Chapter 2
Short Description
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02 NCM7 2nd ed SB TXT.fm Page 32 Saturday, June 7, 2008 2:53 PM
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.
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02 NCM7 2nd ed SB TXT.fm Page 33 Saturday, June 7, 2008 2:53 PM
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• complementary angles5678 • label and name images Two angles that add 1234567890123456789 234567890123 45678901234 0123456789 to 90°. • estimate, measure and construct angles 0123456789 56789012 456789012345678901234567890123456789012345678 56789012 8 7890123456789012 3456789012345678 • parallel lines Lines that point in the same • classify901234567890123456789012345678901234567890123456789012345678 angles as right, acute, obtuse, reflex, 789012345678 direction and do not intersect. 567890123456789012345678901234567890123456789012345 straight or a revolution 0123456789 56789012345678901234567890 1234567890123456789 • perpendicular lines Lines that intersect at right 78901234 5678 angles, vertically 234567890123 45678901234 • identify and name adjacent angles. opposite angles, straight angles 7890123456789012 and0123456789 angles of 45678901 890123456789012345678901234567890123456 345678901 456789012 9012345 6789012345678901234567890123456789012345678901234567890123456789012 complete 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02 NCM7 2nd ed SB TXT.fm Page 34 Saturday, June 7, 2008 2:53 PM
Start up Worksheet 2-01
1 In this diagram, each gap represents 1° of angle size. I
Brainstarters 2
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 ?
2 In the diagram in Question 1, find one pair of labelled lines which have a 19° angle between them. 3 In the diagram in Question 1, find two pairs of labelled lines which have a 90° angle between them. 4 In the diagram in Question 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. Skillsheet 2-01 Types of angles
6 Decide whether each of these angles is: i acute ii obtuse a b
34
NEW CENTURY MATHS 7
iii reflex c
d
02 NCM7 2nd ed SB TXT.fm Page 35 Saturday, June 7, 2008 2:53 PM
e
f
g
h
i
j
k
l
m
n
o
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 Gˆ
{
• ∠PGH or ∠HGP • PGˆ H or HGˆ P
P arm
vertex G
arc
H
The middle letter is always the letter that labels the vertex of the angle.
Example 1 Name the angle marked with Y a
in each of these diagrams. b Q
X
Z
P S 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.
CHAPTER 2 ANGLES
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02 NCM7 2nd ed SB TXT.fm Page 36 Saturday, June 7, 2008 2:53 PM
Exercise 2-01 Ex 1
1 Name each of these angles in two different ways. P a b C
c G R
Q
d
K
V
O
e
G
D
f
R
E C
A
Q
P
T
D
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
A C
B
3 Name the angle marked with a
in each of these diagrams. b
B
D
c
M
P
S Q
C T
D A
N
d
P
e
Y
B
A
F
G D
E
C H
Z
4 Draw each of these angles, labelling them correctly. a ∠POT b ∠TAF c ∠AFE
36
f
E
X
W
R
Q
NEW CENTURY MATHS 7
d ∠H
02 NCM7 2nd ed SB TXT.fm Page 37 Saturday, June 7, 2008 2:53 PM
5 a There are 13 different angles inside this diagram. Name them all. b What type of angle is ∠NCY?
N A D
C
Y
6 Name the angles marked A
a
and x in each of the following diagrams. P
b
M
c
N
x R
x D
Q
S
x
B C E
d
e
I
Q
R
W
f
D E
x C
F G
P
F
x B
x H
H G
X
A
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.
Y
Z
A P
Worksheet 2-02 Comparing angle size
arm
Worksheet 2-03 360° scale
M
N
2-02 Measuring angles
Worksheet 2-04 Make your own protractors
A protractor is an instrument used to measure angles.
Worksheet 2-05 A page of protractors
Outside scale
01 40
13
0
50
Starting The Geometer’s Sketchpad Skillsheet 2-03
70 180 60 1 0 1
30
Skillsheet 2-02
15
Starting Cabri Geometry
10 20
Base line Centre mark
Geometry 2-01
0
180 170 1 60 15
0
30
Inside scale
12
40
10 2 0
1
30
100 90 80 70 110 60 0 12
0
40
50
80 90 100 11 0
14
0
60
70
CHAPTER 2 ANGLES
Making a protractor
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02 NCM7 2nd ed SB TXT.fm Page 38 Saturday, June 7, 2008 2:53 PM
Example 2 1 Measure angle AOB.
A
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 0 7 12 0 60 13 0 90 80 7 Angle AOB = 54° 0 1 0 0 0 1 01
5
0
14 0
30
15
180 170 1 60
20
10
10 2 0
70 180 60 1 0 1
30
0
0
15
40
0
50
0
40
60
12
14
0 13
O
2 Measure ∠PMQ.
B
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 ∠PMQ = 155° 90 80 13 0 0 1 7 0 0 10 01
5
14
0
180 170 1 60 15
0
30 10 2 0
10
0
20
NEW CENTURY MATHS 7
70 180 60 1 0 1
38
30
Q
15
40
0
1
0
50
0
40
60
12
14
30
M
P
02 NCM7 2nd ed SB TXT.fm Page 39 Saturday, June 7, 2008 2:53 PM
3 Measure ∠TEX.
X
E Solution • 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 X 70 • Use the inside scale. 60 100 90 ∠TEX = 134° 110 50 40
100 1 10 80 7 0
0
60
14 0 15
180 170 1 60
170 180
10
10 2 0
20
60 0 1
30
0
0
15
40
30
13 0
50
0
1
12
0 14
0
13
20
T
E
T
Example 3 Measure the reflex angle ∠GHK. H
G
0
10 30
20 40
13
40
0
50
60
12
00 90 80 70 10 1 60 01
70
80 90 100 11 0
0 14
10 2 0
0
50
12 0
60 0 1 15
G
30
14
13
0
K
0
H
180 170 1 60 15 0
170 180
K
Solution • Actually measure the obtuse angle ∠GHK first (140°). • Subtract 140° from 360°. 360 − 140 = 220 Reflex ∠GHK = 220°
CHAPTER 2 ANGLES
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02 NCM7 2nd ed SB TXT.fm Page 40 Saturday, June 7, 2008 2:53 PM
Exercise 2-02 1 Find the size of each of these angles. B a
E 40 0
40 0 14
180 170 1 60 15 0
14
30
10 2 0 0
0
40 0 14
0 15
0 14
0
14
30
13
0
50
0
20
180 170 1 60
15
30
13
40
10
0
70 180 60 1 01
0
10 2 0
60
15
0
100 90 80 70
40
30
14
40
10 0 1
12
80 90 100 11 01 20
0
30
F
14
40
10 20
70
60 0 13
0
0
0
14
30
10 2 0
0 14
30
10 2 0
180 170 1 60 15 0
40 30
10 2 0
180 170 1 60 15 0
40 30
30
0
0
50
0
12
60
0 90 80 7 0 10 10 60 0 1
I
170 180
10
10 2 0
20
180 170 1 60 15
0
30
0
13
50
A
K
0
15
70
80 90 100
0 14
0
60
60 01
0
100 90 80 70
U
10 2 0
180 170 1 60
170 180
10 0 1
12
15
L
0
60 01
0
70 180 60 1 01
15
0 13
80 90 100 11 01 2
40
10
15
170 180
0
70
60
h
B
60 01
0
0
20
70 180 60 1 01
30
H
15
40
g
13
50
O
50
U
0
14
50
12
10 20
13
0
60
1
60
f
Y
0
0 13
100 90 80 70
100 90 80 70
30
10 0 1
12
110 20
40
0
80 90 100 11 01 70 20 60
14
G
70 180 60 1 01
10 20
e 50
P
15
15
30
O
0 13
80 90 100 11 0
70
60
50
40
D
T
0 14
50
0
10 20
1
13
0
60
13
M
0 14
0 13
20
0
50
30
0
50
100 90 80 70 110
12
60
O
d
80 90 100 11 01 20
70
60
100 90 80 70
110
40
70 180 60 1 01
10 20
A
N
c
0 12
0 14
15
30
O
0
13
80 90 100 11 0
70
60
50
0
40
0
13
0
0
180 170 1 60 15 0
12
0 90 80 7 0 10 10 60 0 1 12 50 0 13
0 14
40
60
b
10 2 0
50
A page of angles
80 90 100 11 0
70
180 170 1 60
Worksheet 2-06
50
Ex 2
R
50
0
0
110 12
13
2 Estimate the size of each of these angles. Name each angle and use a protractor to measure the angles accurately. a
B
b
P
O Q A
40
NEW CENTURY MATHS 7
D
02 NCM7 2nd ed SB TXT.fm Page 41 Saturday, June 7, 2008 2:53 PM
Y
c
N
d
P X
A
e T
M
S
Q
M Z
f
g
Y
D
X
L N A
h
i
F D
G B
j M P C
k
B
Z F
E
l
A G CHAPTER 2 ANGLES
41
02 NCM7 2nd ed SB TXT.fm Page 42 Saturday, June 7, 2008 2:53 PM
Ex 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 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 a
and x on each of these diagrams. b
x
x
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NEW CENTURY MATHS 7
02 NCM7 2nd ed SB TXT.fm Page 43 Saturday, June 7, 2008 2:53 PM
c
d
x
x
e
f
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.
CHAPTER 2 ANGLES
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02 NCM7 2nd ed SB TXT.fm Page 44 Saturday, June 7, 2008 2:53 PM
2-03 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
80 90 100 11 0
0
M
14 0 15
0
180 170 1 60
10
10 2 0
20
70 180 60 1 0 1
30
0
15
0
13 0 40
30
12
00 90 80 70 10 1 60 01 12 50 0 13
0 14
40
50
70
P
choose scale with 0° near M
• Draw a line from P through this mark. Label the end of this line K. You have now drawn angle KPM, measuring 76°.
K
line ruled from P through mark at 76°
P
M
Exercise 2-03 Ex 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. 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°
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NEW CENTURY MATHS 7
02 NCM7 2nd ed SB TXT.fm Page 45 Saturday, June 7, 2008 2:53 PM
2-04 Classifying angles Angles may be classified according to their size as shown below. Angle
Type acute
Worksheet 2-07
Description
Angle cards
less than 90° Skillsheet 2-01 Types of angles
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 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° CHAPTER 2 ANGLES
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3 List the following angles from smallest to largest. d
b a
c
g
f
h e
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 reflex b obtuse and a revolution c acute and a revolution d acute and reflex
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NEW CENTURY MATHS 7
n°
m°
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2-05 Angle relationships
Geometry 2-02
In 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.
Angle vocabulary
Geometry 2-03 Revolutions and straight angles
Exercise 2-05 1 Copy and complete the information below each of these diagrams. Use your protractor to measure the angles. Y a A b X D
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°.
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°? g 42°? h 66°? i 11°?
d 38°? j 74°?
e 89°? k 1°?
f 57°? l 12°?
4 Copy and complete the information below each of these diagrams. Use your protractor to measure the angles. D a b Q
R P A
B
C
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°.) CHAPTER 2 ANGLES
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!
Supplementary angles add to 180°.
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°? g 111°? h 173°? i 54°?
d 125°? j 132°?
e 62°? k 8°?
f 87°? 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 A B E
A
D
C
D
∠ADB = ∠ADC = ∠BDC = ∠ADB + ∠ADC + ∠BDC = (These angles all meet at a point.)
! Geometry 2-02 Angle vocabulary
B
C
∠AEB = ∠BEC = ∠CED = ∠DEA = ∠AEB + ∠BEC + ∠CED + ∠DEA =
Angles at a point (in a revolution) add to 360°.
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. a b c d q° 150°
70°
m°
170°
62°
160°
87°
95° 120° y°
x°
e
f 102° 25°
a° 135°
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NEW CENTURY MATHS 7
g 105° 110° 55° w°
71° 116° d°
h
22°
132° 123° 48° f°
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i
j
k
l
303°
118° n°
30°
152° k°
t°
h°
47° 15°
220°
10 Find the value of d. Select A, B, C or D. A 122 B 61 C 142 D 81
38° d° d° 160°
2-06 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° c°
b°
Example 5 ∠WKZ is vertically opposite and equal to ∠XKY.
X
W
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.
K Y
Z
!
Vertically opposite angles are equal.
Example 6 Find the size of the angles shown by the letters in this diagram. Solution k = 130 m = 50
130° 50°
m°
k°
(vertically opposite angles) (vertically opposite angles)
CHAPTER 2 ANGLES
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Exercise 2-06 Ex 5
1 What angle is vertically opposite to: a the angle marked a°? b the angle marked w°? a°
v°
w°
c the angle marked c°?
u°
u°
a° b°
b°
d°
d° c°
c°
d the angle marked h°?
e° h°
Ex 6
e the angle marked k°?
h°
f the angle marked m°?
i°
f° k°
g°
l°
p°
m°
n°
d°
2 Without measuring, find the size of the angle shown by the letter each time. a b c 70°
k°
110° m°
a°
d
85°
e
f f°
25° 90°
135°
m°
g
x°
h
i 29° 62°
q°
w°
133° n°
j
k h°
t° 163°
50
l
NEW CENTURY MATHS 7
160° g°
90° 20°
q°
s° r°
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2-07 Angle geometry Types of angles Adjacent angles
Meaning
Diagram A
Angles that share a common arm and a common vertex. (∠ABD and ∠DBC are adjacent angles.)
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 = c, b = d)
Angles at a point
a°
Form a revolution and add to 360°. (a + b + c = 360)
b°
n°
b° d°
c°
a° c°
Example 7 Calculate the size of the angle shown by the letters in these diagrams. a b 60° y° 130°
x°
Solution a x + 130 = 180 (angles in a straight line) x = 180 − 130 = 50
b y + 60 + 90 = 360 (angles at a point) y = 360 − 60 − 90 = 50
CHAPTER 2 ANGLES
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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 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.
M N D
Q P
3 Which of the following is an angle adjacent to ∠AXB? Select A, B, C or D. A ∠BXC B ∠DXE C ∠DXC D ∠CXE
A
B
X C E D Q
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? Ex 7
P
23°
67°
R S
5 Calculate the size of the angle shown by the letter. State which type of angles you used. a b c 100° a° a° 120°
d
y°
70°
e
f
m° 45°
100°
100° 40° a°
p° 150°
g
h
i
m° 19°
f° x°
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NEW CENTURY MATHS 7
15°
41°
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j
k
l
a° a°
h°
170° t°
m
n b°
t°
o
32° 135°
a°
y° 82°
d° d°
p
q
r
20°
e°
x°
e°
y°
s
t 118° x° y° 75°
j° 48°
112° f°
k°
l°
u p° 155°
85°
e° e° e°
p°
2-08 Lines in geometry A line is named using two points on the line. For example, this is the line AB.
B
A
When two lines cross, we say that they intersect. Two lines intersect at a point. For example, in this diagram, line DE intersects with line FG at point H.
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 stands for ‘is perpendicular to’.
X P
Q Y
CHAPTER 2 ANGLES
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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. This is written as ‘MN II RS’, where the symbol II stands for ‘is parallel to’.
N S
M R
indicates these lines are parallel
Transversal A line that crosses two or more other lines is called a transversal. Transverse means ‘crossing’. transversal transversal
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.
C
A
H
B
G
C F D
E
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 the diagram on the right, name two angles that are: a adjacent b vertically opposite c supplementary.
C
A E
D
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NEW CENTURY MATHS 7
B
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6 Which line is parallel to line FG? Select A, B, C or D. A CD B LM C AB C
L
D PQ P
A
F B G
Q
M
D Road
Rosalia Road
Christina
Dan iel Stre et
7 In the diagram on the right, Frank Road is perpendicular to which of the following? Select A, B, C or D. A Emilia Parade B Rosalia Road C Daniel Street D Christina Road
Fra
nk R
oad
Emilia
Parade
8 State all the examples of parallel lines, perpendicular lines and intersecting lines you can find in the photograph below.
CHAPTER 2 ANGLES
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Mental skills 2
Maths without calculators
Changing the order Have 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 c 18 + 91 + 9 + 20 e 24 + 16 + 80 + 44 + 10 g 100 + 36 + 200 + 10 + 90
b d f h
38 + 600 + 50 + 12 + 40 75 + 33 + 7 + 25 56 + 5 + 20 + 15 + 4 54 + 27 + 9 + 16 + 3
2-09 Alternate angles on parallel lines Alternate 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 alternate angles as shown. Draw in the broken line and cut along it. transversal 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.
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Alternate angles on parallel lines
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.)
Exercise 2-09 1 Which angle is alternate to the marked angle each time? a b
c
a° b° d° c° a° b°
d° c°
g°
a°
b° c°
d°
e° f°
e°
e°
f ° g°
f°
g°
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° c°
b°
a° e°
d° f°
g°
CHAPTER 2 ANGLES
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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. a b 110°
a°
c n°
50°
80°
m°
d
e
122°
f
n° m°
20° h°
b°
p°
g
h b°
50°
i
130° a°
40° a°
b°
c°
44° a° b°
2-10 Corresponding angles on parallel lines Corresponding angles are on the same side of the transversal and are both either above or below the other two lines. ‘Corresponding’ means ‘matching’.
!
Corresponding angles on parallel lines are equal.
Corresponding angles on parallel lines x
x
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We can prove that corresponding angles on parallel lines are equal. 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°
f ° e° g°
b°
d° c° e°
f°
g°
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.
D
A
b° a°
C
B
c°
5 Write the size of each angle shown by a letter. a b 120°
c m°
y° 63°
a° 28°
CHAPTER 2 ANGLES
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d
e
f t°
50°
108°
a° b°
74° c°
60°
g
h
i
y° a°
m° y° 110°
x°
140°
105°
c°
d°
n°
6 Without measuring, find the size of the other seven angles in this diagram.
f° e°
a°
g°
d°
b° c° 105°
2-11 Co-interior angles on parallel lines Co-interior angles are on the same side of the transversal but between the other two lines. ‘Co-interior’ means ‘together inside’.
!
Co-interior angles on parallel lines are supplementary. They add to 180°.
Co-interior angles on parallel lines
x
Measure the following pairs of angles and see if they really are supplementary. x
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We can also show that co-interior angles on parallel lines add to 180° using the following method. They are angles on a straight line. a + b = 180 a =c They are alternate angles. So c + b = 180
a°
b° c°
Example 8 1 Find the size of the angle marked a° in this diagram. a° 80°
Solution a + 80 = 180 a = 180 − 80 = 100
(co-interior angles on parallel lines)
2 Find the size of the angle marked m° in this diagram. 55° m°
Solution m + 55 = 180 m = 180 − 55 = 125
(co-interior angles on parallel lines)
Exercise 2-11 1 Which angle is co-interior with the marked angle each time? a b c b°
b°
a° d° c°
a° g° g°
e° f°
f°
e°
d°
c°
e° a° b° d° c°
CHAPTER 2 ANGLES
f° g°
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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. a b c
4 Which angle is co-interior with the marked angle? Select A, B, C or D. A d° B b° C e° C g°
c° b° a° d° e° g° f°
Ex 8
5 Without the use of instruments, find the size of the angles shown by letters. a
b
c
50°
b°
m° 90°
a°
d
75°
e 112°
f 68°
b°
d°
98°
a°
m°
g
h
i c°
j° 130°
55° f°
b°
k°
g°
a° 51°
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Just for the record
The Leaning Tower of Pisa 4.1 m
55 m
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 m above the ground. 2 Research how engineers prevented the tower from leaning further. Use the library or the Internet to conduct your research.
2-12 Angles on parallel lines
Worksheet 2-08
Below is a summary of all we have found out about the angles in parallel lines.
Matching 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°).
!
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
Find the missing angle
S X
Q Z CHAPTER 2 ANGLES
Worksheet 2-09
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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°
a°
g
h 28°
i 85°
72°
s°
x°
k°
j
k
l
p° y°
150°
81°
w°
93°
m
n
o
128°
q°
66°
j° d°
109°
3 Without measuring, find the size of all angles labelled with letters in these diagrams. a
b
c 133°
b°
67°
p° n°
j°
a°
k° l°
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NEW CENTURY MATHS 7
m° 52°
02 NCM7 2nd ed SB TXT.fm Page 65 Saturday, June 7, 2008 2:53 PM
d
e
f
c°
b°
m° 95°
42°
45°
y°
30°
l°
z°
g
h
i
75° q°
p°
k°
p°
63°
w°
m° 85°
j
k
l a°
k° x°
72°
y° 130°
62°
m
n n°
b°
55°
o
27°
g°
p°
a°
132°
b°
83° m°
4 Which of the following does y equal? Select A, B, C or D. A 28 B 47 C 77 D 152
c°
y° 28°
105°
Using technology
Constructing angles using geometry software
Starting The Geometer’s Sketchpad
Note: 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.
CHAPTER 2 ANGLES
Skillsheet 2-02
Skillsheet 2-03 Starting Cabri Geometry
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c Measure the size of each angle you have drawn, correct to the nearest degree. A Example m∠ABC = 52° The diagram on the right shows acute angle ∠ABC = 52°. B
C
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. A Example Acute angle m∠ABC = 52°
B
C
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 D
27° 27°
23° 102°
A
B
c A pair of: i complementary angles ii supplementary angles iii corresponding angles of 28°, on parallel lines iv alternate angles of 65°, on parallel lines v co-interior angles on parallel lines, as shown on the right, where one of the supplementary angles is 130°
130°
2-13 Proving lines are parallel We 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°).
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Example 9 1 Is AB parallel to CD in the diagram on the right? X A
B
75° 75°
Solution ∠AXY is alternate to ∠DYX. ∠AXY = ∠DYX = 75° ∴ AB II CD since a pair of alternate angles are equal. (∴ means ‘therefore’)
C
2 Is MN parallel to PQ in the diagram on the right?
D
Y
X
M
N
110° 80°
P
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.
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 a b A c C
E A
B
F
64°
32°
E
H
G
35° F
100°
D
C
D
100°
64°
Ex 9
B
A
D
C
2 In each diagram below, name a pair of corresponding angles and use them to decide if AB is parallel to CD. A a b c G C A G 117°
82° E
A
79° C
B 63° D
E
63° F
C G
110°
F
B
E D
F B
D CHAPTER 2 ANGLES
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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
120° C
B
E 100°
A
B
90°
C
D
F
E
D
85°
60°
90° F
F
B
D
4 For each diagram below, determine if line PQ is parallel to line MN. Explain your reason. P M A a b M
B
81°
N
X
P
Y
99° C
Q
c
K
I
G
E
87°
120° A
M
78° Q
87°
78°
P
L
J
e A
C
95°
80°
80°
N
65°
Q
C
M
75°
75°
105°
P P
85°
85° F
E
D
L
B
P
f B
N
65°
K
78°
H
F
M
D
d N
102° M
Q
N
D
N
Q
Q
X
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)
H G
B A
91° 89°
89° 91° 91° D
E C
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NEW CENTURY MATHS 7
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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° 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°
g
c°
h
i k°
35°
m°
x°
120° 95°
50° 45°
20°
CHAPTER 2 ANGLES
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Chapter 2 review
Angles crossword
adjacent complementary obtuse supplementary
alternate corresponding revolution transversal
arc degree right angle vertex
arm intersecting scale 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. Acute
Revolution Vertically opposite
20
40
50 60 70 80
60
50
NEW CENTURY MATHS 7
C
Parallel
H
F Perpendicular
0
70
Transversal
G
or
40 30 20 10
D A
Co-interior
E
B
x
act
0 150 160 17 30 14 01 01 80
LINES
0
70
otr
Corresponding
10
12
Pr
Alternate
90
120 110 10 0 130 0 90 0 14 15 80
0
10
0 16
0 11
ANGLES
30
17
Worksheet 2-10
acute co-interior line straight angle
0
Matching angles
Language of maths
18 0
Worksheet 2-08
02 NCM7 2nd ed SB TXT.fm Page 71 Saturday, June 7, 2008 2:53 PM
Chapter revision
Topic test 2
1 Draw labelled diagrams of each of these angles. a ∠BKT b ∠FPR
Exercise 2-01
c angle MZQ
2 Use a protractor to measure each angle you drew in Question 1. Name the smallest angle and the largest angle.
Exercise 2-02
3 Use a protractor to draw these angles. a ∠JUG = 84° b ∠QRA = 117° d ∠DGE = 150° e ∠SAR = 96° g ∠MNB = 195° h ∠PLO = 270°
Exercise 2-03
c ∠POT = 41° f ∠XDW = 210° I ∠AMP = 300°
4 Write the name of each of these angles. Then label each one as acute, obtuse, right, reflex or straight. A R a W b c G I
Exercise 2-04
L
H
D
V
d
U
P
e
f
S M
A N P
E
R
g
h V
i M
Z
Y
Q
T P
M X
5 a What is the complement of each of these angles? i 35° ii 78° b What is the supplement of each of these angles? i 45° ii 100°
Exercise 2-05
iii 4° iii 178°
c Without measuring, find the size of the angle shown by each letter. i ii iii 70° a°
70°
25° m°
35° y°
CHAPTER 2 ANGLES
71
02 NCM7 2nd ed SB TXT.fm Page 72 Saturday, June 7, 2008 2:53 PM
Exercise 2-06
6 Find the size of each angle shown by a letter. Do not use a protractor to measure the angle. a b c 100°
m°
Exercise 2-08
a°
95°
a°
44°
b°
7 Without measuring, find the size of each angle shown by a letter. a
b
28°
c k°
m°
x° y° 122°
47°
d
e
f
140° p°
75°
110°
f°
48° x°
g
h
i r°
105°
82° t°
Exercise 2-08
8 In this diagram, name two lines that: a are parallel b are perpendicular c intersect.
q° p°
25° x° x°
G
D
A
F
H
C E Exercise 2-09
9 a Copy each diagram and mark in the alternate angle to the one shown. i ii
72
NEW CENTURY MATHS 7
x°
B
02 NCM7 2nd ed SB TXT.fm Page 73 Saturday, June 7, 2008 2:53 PM
b Without the use of instruments, find the size of each angle shown by a letter. i ii iii 38°
a°
b°
b°
a° 120°
126°
10 a Copy each diagram and mark in the corresponding angle to the one shown. i ii
Exercise 2-10
b Without the use of instruments, find the size of each angle shown by a letter. i ii iii 112°
p°
n°
117° p°
150°
a°
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 k°
112°
Exercise 2-11
iii 82°
72° x°
y° x°
CHAPTER 2 ANGLES
73
02 NCM7 2nd ed SB TXT.fm Page 74 Saturday, June 7, 2008 2:53 PM
Exercise 2-12
12 Label the marked pairs of angles as alternate, co-interior or corresponding. a b c x
x
d
e
f x x
Exercise 2-13
13 Find the size of each angle shown with a letter. a b a°
c
35°
65° k°
m°
115°
d
e
f x°
125°
q°
130° 62°
d°
g
h
i
37°
112° y° x°
74
NEW CENTURY MATHS 7
z° 62°
m°
p°
t° d°
a°
02 NCM7 2nd ed SB TXT.fm Page 75 Saturday, June 7, 2008 2:53 PM
14 Find the size of each angle shown with a letter. a
b
Exercise 2-13
c
64°
c°
x° x°
e
38°
38°
70°
m°
y°
d
z°
a°
130°
x° y° 57°
a° z° 145° c°
15 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
Exercise 2-13
16 In each diagram below, is AB parallel to CD? Give a reason for your answer each time. a b c A E E E B
Exercise 2-13
A
45° F
B
F 110°
74°
D C
C
135° G H
D
A
112° G C
H
F B
74° G H
CHAPTER 2 ANGLES
D
75
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