 15

80 7 0

0

M

14

0 13

120

100 90 110

13

50

20

50

13

d

80 90 100 11 0

70

60

120

60

30

20

A

N

c

80 7 0

40

70 180 60 1 0 1 15 10 0

30

O

0

100 90

110 120

0

50

13

50

0

40

30

13

150

60

10 2 0

70

80 90 100 11 0

70

60

14

40

100

0

0

110 120

180 170 1 60

50

E 14

Worksheet 2-05 A page of angles

0

13

SkillBuilders 24-02 & 24-03 Measuring angles

b

B 80 90 100 11 0 1 70 20 60 90 80

180 170 1 60

a

50

Example 2

R

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

B

O

P

Q A

40

b

NEW CENTURY MATHS 7

D

c

d

Y

N

P X

A

e T

f

M

S

Q

M

g

Z

Y

X

L

D

N

h

i

j F

A

B

D

G P M Z

C B

k

A

l

F

E

G

ANGLES

41

CHAPTER 2

Example 3

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

b

X

A

B

Z

C

N Y

c

d G

H M L

K

4 Measure the angles marked with a

and x on each of these diagrams. b

x

x

c

d

x

42

NEW CENTURY MATHS 7

x

e

f

x

x

Geometry

Using technology Estimating angles

Skillsheet 2-01

1 Using a drawing package and estimation skills, draw and label each of the following angles. a ∠ABC = 45° b ∠DEF = 30° c ∠GHI = 60° d ∠JKL = 90° e ∠MNO = 120° f ∠PQR = 155°

Starting Geometer’s Sketchpad

2 Print out the six angles that you have drawn. Measure the angles with a protractor. Skillsheet 2-02

3 What was the error each time between the angle you drew and the actual angle requested?

Starting Cabri Geometry

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.

ANGLES

43

CHAPTER 2

Drawing angles You can also use your protractor to draw angles.

Example 4 Use a protractor to draw angle KPM which measures 76°.

Solution M P • 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. mark 76°

60

50

90 100 1 10

100 90 80 70

0

0

0

30

15

180 170 1 60

20

10 2 0

30

70 180 60 1 0 1 15 10 0

40

0

13

50

14 0

1

60

12

0

40

10 0 1 12

80

14

30

70

P

M

choose scale with 0° near M K

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

line ruled from P through mark at 76°

P

M

Exercise 2-04 Example 4

1 Accurately draw these angles, using your protractor: a 35° b 115° c 150° e 15° f 170° g 117°

d 40° h 200°

2 Use your protractor to accurately draw and label these angles: b ∠BGH = 145° c ∠GRT = 32° a ∠DRE = 65° e ∠SAQ = 110° f ∠NMH = 265° d ∠ABC = 45° h ∠LMN = 180° i ∠LKY = 90° g ∠KLY = 28° Geometry 2-01 Making a protractor

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

44

NEW CENTURY MATHS 7

Angle geometry

Worksheet 2-06 Angle cards

Classifying angles Angles may be classified according to their size: Angle

Type

Skillsheet 2-03 Types of angles

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

revolution

greater than 180° but less than 360°

360° (complete turn)

Exercise 2-05 1 Draw two different examples of: a an acute angle b an obtuse angle d a reflex angle e a straight angle

c a right angle f a revolution

2 Classify each of the following angles: a 37° b 107° d 195° e 79° g 163° h 179° j 5° k 345° m 14° n 299° p 205° q 126°

c f i l o r

252° 180° 360° 91° 90° 44° ANGLES

45

CHAPTER 2

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

c

d

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

B

C

Z

∠ABD = ∠XYZ = ∠CBD = ∠XZY = ∠ABD + ∠CBD = ∠XYZ + ∠XZY = (The angles you measured are called complementary angles. They complement each other to form 90°.) Complementary angles add to 90°. 5 Look up ‘complement’ in a dictionary. Write one non-mathematical meaning you find.

SkillBuilder 24-05 Complementary angles

6 What is the complement of: a 30°? b 70°? e 89°? f 57°? i 11°? j 74°?

46

NEW CENTURY MATHS 7

c 25°? g 42°? k 1°?

d 38°? h 66°? l 12°?

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

R P A

C

B

S

∠ABD = ∠PQR = ∠CBD = ∠SRQ = ∠ABD + ∠CBD = ∠PQR + ∠SRQ = (These pairs of angles are said to be supplementary. They supplement each other, together forming 180°.) SkillBuilder 24-04 Supplementary angles

Supplementary angles add to 180°. 8 Look up ‘supplement’ in your dictionary. Write a non-mathematical meaning for it. 9 What is the supplement of: a 18°? d 125°? g 111°? j 132°?

b e h k

150°? 62°? 173°? 8°?

c f i l

35°? 87°? 54°? 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 B A

B

E A

D

C

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

D

C

∠AEB = ∠BEC = ∠CED = ∠DEA = ∠AEB + ∠BEC + ∠CED + ∠DEA =

Angles at a point (in a revolution) add to 360°. 11 Use Cabri Geometry or Geometer’s Sketchpad to illustrate the meaning of as many angle words as you can.

ANGLES

47

Geometry 2-02 Angle vocabulary CHAPTER 2

12 Use the given information to find the size of the angle shown by the letter each time. a b c d q° 150°

70°

m°

170°

62°

160°

95° 120° y°

87° x°

e

f

g

h 132° 123°

102°

71° 116°

25° a° 135°

i

d°

105° 110° 55° w°

22°

j

48° f°

k

l 118°

n°

30°

152° k°

t°

h° 47° 15°

220°

Geometry 2-03 Revolutions and straight angles

303°

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

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

a° d°

b° c°

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 on diagrams with the same type of arc or symbol.

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NEW CENTURY MATHS 7

X

W

K Z

Y

Vertically opposite angles are equal.

Example 6 Find the size of the angles shown by the letters in this diagram.

130° 50°

m°

k°

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

Exercise 2-06 1 What angle is vertically opposite to: a the angle marked a°? b the angle marked w°?

a°

w°

b°

d°

v°

Example 5

c the angle marked c°?

a° b°

u°

u°

d° c°

c°

d the angle marked h°?

e° h°

e the angle marked k°?

f

the angle marked m°?

h° i°

f°

g°

k°

p°

l° n°

m°

d°

2 Without measuring, find the size of the angle shown by the letter each time. a b c 70°

110°

k°

m°

85°

a°

d

e

f 25°

90° m°

Example 6

135°

f°

x°

ANGLES

49

CHAPTER 2

g

h

i 29° 62°

q°

w° 133° n°

j

k

l h°

t° 163°

160° g° 20°

90° s°

q° r°

Angle facts Types of angles

Meaning

Adjacent angles

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

Diagram A

x

B

D C

Complementary angles

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

Supplementary angles

b°

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

Vertically opposite angles

Formed when two straight lines cross. Vertically opposite angles are equal.

a°

n°

b°

c°

d°

(a = c, b = d) Angles at a point

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

b°

Remembering these facts will help you complete the next exercise.

50

NEW CENTURY MATHS 7

a° c°

Exercise 2-07 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.

M N D

Q

P

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?

Q

P

23°

67°

R

S

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

b

SkillBuilder 24-07 Combination figures

c 100° y° a°

a° 120°

d

70°

e

f

m° 45°

100°

100° 40° a°

p° 150°

g

h m°

i 41° 15°

19° x°

f°

ANGLES

51

CHAPTER 2

j

k

l a° a°

h° 170° t°

m

n

b°

t°

o

32°

135° y° 82°

a°

d° d°

p

q

r

20° y°

s

j° 48°

e° 112° e° f°

x°

k°

t

l°

u p°

118° x° y° 75°

85° 155° p°

e° e° e°

Naming lines A 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.

52

NEW CENTURY MATHS 7

B A D G H F E

Perpendicular lines Lines 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 P stands for ‘is perpendicular to’.

X

Q Y

Parallel lines Lines 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. N This is written as ‘MN II RS’, where the symbol II stands S M for ‘is parallel to’. R

indicates these lines are parallel

SkillBuilder 24-08 Parallel lines

Exercise 2-08 1 Name the six different lines in this diagram. B

A

D

2 In this diagram, name two lines that: a are perpendicular b are parallel c intersect?

H

C

A B

G C F E

D

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 c line PQ parallel to line YZ

b line AB intersecting line CD at point E 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.

ANGLES

53

CHAPTER 2

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

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

transversal

transversal

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.

54

NEW CENTURY MATHS 7

Alternate angles Alternate 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.

Alternate angles on parallel lines are equal. Pairs of alternate angles

x x

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.) SkillBuilder 24-09 ‘Z’ in parallel lines

Exercise 2-09 1 Which angle is alternate to the marked angle each time? a b

c a°

a° b° d° c°

b° c°

d°

a° b° c°

d° e° g° f °

e° f ° g°

e° f°

g°

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

ANGLES

55

CHAPTER 2

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 110°

a°

n° 50° 80°

m°

d

e

f

n°

122°

m°

20° h°

b°

p°

g

h b°

50°

i

130° a°

40°

c°

a°

44°

b° b°

a°

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

Pairs of corresponding angles

x

x

56

NEW CENTURY MATHS 7

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°

Exercise 2-10 1 Which angle is corresponding to the marked angle each time? a b c b° c°

a°

a°

a° b° c° d°

g° d° f° e°

b°

f ° e° g°

d° c° e°

f°

g°

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: a b

c

120°

m° y° 63°

a° 28°

d

e

f t°

50°

108° a°

74° c°

b°

60°

ANGLES

57

CHAPTER 2

g

h

i

y° a°

m° y° 110°

140° c° d°

x° 105°

n°

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

105°

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

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

Pairs of co-interior angles

x

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

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

58

NEW CENTURY MATHS 7

a° b°

c°

Example 7 1 Find the size of the angle marked a° in this diagram. a° 80°

Solution a + 80 = 180 Co-interior angles are supplementary. a = 180 − 80 So a = 100 2 Find the size of the angle marked m° in this diagram. 55° m°

Co-interior angles are supplementary. m + 55 = 180 m = 180 − 55 So m = 125

Exercise 2-11 1 Which angle is co-interior with the marked angle each time? a b c a° b° d° c° e° g° f°

a° g° f°

e°

d°

b° c°

e° a° b° d° c°

f° g°

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. a b c

ANGLES

59

CHAPTER 2

4 Without the use of instruments, calculate the size of the angles shown by letters: a b c b°

50°

m° 90°

a°

d

75°

e

f 68°

112° d°

b° a°

98° m°

g

h

i c° b°

j°

130°

55°

k°

f ° g°

a° 51°

Just for the record

4.1 m

The Leaning Tower of Pisa The 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.

60

NEW CENTURY MATHS 7

55 m

Example 7

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

Worksheet 2-07 Matching angles

Exercise 2-12 1 In the diagram on the right, name the angle that is: a corresponding to ∠VWA b alternate to ∠QXW c co-interior with ∠PWX d supplementary with ∠AWX e alternate to ∠SXV f corresponding to ∠ZXS.

V P

W A

S X

Q Z

2 Without the use of instruments, find the size of each angle shown by a letter: a b c 105°

p°

71°

115° t°

k°

d

e

f 132°

70°

m°

n° 120°

g

a°

h 28°

i 85°

72°

s°

x°

k°

j

k

l

p°

y°

81°

150°

w°

93°

ANGLES

61

CHAPTER 2

m

n

o

128°

q° 66°

j° d°

SkillBuilders 24-10 to 24-12 Combination figures

109°

3 Without measuring, find the size of all angles labelled with letters in these diagrams: a b c 133° b°

p°

j°

67°

n° m°

a° k°

52°

l°

d

e

f

c°

b°

m° 95°

42° y°

45°

30°

l°

z°

g

h

i

75° p°

p°

k°

q°

63°

w°

m° 85°

j

k

l a°

k° x°

72°

y° 130°

62°

m

n n°

m°

Geometry 2-04 Angles and parallel lines

o

p° 83°

b°

55° 27°

a°

g° 132°

b° c°

4 You can use Geometer’s 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.

62

NEW CENTURY MATHS 7

Finding parallel lines We 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?

Solution

A

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

C

X

B

75°

D

75°

Y

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

X N

110°

Solution ∠MXY is co-interior with ∠PYX. P ∠MXY + ∠PYX = 110° + 80° = 190° ≠ 180° Since co-interior angles do not add to 180°, MN is not parallel to PQ.

80° Q

Y

Exercise 2-13 1 In each diagram below, name a pair of alternate angles and use them to decide if AB is parallel to CD. B C a b c A E

A

D

B 64°

100°

64° F

100°

D

C

E 32° 35° F

H

G

A

D

B

C

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

82° E

A

63°

D

79°

E

F

C

63° G

F

C

110°

117°

G

F

B

E D

B

D

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 A C A

E

C

B

E

120°

100°

A

B

85°

60° F

D

C

F

90° D

E B

90° F D ANGLES

63

CHAPTER 2

Example 8

4 For each diagram below, determine if line PQ is parallel to line MN. Explain your reasons. P A M a b M B

81°

N

X

P

Y

99° C

Q N

D

c

K

I

G

E

d

Q

87°

N

e B 80°

A 95°

F

C 80°

E

P

f N

M

L

B 65°

Q

C

75°

75°

105°

P

95°

85°

L

J

H

F

65°

K

78°

78°

P

P

120° A

M

78°

87°

M

D

N 102°

M

Q

D

N

Q

Q

X

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 b c d

Draw any triangle and measure the sizes of all three angles. What is the sum of the angles in any triangle? Draw any quadrilateral and measure the sizes of all four angles. 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°

64

NEW CENTURY MATHS 7

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

m°

b

c

62°

a°

x°

51°

125°

y°

d

e

f

145°

35°

82°

k°

m° 80° y° 40° 250°

c°

g

h

i

35°

k°

m°

x°

120° 95°

50° 45°

20°

Language of maths acute arm degree revolution supplementary

adjacent co-interior intersecting right angle transversal

alternate complementary line scale vertex

arc corresponding obtuse straight angle vertically opposite

1 Name the two parts of an angle.

Worksheet 2-07 Matching angles Worksheet 2-08 Angles crossword

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 perpendicular

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

65

CHAPTER 2

Topic overview • • • • •

Give three examples of where angles are used. Do you think this chapter is very useful to you? Why? How confident 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.

Acute

Supplementary

Revolution

Vertically opposite

Complementary

Adjacent

ANGLES 20

0

10

0

80

18

50

130

60

0

17

70

120

0

90

0

10

90

0

15

130 120 110 100 140

11

0 16

50 60 70 80

40

30

40

140 150 160 170 18 0

30

Pro

20

tra

10

cto

LINES

B D

A Alternate

C

Corresponding

Parallel E

x

Co-interior

66

NEW CENTURY MATHS 7

G Transversal

H F

Perpendicular

0

r

Chapter 2

Review

Topic test Chapter 2

1 Draw labelled diagrams of these angles: a ∠BKT

Ex 2-01

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.

Ex 2-03

3 Use a protractor to draw these angles.

Ex 2-04

a ∠JUG = 84°

b ∠QRA = 117°

c ∠POT = 41°

d ∠DGE = 150°

e ∠SAR = 96°

f

g ∠MNB = 195°

h ∠PLO = 270°

I ∠AMP = 300°

∠XDW = 210°

4 Write the name of each of these angles. Then label each one as acute, obtuse, right, reflex or straight. I A R a W b c G

Ex 2-05

L

H V

d

D U

P

e

f S M

A N P

E

R

g

h

V

Z

Y

i

M

Q

T M P X

5 Without measuring, find the size of each angle shown by a letter: a b c 28° k°

m°

x° y° 122°

47°

d

e

Ex 2-07

f

140° p°

75°

110°

f°

48° x° ANGLES

67

CHAPTER 2

g

h

i r°

105°

82°

q° p°

t°

25° x° x°

Ex 2-12

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

x°

x

x

d

e

f

x x

Ex 2-12

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

c

35°

65°

k°

m°

115°

d

e

f x°

q°

125° 130°

62°

d°

g

h

i

37°

112° y° x°

68

NEW CENTURY MATHS 7

z° p° 62° m°

t° d° a°

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

b

Ex 2-12

c

64°

c°

x° x°

70°

m°

y°

d

z°

a°

130°

e

38°

x°

38°

y° 57°

a° z° 145° c°

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

45° F

F 110°

B

74°

D

F

C C

135° G H

D

B 74°

A

112° G C

Ex 2-13

H

G

H

D

10 Draw a neat diagram to illustrate each of the following: a an acute angle b supplementary angles c a straight angle d vertically opposite angles e alternate angles f an obtuse angle g corresponding angles h a reflex angle i complementary angles j co-interior angles

ANGLES

69

CHAPTER 2