CHAPTER
10 Preparation
Useful skills for this chapter:
• previous experience in identifying lines and angles
G
ES
• the ability to draw lines and angles using a ruler
• the ability to identify and draw vertical, horizontal and oblique lines.
Use 8 craft sticks to make this fish. Show the fish
swimming to the left by moving only 3 craft sticks.
FI
N
AL
PA
K
K I CF F
O
Show what you know
Look around your classroom to find an object that has 2 lines that are parallel.
Draw the object. Now find and draw another object that has 2 perpendicular lines.
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10
CHAPTER
G
ES
Measurement and Geometry
Lines and angles
PA
Lines and angles are all around us. We use them when we draw, we use them
when we build and we even use them in games. Architects and builders use lines
and angles in their work.
In mathematics, a line is always a straight line. It does not include curves such as
circles or squiggles.
AL
Lines go on forever in both directions. It is impossible to draw a line that goes on
forever because we eventually run out of paper. So we usually draw part of a line
and imagine that it goes on forever. Sometimes we add arrows to show this.
line
line
N
An angle is the measurement of a turn. If you turn
through one revolution, you have turned 360°.
There are 360° of turn in a circle.
FI
360°
When two lines meet (or intersect) at a point, we
measure the number of degrees you would need to
turn from one line to the other. For example, the angle
between these two lines is 35°.
35°
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10A
Lines and angles
Lines can be horizontal, vertical or oblique.
horizontal
vertical
oblique
oblique
G
ES
Do you know how to check whether an edge is horizontal? People often use a tool
called a spirit level. Have you ever seen or used a spirit level? How does it work?
PA
Do you know how to make a vertical line? Builders sometimes use a tool
called a ‘plumb bob’. A plumb bob is a string with a pointed weight on one
end. The weight used to be made of a heavy metal called lead. The Latin
name for lead is plumbum and this is where plumb bob gets its name.
AL
To use a plumb bob, start by finding a mark on the wall, such as a nail hole.
Then hold one end of the string level with that mark. When the plumb bob
stops moving, have a friend mark the floor beside the pointed end of the
plumb bob. Then rule a line from the mark on the wall to the mark on the
floor and you have a vertical line.
Parallel and perpendicular lines
FI
N
Pairs of lines can be related. Two important relationships are parallel lines and
perpendicular lines.
parallel
perpendicular
Two lines are parallel if they will not cross no matter how far they are extended.
When we draw parallel lines, we draw two small arrows to show that the lines are
parallel. For example, these pairs of lines are parallel.
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These pairs of lines are not parallel.
G
ES
You might be able to see pairs of parallel lines
in your classroom. Choose a wall and look at
its side edges. Now imagine extending those
two edges into the air. The edges are parallel
and will not meet.
Two lines are perpendicular if they are at right angles ( 90°) to each other.
PA
When one line is perpendicular to the other, we draw a small right angle where the
lines intersect to show that the lines are at a 90° angle to each other.
AL
A builder makes sure that the walls are perpendicular to the floor and adjacent walls
are perpendicular to each other.
Line segments
FI
N
We have seen that we can draw part of a line to mean a line that goes on forever.
However, sometimes we draw part of a line and really mean only the piece instead of
the whole line. A piece of a line is called a line segment. The word ‘segment’ means
part.
line segment
Sometimes we draw part of a line and really mean half of a line. A half line is called a
ray. A ray starts at a point and then goes on forever in one direction. Think of rays of
sunshine.
ray
The arrows on the ends of rays and lines are not always used, but sometimes it is
helpful to use them.
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Angles
To make an angle we need two rays meeting at a point. We draw this by showing
two line segments meeting at a point. The rays or segments are called the arms of
the angle. The point where the arms of the angle intersect is called the vertex. It is
sometimes labelled O.
angle
G
ES
O
arms
vertex
PA
Turning the whole picture around doesn’t change the angle.
AL
Changing the length of the arms doesn’t change the angle.
angle between
the arms
angle outside
the arms
FI
N
When we cut the first slice from a cake,
we make two cuts. Those two cuts
form the arms of an angle. In fact, they
form the arms of two angles; a smaller
one between the arms and a larger one
outside the arms.
325°
Any two arms will produce two angles. We draw
a small curved arrow to mark the angle we are
talking about.
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There are many types of angles.
straight angle (180°)
acute angle
(less than 90°)
G
ES
right angle (90°)
PA
reflex angle
(between 180° and 360°)
obtuse angle
(between 90° and 180°)
Example 1
N
AL
Which angle is a right angle?
A
B
FI
C
Solution
Look at angle A. The intersecting lines are perpendicular,
so angle A is a right angle. Now look at the other angles.
Angle B is less than a right angle.
Angle C is greater than a right angle.
A
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10A
Whole class
CONNECT, APPLY AND BUILD
1 Draw lines to show angles a to f.
a Right angle
b Straight angle
c Acute angle
d Obtuse angle
e Reflex angle
f Look around the room to find an
example of each angle
PA
G
ES
2 Draw these angles, then group them according to their type.
AL
3 You can use the hands of an analogue clock to show angles. What type of angle
do the hands of a clock make when the clock shows:
a 6:00 am?
b 9:00 pm?
c 5:00 pm?
d 11:00 am?
e 2:00 pm?
f 8:00 am?
N
4 Write the following compass directions on cards and place them on the
appropriate walls of the classroom.
• North
FI
• North-east
• North-west
• South
• South-east
• South-west
• East
• West
254
Face north, then turn to face east. What turn have you made? (A right-angle
turn.) Repeat with other compass directions to show turns that are a straight
angle, a reflex angle, an acute angle and an obtuse angle.
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10A
Individual
1 Name each type of angle.
a
c
b
e
f
G
ES
d
e
AL
N
d
PA
2 Name the type of angle made by the hands of each clock.
a
b
c
FI
3 Name the angle that matches each clue. What type of angle am I?
a I am half a right angle.
b I am twice as big as a right angle.
c I am three times the size of a right
angle.
d I am half a straight angle.
e I am half a revolution.
f I am more than a right angle added
to a straight angle.
4 Draw an example of:
a a straight angle
b an obtuse angle
d a right angle
e an acute angle
c a reflex angle
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10B
Using a protractor
How do we measure the angle made when two lines intersect?
2 cm
G
ES
2 cm
Look at these two angles. We cannot use a ruler to measure the distance between the
arms; the measurement could be the same, but we know that one angle is 90° and
the other is an acute angle, which is less than 90°.
Also, if you move the ruler up or down the angle, the length changes.
PA
The best way to measure an angle is to use a protractor.
Measuring acute angles
A protractor has two sets of numbers: one set on the inside edge and one set on the
outside. The inside numbers measure angles from the right.
AL
To measure an angle, we put the 0° line
along one arm and the centrepoint of the
0° line on the vertex of the angle.
The angle is 38°.
FI
N
For example, this angle measures 38°.
The outside numbers are for measuring
angles from the left.
One arm is on the 0° line. We measure
the angle from the left using the outside
numbers.
For example, this angle measures 30°.
The angle is 30°.
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The angle is 110°.
Both of these angles are greater
than a right angle.
Measuring reflex angles
The angle is 110°.
PA
To measure a reflex angle, you
may need to rotate the protractor.
G
ES
Measuring obtuse angles
This gives you part of the
angle ( 55°). To find the full size
of the angle, you now need to
add 180° to the number of
degrees shown on the protractor.
AL
180° + 55° = 235°
CONNECT, APPLY AND BUILD
Example 2
FI
N
Whole
class
What is the size
of these angles?
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257
Solution
AL
PA
G
ES
Use a protractor to measure the angle. Place the 0° line of the protractor on the
horizontal arm of the angle. Make sure that the centrepoint of the 0° line is at the
vertex of the angle. Read the number where the other arm of the angle is pointing.
Both of these show 45° angles.
N
Example 3
FI
What is the size of this angle?
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Example 4
PA
Rotate the protractor
so the 0° line on the
protractor is on the
oblique arm of the
angle. Make sure the
centre of the 0° line
is at the vertex. Read
the number where the
other arm of the angle
is pointing. The angle
turns from the right,
so we use the inside
scale. This is a 60° angle.
G
ES
Solution
AL
What is the size of this angle?
Solution
FI
N
Turn the protractor upside
down so that the 0° line on
the protractor is on the
horizontal arm of the angle,
with the arc below the line.
Make sure the centre of
the 0° line is at the vertex.
Measure how much bigger
than 180° the angle is by
reading the inside scale. The
arm is pointing to a 60° angle.
Now add 180° to 60°.
180° + 60° = 240°
This is a 240° angle.
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259
10B
Whole class
CONNECT, APPLY AND BUILD
1 Work in pairs. One student draws an acute angle and their partner draws an
obtuse angle. Swap angles with your partner and use a protractor to measure each
other’s angles.
2 Work in pairs. Draw a reflex angle each. Swap angles with your partner and use a
protractor to measure each other’s angles.
G
ES
3 Draw a sketch of these angles then use a protractor to measure them. How close
was your sketch to the correct angle?
a 60°
b 180°
c 120°
d 270°
e 210°
f 45°
g 300°
h 155°
i 20°
j 100°
4 Make your own protractor
• Take a paper semicircle. The base is a straight angle. Write 0° at one end of the
straight angle and 180° at the other.
PA
• Fold the semicircle in half, mark the middle and draw a perpendicular line to
show a right angle. What will this angle be? Write 90°.
• Find the mark halfway between 0° and 90°. What will this angle be?
Write 45°.
AL
• Repeat this step for the mark halfway between 90° and 180° to show135°.
• Then divide the arc into increments of 10° and write the number of degrees.
Individual
FI
N
10B
1 Write the size of each angle, then name the type of angle.
a
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d
c
G
ES
2 Estimate the size of these marked angles, then use a protractor to measure
each one.
b
a
PA
c
N
AL
d
FI
3 Use a protractor to draw two intersecting lines that make these angles. Mark the
angle with a curved arrow to show clearly which angle is the answer.
a 80°
b 135°
c 25°
d 230°
e 190°
Homework
1aUse your angle marker and a protractor to find a right angle,
an acute angle, an obtuse angle and a reflex angle in your home.
bDraw each object that has those angles, then write the size of the angle.
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261
10C
Finding unknown
angles
You do not need to use a protractor to find the size of every angle.
Sometimes, the angle you need to measure is related to one you already know about.
G
ES
When two lines cross, we can see four angles at a point.
We will investigate how angles at a point are related.
A
In mathematics we use letters of the alphabet
to label angles because it helps us know which
angle we are talking about.
PA
Complementary angles
D
B
C
The three lines on the right intersect and we can
see three angles, marked A, B, and C.
We can see that angle A is a right angle ( 90°).
B
C
AL
Angles B and C make up another right angle ( 90°).
They are called complementary angles
because the two angles together make a right angle.
A
If we know the size of angle C, then we can work out the size of angle B.
N
If angle C is 35°, angle B is 90° − 35° = 55°.
Supplementary angles
FI
When two lines cross, we can see four angles.
Look at the horizontal line.
We can see that angles A and B make a straight
angle (180°).
A
D
B
C
Angles A and B are called supplementary angles because
the two angles add up to 180°. If angle A is 135°, then angle B is 180° − 135° = 45°.
Angles C and D are also supplementary angles.
Now look at the other (oblique) line. The angles A and D are supplementary because
they make a straight angle. So are the angles B and C.
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Opposite angles
When two lines intersect, the two opposite angles
are equal.
A
D
Angles A and C are the same size.
B
C
This is because:
angle A + angle D = 180° and angle C + angle D = 180°
Angles B and D are also the same size.
If angle A is 145°, then angle C is 145°.
Angles about a point
G
ES
If angle D is 35°, then angle B is 35°.
When three rays meet at one point, we get three angles.
The three angles add up to 360°.
angle A + angle B + angle C = 360°
B
A
C
If angle A is 60° and angle B is 160°, then angle C is:
PA
360° − 60° − 160° = 140°
Example 5
FI
N
AL
a What is the size of angle A?
b What is the size of angle B?
B
45°
70°
A
Solution
a Angle A + 70° = 90°
So angle A = 90° − 70° = 20°
b Angle B + 45° = 180°
So angle B = 180° − 45° = 135°
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263
Example 6
a What is the size of angle A
and angle B?
b What is the size of angle A?
120°
145°
60°
A
A
B
G
ES
160°
Solution
Remember
PA
a Angle A is opposite an angle that measures 60°, so angle A = 60°.
Angle B is opposite an angle that measures 120°, so angle B = 120°.
b Angle A + 145° + 160° = 360°
So angle A = 360° − 160° − 145° = 55°
Two angles that add up to 90° are called complementary angles.
AL
Two angles that add up to 180° are called supplementary angles.
When two lines cross each other, the opposite angles are equal.
FI
N
When three lines meet at one point, the angles they make add
up to 360°.
10 C
Whole class
CONNECT, APPLY AND BUILD
1 Work in pairs.
• The first student draws a right angle.
• Their partner draws a line from the vertex cutting the right angle into two
angles and measures one of the new angles.
• The first student then works out the size of the complementary angle by
subtracting from 90°.
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2 Work in pairs.
• The first student draws a straight angle and marks a vertex on it.
• Their partner draws a line out from the vertex and measures one of the
two angles.
• The first student then works out the supplementary angle by taking away
from 180°.
3 Work in pairs.
• The first student draws two lines crossing each other.
• Their partner measures one of the angles.
G
ES
• The first student then works out which angle is the same size, using the
principle that opposite angles are equal.
• How many pairs of opposite angles can you find?
• Measure the angles to check that they are equal.
4 As a class, discuss how you can find the size of all the angles if you know the size
of one of the angles. Draw this diagram and mark
A
the angles.
PA
a How many degrees are
there when all of the angles
are added together?
b Find angle A.
c Find angle C.
B
50°
C
AL
d Find angle B. Can you do this in more than one way?
Individual
FI
N
10C
1 Name the complementary angles.
a
b
c
E
B
A
C
F
G
I
H
d
J
K
D
L
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265
2 Name the supplementary angles.
a
b
Z
G
ES
c
W
O
P
C
D
N
M
B
A
X
Y
35°
X
PA
3 These angles are not drawn to scale. Find the unknown angle without using a
protractor.
b
a
75°
X
AL
c
d
X
45°
X
N
160°
135°
FI
160°
135°
4 Without using a protractor, write the size of the angle B, then work out the size
of angles C and D.
C
50°
B
D
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10D
Review questions
1 Draw a simple line drawing of a house. Use one or more of these words to
describe the lines in your drawing.
horizontal
vertical
parallel
perpendicular
G
ES
2 Choose from the following words to describe the angles a to f below.
right angle
obtuse angle
acute angle
straight angle
reflex angle
a
b
c
PA
d
e
f
AL
3 Draw two rays to make these kinds of angles. Mark the angle with a
curved arrow.
N
a Acute angle
d Reflex angle
b Straight angle
c Right angle
e Obtuse angle
FI
4 Use a protractor to measure these angles.
a
b
c
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267
e
d
5 Use a protractor to draw these angles.
b 120°
c 45°
d 285°
e 310°
f 20°
g 25°
h 300°
i 340°
j 205°
G
ES
a 90°
6 a Name the complementary angles.
PA
b Name the supplementary angles.
B
C
D
A
AL
E
c If angle C is equal to 45°, what size is angle D?
N
d If angle B is equal to 38°, what size is angle A?
7 For a to f, describe the kind of angle shown and write down its size.
FI
a
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d
c
f
G
ES
e
PA
g Which is the smallest angle?
h Which is the largest angle?
i What is the difference between the largest and smallest angles?
AL
8 Find the unknown angle without using a protractor.
120°
N
?
FI
9 Write the size of angle B, then work out the size of angle C and angle D.
C
77°
B
D
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