5 Angles and

constructions
Getting started
1 Estimate the size of each angle.
a b
a

c d d
c

2 State whether each of the angles in Question 1 is acute, right, obtuse or reflex.
3 ABC is a straight line.
Calculate the value of x. Show how you worked
out your answer. A
20°
4 Two angles of a triangle are 50 ° and 72 °. x°
54°
a Calculate the third angle. B
b Explain how you worked out the third angle.
C

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 5.1 A sum of 360 °

A length is the distance between two points.

You can use units, for example, metres, kilometres and
millimetres, to measure lengths.
When you change direction, you turn through an angle.
You measure the size of an angle in degrees.
A whole turn is 360 degrees. You write this as 360 °.
Why is a whole turn 360?
The Babylonians and ancient Egyptians divided a whole
turn into 360 parts as long ago as 1500 bce. This clay tablet
excavated in Shush in modern-day Iran shows this.
The Babylonians and ancient Egyptians may have used 360 parts
because some calendars at that time divided the year into 360 days.
360 is a useful number because many simple fractions of 360 are whole
numbers, including 1 , 1 , 1 , 1 and 1 .
2 3 4 5 6
You already know that the sum of the angles on a straight line is 180 °.
You also know that the sum of the angles of a
triangle is 180 °.
In this unit you will discover other useful angle facts
and use angles to solve problems.

5.1 A sum of 360 °

In this section you will … Key words
• use the fact that the sum of the angles around a point is 360 ° quadrilateral
• show and use the fact that the angles of any quadrilateral sum
add up to 360 °.

The sum of the angles on a A whole turn is 360 °. The sum of

straight line is 180 °. the angles around a point is 360 °.

53°
72° 63°
45° 65° 107°
135°

45 ° + 72 ° + 63 ° = 180 ° 65 ° + 53 ° + 107 ° + 135 ° = 360 °

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5 Angles and constructions

You can apply your algebra skills to find unknown angles, represented
by letters.
Tip
See Unit 2 for a
Worked example 5.1 reminder on using
algebra.
Here are three angles around a point.

142°
77°
a

Answer
142 ° + 77 ° = 219 °
The sum of the three angles is 360 °, so a = 360 ° – 219 ° = 141 °.

The sum of the angles of a triangle is 180 °.

A quadrilateral has four straight sides and four angles.

You can draw a straight line to divide the quadrilateral into two triangles.

The six angles of the two triangles make the angles of the quadrilateral.
The sum of the angles of each triangle is 180 °.
The sum of the angles of the quadrilateral is 2 × 180 ° = 360 °.
This result is true for any quadrilateral.
You can use the geometrical properties of shapes to calculate missing angles.

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 5.1 A sum of 360 °

Worked example 5.2

Three of the angles of a quadrilateral are each equal to 85 °. Work out the fourth angle.

Answer
3 × 85 ° = 255 °
The sum of three of the angles is 255 °.
All four angles add up to 360 °.
The fourth angle is 360 ° − 255 ° = 105 °.

Worked example 5.3

135°

80° a

This shape is a kite. Calculate the missing angles.

Answer
There is a vertical line of symmetry, so angle a = 80 °.
The four angles add up to 360 °.
135 ° + 80 ° + 80 ° = 295 °, so angle b = 360 ° – 295 ° = 65 °

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5 Angles and constructions

Exercise 5.1
Throughout this exercise, you need to apply your algebra skills to find
Tip
unknown values, represented by letters.
1 Work out the size of the angle that has a letter. See Unit 2 for a
a b reminder on using
algebra.

55°
116° a
b

c d

34°
60° c 24°
d

2 Calculate the size of each angle that has a letter.

a b
130°
120° a
155° 37°

c d

68° 52°
36° 42° d
c

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 5.1 A sum of 360 °

3 The angles in each of these diagrams are all the same size. What is
the size of each angle?
a b

4 Calculate the size of angle B in each of these triangles.

a   B b           
A C c C
28°
38°
A 25°
A 57° 49° C B B

5 Three angles of a quadrilateral are 60 °, 80 ° and 110 °. Work out

the fourth angle.
6 In these quadrilaterals, calculate the size of the angles that have a
letter.
a   b            c
63° 100°
40°
172°
95° b
35° 35°

110° a 62° c

7 All the angles of a quadrilateral are equal. What can you say about
the quadrilateral?
8 Sofia measures three of the
angles of a quadrilateral. The angles
are 125 °, 160 °
Sofia says: and 90 °.
a Show that she has made
a mistake.
b Show your answer to part a to another learner.
Is your answer clear? Could you improve your
answer?
9 One angle of a quadrilateral is 160 °. The other angles are all the
same size. x
Work out the size of the other three
angles. 68°
y
10 This shape is a parallelogram.
Work out angles x, y and z. z

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5 Angles and constructions

11 ABCD is a quadrilateral. A B
60° 50°
Angle A = 60 ° and angle B = 50 °.
Calculate angles C and D.
C

Think like a mathematician

12 All the angles of a quadrilateral are multiples of 30 °.
a When all the angles are different, show that there is only one possible set of
angles.
b If one of the angles is 90 °, find the other three angles. Show that you have
found all possible answers.

13 This is a rectangle. 30° 40°

Work out the angles that have a letter.
a
b 140°
In what order did you find the angles?
Could you find the angles in a different order?
c d
e
14 Here are two identical triangles. 20° 20°

30° 30°

60° 60°

You can put the triangles together to make 60°

a quadrilateral, as shown.
a i Find the angles of this
quadrilateral.
ii Show that the sum of the angles
is 360 °. 60°

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 5.2 Intersecting lines

b Find all the different ways of putting

the two triangles together to make a
quadrilateral. You can turn the triangle over, 30°
as shown, if you prefer.
c i Find the angles of your quadrilaterals.
ii Show that the sum is 360 ° for each
quadrilateral. 60°

Compare your answers with a partner’s answers. Have you got the
same answers? Are your diagrams the same or are they different?

Summary checklist
I know that the sum of the angles around a point is 360 °. I can use this fact to
calculate missing angles.
I know that the sum of the angles of a quadrilateral is 360 °. I can use this fact to
calculate missing angles.

5.2 Intersecting lines

In this section you will … Key words
• recognise the properties of angles on perpendicular lines intersect
and intersecting lines
opposite angles
• recognise the properties of angles on parallel lines. perpendicular
parallel
These two lines intersect at A.
transversal

When two lines intersect, opposite angles are equal.

b
a c
d

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5 Angles and constructions

In this diagram, a and c are opposite angles. Angles a and c are equal.
Tip
Also b and d are opposite angles. Angles b and d are equal.
You can use your algebra skills to find unknown angles, represented See Unit 2 for a
by letters. reminder on using
algebra.
Worked example 5.4
Work out angles p, q and r.
p
q
53° r

Answer
53 ° and p are angles on a straight line.
The sum of 53 ° and p is 180 °.
So p = 180 ° – 53 ° = 127 °.
53 ° and q are opposite angles, so q = 53 °.
p and r are opposite angles, so r = 127 °.

These two lines intersect, as shown. The angle

between the two lines is a right angle. They are
perpendicular lines.
AB and CD are two lines that do not intersect.
They are parallel.

A
The arrows show that the lines are parallel.
F
The line EF crosses the parallel lines.
This line is called a transversal.
C B

E
D

In the previous diagram there are only two

different sizes of angle. 110°
70° 70°
The four angles at the top are the same as 110°
the four angles at the bottom.
110°
70 ° + 110 ° = 180 ° 70° 70°
70 ° + 110 ° + 70 ° + 110 ° = 360 ° 110°

110
 5.2 Intersecting lines

Worked example 5.5

Work out the unknown angles, a, b, c and d, in this diagram.

b
82° a c
d

Answer
82 ° and a are angles on a straight line. The sum is 180 °. Compare the angles at the two
a = 180 ° – 82 ° = 98 ° points where the transversal
crosses the parallel lines.
82 ° and d are opposite angles. They are equal. d = 82 °
a and b are in the same position. They are equal. b = 98 °
c and b are opposite angles. They are equal. c = 98 °

Exercise 5.2
Throughout this exercise, you need to apply your algebra skills to find
unknown values, represented by letters.
1 Work out the angles that have a letter.
a b

y 114°
53° x
w z

2 Two straight lines are shown.

There are four angles. One of the angles is 87 °.
Work out the other three angles.

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5 Angles and constructions

3 Three straight lines meet at a point.

61°
d 46°
a
c
b

Calculate the values of a, b, c and d. Give reasons for your answers.

4 There are two parallel lines in this diagram. One angle is 42 °.

42°

Copy the diagram and write in the size of all the other angles.
5 Work out the unknown angles a, b and c.

113°
a

b c

6 Lines WX and YZ are parallel.

W Y

77°
c a
b

X Z

One angle is 77 °. Find a, b and c.

112
 5.2 Intersecting lines

7 AB and CD are parallel lines. Calculate s and t.

B D

t 75°
s

A C

8 Look at the diagram. 56°

a Explain why these two lines
cannot be parallel.
b Give your answer to part a to
a partner to read. Can your 126°
answer be improved?
9 This shape is made from eight a c
identical triangles.
a Sketch the diagram and label the b
other angles equal to a, b or c.
b Use arrows to mark any parallel
lines.
10 The diagram shows angle X is 45 °. X
a Calculate a. 45°
b Angle X is increased to 90 °. a
Find the new value of a.
c Angle X is increased to 119 °.
Find the new value of a. 60°
d Can angle X be more than 119 °?
Give a reason for your answer.
11 This trapezium has a pair of parallel sides. Use this fact to calculate
the missing angles.
D

A 45°
C

67°
B

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5 Angles and constructions

Think like a mathematician

12 These shapes are an equilateral triangle, a rhombus and a square.

120°

60° 60°

60°

All the sides are the same length.

Two squares and three triangles can be placed around a point, as shown.

a How do you know that the shapes fit exactly around a point?
b Find a different way to fit two squares and three triangles around a point.
c Show how to fit only triangles around a point.
d Find all the possible ways of fitting only rhombuses around a point.

Can you be sure you have found all the possible ways in part d?
Look back through this exercise. What facts do you need to remember?
Make brief notes, with diagrams, to help you remember these facts.

Summary checklist
I know the angle properties of perpendicular lines.
I know the angle properties of intersecting lines.
I know the angle properties of parallel lines and transversal lines.

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 5.3 Drawing lines and quadrilaterals

5.3 Drawing lines and quadrilaterals

In this section you will … Key words
• draw quadrilaterals, perpendicular lines and parallel lines. protractor
perpendicular
You can use a ruler and a protractor or a set square to make accurate parallel
drawings.
quadrilateral
1 Here is a line.
set square

This diagram shows how you can use a set square to draw a second
line at A that is perpendicular to the first line. Put one edge of the
set square on the line. Draw along the other edge.

A
cm
1
2
3
4
5

The next diagram shows how you can also use a protractor to draw
the same line. Put the centre mark of the flat edge of the protractor
at A so that the 90 degree marker is on the line. Draw along the flat
edge of the protractor.

30 40 50
20 60
10 140 130 70
160 150 120
170 11 80
0
10
90 0
100
80

110
70 60 50 40 3

120 130 14

A
0 1
0
50
20
16

10
0
17
0

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5 Angles and constructions

2 Here are a line and a point B.

B

This diagram shows how you can use a set square to draw a second
line through B that is perpendicular to the first line. Draw a line
along the edge of it.

8
7
6
5
4
3
2
cm 1

The next diagram shows how you can also use a protractor to draw
the same line. Draw a line along the protractor’s flat edge, with the
original line aligned with the 90 ° mark.

B
10
17

20
01
601
30 40
5014013012011 80
50 60
7
010
0
90 0

80
70 10 10
01 60 50 40 30 20 70
10 0 1
120 13 6
0 140 150 1

3 Here is a line.

You want to draw a second line that is parallel to the first line.
The lines must be 5 cm apart.
Draw a perpendicular line with a set square or a protractor.
Measure 5 cm.

5 cm

116
 5.3 Drawing lines and quadrilaterals

Use a protractor to draw the parallel line, as shown.

100 1
60
70
80
90 10
12
0
0 80 7
110 10 0 13
50 20 60 0
01 50
13

14
40

0
0
14

40

150
30
50

30
60 1

160
5 cm
20

20 10
170 1

170
10

You can also use a set square to draw the same line.
If you want to draw a quadrilateral, you need to know some of the sides
and angles. You don’t need to know all of the sides and angles.
Worked example 5.6 shows you how to draw a quadrilateral using a
ruler and a protractor.

Worked example 5.6

This is a sketch of a quadrilateral.
B

3 cm

A
130°

4 cm

110°
D

6 cm

C
Make an accurate drawing of the quadrilateral.

117
5 Angles and constructions

Continued
Answer
A
Draw a line whose length you know. Choose, for example, AD.

4 cm

B
Place the protractor at vertex A. Draw a line at an angle of
3 cm 130 °. Measure 3 cm and label the end point of the line B.

A
130°

B Now put the protractor at D. Draw a line at an angle of 110 °.

Measure 6 cm and label the end point of the line C.

110°
D

6 cm

118
 5.3 Drawing lines and quadrilaterals

Continued
B
Now join points B and C.
3 cm

A
130°

You did not need to know the angles at B and C

or the length of BC.
4 cm

110°
D

6 cm
If you started by drawing line CD, what would be
the next step?
C

Exercise 5.3
1 a Make an accurate drawing
D
of this line.
b Draw a line at B that is C 3 cm
perpendicular to AD.
c Draw a line at C that is B 4 cm
perpendicular to AD.
d Your lines from parts b and c 3 cm
A
should be parallel. Are they?
2 a Make an accurate drawing of this diagram. X
b Draw a perpendicular line from X to line YZ. Label the 6 cm
intersection as P.
c Measure: i XP ii YP
Y 52°
d Compare your answers to part c with a learner’s answers.
Do you have the same answers? If not, check your
accuracy. 7 cm
Z

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5 Angles and constructions

3 a Make an accurate drawing of this diagram. The length of

PC is 4.5 cm.

B
4.5 cm
63°
C
A

b Draw a line through P that is parallel to AB.

4 a Make an accurate drawing of this quadrilateral.
A 6.9 cm B
120° 110°
5.1 cm
8.2 cm

b Measure CD.
c The length of CD should be 13.3 cm. Is your measurement in
part b close to 13.3 cm? If not, check your drawing.
5 This diagram has two pairs of parallel lines. 4 cm
a Make an accurate drawing of the diagram.
A 70° B
b Draw the line AC and measure the length of
this line.
3 cm
c The length of AC should be 5.4 cm. Is your
measurement in part b close to 5.4 cm? If not,
check your drawing. D C

6 Three angles of a quadrilateral are 60 °, 75 ° and 130 °.

a Calculate the fourth angle of the quadrilateral.
b Draw a quadrilateral with these four angles. The
60 ° angle must be opposite the 75 ° angle, as
shown in this diagram. opposite
75°
c Draw a different quadrilateral with the same four angles 60°
angles. This time put the 60 ° angle opposite the
130 ° angle.

120
 5.3 Drawing lines and quadrilaterals

Compare your quadrilaterals with a partner’s quadrilaterals.

In what way are your diagrams the same? In what way are
your diagrams different?

7 Try to draw a quadrilateral where three of the angles are 120 °.

What happens? Why?

Think like a mathematician

8 a Make an accurate drawing of this quadrilateral. 6 cm
b You were given the size of every side and
125°
angle. Did you need all this information to
draw the quadrilateral? What is the least 7.5 cm 8 cm
number of measurements you need to draw
the quadrilateral accurately?
c Describe the least set of measurements
you need in general to draw a quadrilateral 65° 80°
accurately. 10.5 cm

9 Use the measurements shown to make an accurate drawing of this

quadrilateral.

35°
8 cm

122°

8 cm 35°

Summary checklist
I can draw perpendicular lines.
I can draw parallel lines.
I can draw quadrilaterals.

121
5 Angles and constructions

Check your progress

1 Here are three angles. Two of the angles are equal.
The diagram shows one angle of 100 °. Calculate the other two angles. 100°
There are two possible answers.
2 a Calculate angle C. 4 cm
A B
b Make an accurate drawing of the 118° 106°
quadrilateral. 3 cm

c Measure CD.
D

C
3 Work out the unknown angles a, b, c and d.

72°
55°

c d
a b

4 a Make an accurate drawing of this diagram.

b Measure angle C.
P

5.5 cm
Q

3 cm
A
4 cm
B

122
 Project 2
Clock rectangles
The diagram shows a clock face with 12 equally spaced points
around a circle.
Can you find four points that join together to make a rectangle?
How do you know it is a rectangle?
Can you find more than one rectangle?
How many different rectangles is it possible to draw on
a clock face?
Once you have explored rectangles on a clock face, here are some more questions you
might like to explore.
Consider a clock face that has only eight equally spaced points around the edge.
How many rectangles can you find now?
What about a clock face with 18 equally spaced points?
Is there a way to predict the number of rectangles for a clock face with any number of
equally spaced points?
What other quadrilaterals can you find on different clock faces?
Which of these is it possible to draw?

• kite
• parallelogram
• rhombus
• trapezium
Do your answers depend on
the number of equally
spaced points?

123
6 Collecting data
Getting started
1 a What is a questionnaire?
b Why would you use a questionnaire?
2 You are planning an investigation of the vehicles using a road.
a List four statistical questions you could ask.
b List a prediction you could make for each question.
3 Here are two predictions:
• 12-year-old boys are heavier than 12-year-old girls.
• 12-year-old boys are taller than 12-year-old girls.
a How could you collect data to test this prediction?
b How would you analyse the data?
4 Here are 20 test marks.
7 8 8 9 12 14 14 14 15 15 15 16 16 17 18 18 18 18 18 20
a Which mark is the mode?
b Which mark is the median?
c The sum of the marks is 290. Work out the mean mark.

Look at the following three situations.

1.  eople use a website to make hotel reservations online.

P
The company that manages the website wants to know if
people find the site easy to use. The company also wants
feedback on the quality of the hotels.

2. A drug company is testing a new drug to help people sleep.

There are already drugs available for this. The company
wants to know whether people think the new drug is better
than existing drugs.

124
 6.1 Conducting an investigation

3.  eachers think that drivers are exceeding the speed limit

T
when they drive past a school. This is dangerous for the
learners. The teachers want to find out if the drivers are
speeding.

In all three cases you need to collect data. You can collect data in
different ways.
In the first case, the company could use an online questionnaire.
In the second case, the drug company could interview people using the
new drug and ask them about their experience.
In the third case, the teachers could record the speeds of cars as they
drive past the school.
In all three cases you have statistical questions. You collect data to
answer the questions.
You need to decide:
• what sort of data you want to collect
• how you will collect the data.

6.1 Conducting an investigation

In this section you will … Key words
• learn how to collect data to investigate statistical questions. continuous data
categorical data
Look at these examples of statistical questions. data
1 How many brothers do the learners in your class have? discrete data
2 What is the average mass of a baby born in your country?
prediction
3 What sports do learners in your school like to watch?
statistical
To answer a statistical question you need to collect data. question
There are different types of data. The number of brothers you have, the
mass of a baby and a sport you watch are all examples of different types
of data.
The type of data needed to answer Question 1 is discrete data.
The values can be only 0, 1, 2, … Discrete data can take particular
values only.

125