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EMVIC73ed Ch07

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7

Geometry

Maths in context: Cable stayed bridges

Skills in geometry are essential for all the practical concrete panels forming the bridge deck, supported
occupations including animators, architects, by inverted triangular steel trusses. It is about 100 m
astronomers, bricklayers, boilermaker, builders, above ground and suspended by eight cables from
carpenters, concreters, construction workers, single pylon. The pylon is 81.5 m high and must bear
designers, electricians, engineers, jewellers, the total load of the bridge via the cables. Engineers
navigators, plumbers, ship builders and surveyors. had to solve the critical problem of how to balance
the immense weight of the bridge from a single point
Cable stayed bridges use pylons with attached cables
at the top of the pylon.
that fan out to support the bridge deck, such as the
Langkawi Sky Bridge in Malaysia. Here we can see Cable stayed bridges in Australia include the Anzac
the application of angles at a point, angles formed Bridge in Sydney, the Eleanor Schonell Bridge and
by parallel lines and angles in a triangle. The bridge the Kurilpa Bridge in Brisbane, the Seafarers Bridge
is a 125 m curved pedestrian bridge with steel and in Melbourne and the Batman Bridge in Tasmania.

Essential Mathematics for the Victorian Curriculum ISBN 978-1-009-48064-2 © Greenwood et al. 2024 Cambridge University Press
Year 7 Photocopying is restricted under law and this material must not be transferred to another party.
Chapter contents
7A Points, lines, intervals and angles (CONSOLIDATING)
7B Adjacent and vertically opposite angles
7C Transversal and parallel lines
7D Solving compound problems with parallel lines (EXTENDING)
7E Classifying and constructing triangles
7F Classifying quadrilaterals and other polygons
7G Angle sum of a triangle
7H Symmetry
7I Reﬂection and rotation
7J Translation
7K Drawing solids
7L Nets of solids

Victorian Curriculum 2.0

This chapter covers the following content descriptors in the
Victorian Curriculum 2.0:

SPACE
VC2M7SP01, VC2M7SP02, VC2M7SP03, VC2M7SP04

MEASUREMENT
VC2M7M04, VC2M7M05
Please refer to the curriculum support documentation in the
teacher resources for a full and comprehensive mapping of this
chapter to the related curriculum content descriptors.
© VCAA

Online resources
A host of additional online resources are included as part of your
Interactive Textbook, including HOTmaths content, video
demonstrations of all worked examples, auto-marked quizzes and
much more.

7A Points, lines, intervals and angles CONSOLIDATING

LEARNING INTENTIONS
• To know the meaning of the terms point, vertex, intersection, line, ray, segment and plane
• To know the meaning of the terms acute, right, obtuse, straight, reﬂex and revolution
• To understand what collinear points and concurrent lines are
• To be able to name lines, segments, rays and angles in terms of labelled points in diagrams
• To be able to measure angles using protractors
• To be able to draw angles of a given size

The fundamental building blocks of geometry

are the point, line and plane. They are the basic
objects used to construct angles, triangles and other
more complex shapes and objects. Theoretically,
points and lines do not occupy any area, but we
can represent them on a page using drawing
instruments.

Angles are usually described using the unit of

measurement called the degree, where 360 degrees
(360°) describes one full turn. The idea to divide a
Engineers calculate the angles between the straight
circle into 360° dates back to the Babylonians, who structural supports used on bridges such as the angles seen
used a sexagesimal number system based on the here on the Story Bridge, Brisbane.
number 60. Because both 60 and 360 have a large
number of factors, many fractions of these numbers
are very easy to calculate.

Lesson starter: Estimating angles

How good are you at estimating the size of
angles? Estimate the size of these angles and
then check with a protractor.

Alternatively, construct an angle using

interactive geometry software. Estimate and
then check your angle using the angle-measuring tool.

KEY IDEAS
■ A point is usually labelled with a capital letter. P
■ A line passing through two points, A and B, can be called line AB or B
line BA and extends indefinitely in both directions.
A
• upper-case letters are usually used to label points.

■ A plane is a flat surface and extends indefinitely.

■ Points that all lie on a single line are collinear. C

B
A

■ If two lines meet, an intersection point is formed.

■ Three or more lines that intersect at the same point are concurrent.

■ A line segment (or interval) is part of a line with a fixed length B

and end points. If the end points are A and B then it would be named A
segment AB or segment BA (or interval AB or interval BA).
■ A ray AB is a part of a line with one end point A and passing through
B
point B. It extends indefinitely in one direction.
A
■ When two rays (or lines) meet, an angle is formed at the intersection
arm
point called the vertex. The two rays are called arms of the angle.
vertex
arm
■ An angle is named using three points, with the vertex as
A or
the middle point. A common type of notation is ∠ABC
or ∠CBA. The measure of the angle is a°.
B a°
• lower-case letters are often used to represent the size
of an angle. C

■ Part of a circle called an arc is used to mark an angle. B

A
■ These two lines are parallel. This is written AB ∥ DC. C
D

■ These two lines are perpendicular. This is written AB ⊥ CD. C B

A D

■ The markings on this diagram show that AB = CD, AD = BC, A B

∠BAD = ∠BCD and ∠ABC = ∠ADC. ×

×
D C

■ Angles are classified according to their size.

Angle type Size Examples

acute between 0° and 90°

right 90°

obtuse between 90° and 180°

straight 180°
reﬂex between 180° and 360°

revolution 360°

■ A protractor can be used to measure angles to within an accuracy of about half a degree. Some
protractors have increasing scales marked both clockwise and anticlockwise from zero. To use a
protractor:
1 Place the centre of the protractor on the vertex of the angle.
2 Align the base line of the protractor along one arm of the angle.
3 Measure the angle using the other arm and the scale on the protractor.
4 A reflex angle can be measured by subtracting a measured angle from 360°.

BUILDING UNDERSTANDING
1 Describe or draw the following objects.
a a point P b a line AN
c an angle ∠ABC d a ray ST
e a plane f three collinear points A, B and C
2 Match the words line, segment, ray,
collinear or concurrent to the correct
description.
a Starts from a point and extends
indefinitely in one direction.
b Extends indefinitely in both directions,
passing through two points.
c Starts and ends at two points.
d Three points in a straight line.
e Three lines intersecting at the Why would we use the geometric term rays to describe
same point. the sunlight showing through the trees?

3 Without using a protractor, draw or describe an example of the following types of angles.
a acute b right
c obtuse d straight
e reflex f revolution
4 What is the size of the angle measured with these protractors?
a b

c d

Example 1 Naming objects

Name this line segment and angle.

a b P
A B

Q
R

SOLUTION EXPLANATION
a segment AB Segment BA, interval AB or interval BA are also
acceptable.

b ∠PQR Point Q is the vertex so the letter Q sits in

between P and R. ∠RQP is also correct.

Now you try

Name this ray and angle.
a b B
A A

B
C

Example 2 Classifying and measuring angles

For the angles shown, state the type of angle and measure its size.
a A b G c
D
O

B E E
O F

SOLUTION EXPLANATION
a acute A
∠AOB = 60°

B
O
The angle is an acute angle so read from the
inner scale, starting at zero.

b obtuse G
∠EFG = 125°

E
F
The angle is an obtuse angle so read from the
outer scale, starting at zero.

c reflex D
O
obtuse ∠DOE = 130°
reflex ∠DOE = 360° − 130°
= 230°

E
First measure the obtuse angle before
subtracting from 360° to obtain the reflex angle.

Now you try

For the angles shown, state the type of angle and measure its size.
a B b E c
A O D

O E
G

Example 3 Drawing angles

Use a protractor to draw each of the following angles.

a ∠AOB = 65° b ∠WXY = 130° c ∠MNO = 260°

SOLUTION EXPLANATION
a A Step 1: Draw a base line OB.
Step 2: Align the protractor along the base line
with the centre at point O.
Step 3: Measure 65° and mark a point, A.
Step 4: Draw the arm OA.
B
O
b Y Step 1: Draw a base line XW.
Step 2: Align the protractor along the base line
with the centre at point X.
Step 3: Measure 130° and mark a point, Y.
Step 4: Draw the arm XY.
W
X
c O Step 1: Draw an angle of 360° − 260° = 100°.
Step 2: Mark the reflex angle on the opposite
side to the obtuse angle of 100°.
Alternatively, draw a 180° angle and measure
an 80° angle to add to the 180° angle.

M
N

Now you try

Use a protractor to draw each of the following angles.
a ∠AOB = 30° b ∠WXY = 170° c ∠MNO = 320°

Exercise 7A
FLUENCY 1–3, 4–6(1/2), 7, 8 2, 3, 4–6(1/2), 7, 8 2, 3–6(1/2), 7, 8

1 Name these line segments and angles.

Example 1a a i Q ii X
P Y
Example 1b b i A ii T

B S
U
O
Example 1 2 Name the following objects.
a T b D
C

c d
B

A
C

e f S
Q

P T

Example 1b 3 Name the angle marked by the arc in these diagrams.

a A b B
B

C A
C

D
D

O
c B d O

E
D
C E
A
A B C D

Example 2 4 For the angles shown, state the type of angle and measure its size.
a b c

d e f

g h i

Example 3 5 Use a protractor to draw each of the following angles.

a 40° b 75° c 90° d 135° e 175°
f 205° g 260° h 270° i 295° j 352°

6 For each of the angles marked in the situations shown, measure:

a the angle that this ramp makes with the ground

b the angle the Sun’s rays make with the ground

c the angle or pitch of this roof

d the angle between this laptop screen and the keyboard.

7 Name the set of three labelled points that are collinear in these diagrams.
a b B D

D C
A
B
C A

8 State whether the following sets of lines are concurrent.

a b

PROBLEM-SOLVING 9, 10 9–11 11, 12

9 Count the number of angles formed inside these shapes. Count all angles, including ones that may be
the same size and those angles that are divided by another segment.
a b

10 A clock face is numbered 1 to 12. Find the angle the minute hand turns in:
a 30 minutes b 45 minutes c 5 minutes d 20 minutes
e 1 minute f 9 minutes g 10.5 minutes h 21.5 minutes

11 A clock face is numbered 1 to 12. Find the angle between the hour hand and the minute hand at:
a 6 : 00 p.m. b 3 : 00 p.m. c 4 : 00 p.m. d 11 : 00 a.m.

12 Find the angle between the hour hand and the minute hand of a clock at these times.
a 10 : 10 a.m. b 4 : 45 a.m.
c 11 : 10 p.m. d 2 : 25 a.m.
e 7 : 16 p.m. f 9 : 17 p.m.

REASONING 13 13, 14 14, 15

13 a If points A, B and C are collinear and points A, B and D are collinear, does this mean that points B,
C and D are also collinear? Use a diagram to check.
b If points A, B and C are collinear and points C, D and E are collinear, does this mean that points B,
C and D are also collinear? Use a diagram to check.

14 An acute angle ∠AOB is equal to 60°. Why is it unnecessary to use a protractor to A

work out the size of the reflex angle ∠AOB?

60°
O
? B

15 The arrow on this dial starts in an upright position. It then turns by a given number
of degrees clockwise or anticlockwise. Answer with an acute or obtuse angle. ?
a Find the angle between the arrow in its final position with the arrow in its
original position, as shown in the diagram opposite, which illustrates part i. 290°
Answer with an acute or obtuse angle.
i 290° clockwise ii 290° anticlockwise
iii 450° clockwise iv 450° anticlockwise
v 1000° clockwise vi 1000° anticlockwise
b Did it matter to the answer if the dial was turning clockwise or anticlockwise?
c Explain how you calculated your answer for turns larger than 360°.

ENRICHMENT: How many segments? – – 16

16 A line contains a certain number of labelled points. For example, this line has three points.
a Copy and complete this table by counting the total number of segments for the given number of
labelled points.

C
A B

Number of points 1 2 3 4 5 6
Number of segments

b Explain any patterns you see in the table. Is there a quick way of finding the next number in
the table?
c If n is the number of points on the line, can you find a rule (in terms of n) for the number of
segments? Test your rule to see if it works for at least three cases, and try to explain why the rule
works in general.

7B Adjacent and vertically opposite angles

LEARNING INTENTIONS
• To know the meaning of the terms adjacent, complementary, supplementary, vertically opposite and
perpendicular
• To be able to work with vertically opposite angles and perpendicular lines
• To be able to ﬁnd angles at a point using angle sums of 90°,180° and 360°

Not all angles in a diagram or construction need to be

measured directly. Special relationships exist between
pairs of angles at a point and this allows some angles
to be calculated exactly without measurement, even if
diagrams are not drawn to scale.

Lesson starter: Special pair of angles

By making a drawing or using interactive geometry
People who calculate angles formed by intersecting
software, construct the diagrams below. Measure the lines include: glass cutters who design and construct
two marked angles. What do you notice about the two stained-glass windows; jewellers who cut gemstones
at precise angles; and quilt makers who design and
marked angles? sew geometric shapes.

A A A B
O e°
B a° O
b°
a° C f°
b° C
O D C
B

KEY IDEAS
■ Adjacent angles are side by side and share a vertex and an arm. ∠AOB
∠ AOB and ∠BOC in this diagram are adjacent angles. A

B ∠BOC

C
O

■ Two adjacent angles in a right angle are complementary.

They add to 90°.
• If the value of a is 30, then the value of b is 60 because 30° + 60° = 90°. b°
We say that 30° is the complement of 60°. a°
a + b = 90

■ Two adjacent angles on a straight line are supplementary.

They add to 180°.
If the value of b is 50, then the value of a is 130 because 50° + 130° = 180°. a° b°
We say that 130° is the supplement of 50°. a + b = 180

■ Angles in a revolution have a sum of 360°. a°

b°
a + b = 360

■ Vertically opposite angles are formed when two lines intersect. b°

The opposite angles are equal. The name comes from the fact that a° a°
the pair of angles has a common vertex and they sit in opposite b°
positions across the vertex.
■ Perpendicular lines meet at a right angle (90°). We write AB ⊥ CD. D
B

A
C

BUILDING UNDERSTANDING
a°
1 a Measure the angles a° and b° in this diagram. b°
b Calculate a + b. Is your answer 90? If not, check your measurements.
c State the missing word: a° and b° are ____________ angles.

2 a Measure the angles a° and b° in this diagram.

a° b°
b Calculate a + b. Is your answer 180? If not,
check your measurements.
c State the missing word: a° and b° are
____________ angles.

3 a Measure the angles a°, b°, c° and d° in this diagram.

b What do you notice about the sum of the four angles? a°
c State the missing words: b° and d° are d° b°
_____________ angles. c°

4 a Name the angle that is complementary to ∠ AOB in this diagram. A B

C
O
b Name the two angles that are supplementary to ∠ AOB in this diagram. D
C
O
A
B
c Name the angle that is vertically opposite to ∠ AOB in this diagram. B C
O
A D

Example 4 Finding angles at a point using complementary and

supplementary angles

Without using a protractor, find the value of the pronumeral a.

a b
55°
a° a°
35°

SOLUTION EXPLANATION
a a = 90 − 35 Angles in a right angle add to 90°.
= 55 a + 35 = 90

b a = 180 − 55 Angles on a straight line add to 180°.

= 125 a + 55 = 180

Now you try

Without using a protractor, find the value of a for the following.
a b

A a°
a°
130°
65°

Example 5 Finding angles at a point using other properties

Without using a protractor, find the value of the pronumeral a.

a b a°
47°
a° 120°

SOLUTION EXPLANATION
a a = 47 Vertically opposite angles are equal.

b a = 360 − (90 + 120) The sum of angles in a revolution is 360°.

= 360 − 210 a + 90 + 120 = 360
= 150 a is the difference between 210° and 360°.

Now you try

Without using a protractor, find the value of a for the following.
a b

a° 116° a°
50°

Exercise 7B
FLUENCY 1, 2–5(1/2) 1–5(1/2) 2–6(1/2)

Example 4a 1 Without using a protractor, find the value of the pronumeral a.

a b c
a°
a° 60° a°
20° 15°

Example 4 2 Without using a protractor, find the value of the pronumeral a. (The diagrams shown may not be drawn
to scale.)
a 40° b c
75°
a°
a°
30° a°
d e f
a°
110° 120° a°
a° 45°

g h i
a° a°
a°
50° 60° 49°

Example 5 3 Without using a protractor, find the value of the pronumeral a. (The diagrams shown may not be drawn
to scale.)
a b c
a° a°
77°
115° a°

37°
d a° e a° f

120° 220°
a°

g h i

160° 135° 100°

a° 140°
a° a°

4 For each of the given pairs of angles, write C if they are complementary, S if they are supplementary
or N if they are neither.
a 21° , 79° b 130° , 60° c 98° , 82° d 180° , 90°
e 17° , 73° f 31° , 59° g 68° , 22° h 93° , 87°

5 Write a statement like AB ⊥ CD for these pairs of perpendicular line segments.

a H b S U c W
T
Y
E

F V
X
G
6 Without using a protractor, find the value of a in these diagrams.
a b c
40°
30° a° 30°
a°
a°
110°
65° 100°

d e f

a° 40°
a° 45° 135°
a°

PROBLEM-SOLVING 7 7-8(1/2) 8, 9

7 Decide whether the given angle measurements are possible in the diagrams below. Give reasons.
a b c

60°
140°
25° 40° 50°
310°
d e f
42° 138°
35°
80° 250°

35°

8 Find the value of a in these diagrams.

a a° b c
a° a° a°
(2a)° (3a)°
(2a)°
a°

d e f
(2a)°
(a + 10)° (3a)°
(a − 60)°
(a − 10)° (a + 60)°

9 A pizza is divided between four people. Bella is to get twice as much as Bobo, who gets twice as
much as Rick, who gets twice as much as Marie. Assuming the pizza is cut into slices from the centre
outwards, find the angle at the centre of the pizza for Marie’s piece.

REASONING 10 10, 11 11, 12

10 a Is it possible for two acute angles to be supplementary? Explain why or why not.
b Is it possible for two acute angles to be complementary? Explain why or why not.

11 Write down a rule connecting the letters in these diagrams, e.g. a + b = 180.
a b c
a° b° a°
b°
c° b°
a°

12 What is the minimum number of angles you would need to measure in this
diagram if you wanted to know all the angles? Explain your answer.

ENRICHMENT: Pentagon turns – – 13

13 Consider walking around a path represented by this regular pentagon. All

a°
sides have the same length and all internal angles are equal. At each corner
a° b°
(vertex) you turn an angle of a, as marked.
b° b°
a°

a° b° b°

a°

a How many degrees would you turn in total after walking Regular shape a b
around the entire shape? Assume that you face the same Triangle
direction at the end as you did at the start. Square
b Find the value of a.
Pentagon
c Find the value of b.
Hexagon
d Explore the outside and inside angles of other regular
Heptagon
polygons using the same idea. Complete this table to
Octagon
summarise your results.

Each of the identical shapes that make up this quilt design is called a rhombus.
Four line segments form the sides of each rhombus. How many lines intersect
at each vertex? How many angles meet at each vertex? Can you determine the
size of the angles in each pattern piece?

7C Transversal and parallel lines

LEARNING INTENTIONS
• To know the meaning of the terms transversal, corresponding, alternate, cointerior and parallel
• To be able to identify angles that are in a given relation to another angle (for example, identifying an angle
cointerior to a given angle)
• To be able to ﬁnd the size of angles when a transversal crosses parallel lines
• To be able to determine whether two lines are parallel using angles involving a transversal

When a line, called a transversal, cuts

two or more other lines a number
of angles are formed. Pairs of these
angles are corresponding, alternate or
cointerior angles, depending on their
relative position. If the transversal
cuts parallel lines, then there is a
relationship between the sizes of the
special pairs of angles that are formed.

Surveyors use parallel line geometry to accurately measure the angles and
mark the parallel lines for angle parking.

Lesson starter: What’s formed by a transversal?

Draw a pair of parallel lines using either:

• two sides of a ruler; or

• interactive geometry software (parallel line tool).

Then cross the two lines with a third line (transversal) at any angles.

Measure each of the eight angles formed and discuss what you find.
If interactive geometry software is used, drag the transversal to see if
your observations apply to all the cases that you observe.

KEY IDEAS
■ A transversal is a line passing through two or more
other lines that are usually, but not necessarily, parallel.

transversal transversal

■ A transversal crossing two lines will form special pairs

of angles. These are:
• corresponding (in corresponding positions)
×

• alternate (on opposite sides of the transversal and

inside the other two lines)
×

• cointerior (on the same side of the transversal and

inside the other two lines). ×
×

■ Parallel lines are marked with the same arrow set. B

• If AB is parallel to CD, then we write AB | |CD. D
A

■ If a transversal crosses two parallel lines, then:

• corresponding angles are equal
• alternate angles are equal
• cointerior angles are supplementary (i.e. sum to 180°).
corresponding alternate cointerior

a° b° a°
b° a°
a° b° b° a°
a° b° b°
a=b

a=b a=b a + b = 180 a + b = 180

a=b

BUILDING UNDERSTANDING
1 Use a protractor to measure each of the eight angles in
this diagram.
a How many different angle measurements did
you find?
b Do you think that the two lines cut by the
transversal are parallel?

2 Use a protractor to measure each of the eight

angles in this diagram.
a How many different angle measurements did
you find?
b Do you think that the two lines cut by the transversal
are parallel?

3 Choose the word equal or supplementary to complete these sentences.

If a transversal cuts two parallel lines, then:
a alternate angles are _____________.
b cointerior angles are _____________.
c corresponding angles are ________.
d vertically opposite angles are ______.

Example 6 Naming pairs of angles

Name the angle that is: A H

a corresponding to ∠ ABF
G
b alternate to ∠ ABF
B F
c cointerior to ∠ ABF
C
d vertically opposite to ∠ ABF.
D E

SOLUTION EXPLANATION
a ∠HFG (or ∠GFH) H These two angles are in
corresponding positions, both
G above and on the right of the
F
intersection.

b ∠EFB (or ∠BFE ) These two angles are on opposite

G sides of the transversal and inside
F the other two lines.

c ∠HFB (or ∠BFH ) H These two angles are on the same

side of the transversal and inside
× the other two lines.
B
F

d ∠CBD (or ∠DBC ) These two angles sit in opposite

positions across the vertex B.
B
C

Now you try

Using the same diagram as in the example above, name the angle that is: A H
a corresponding to ∠EFB
b alternate to ∠DBF G
c cointerior to ∠EFB B F
d vertically opposite to ∠GFE. C

D E

Example 7 Finding angles in parallel lines

Find the value of a in these diagrams and give a reason for each answer.
a b c
115°
a° a° a°
55°
110°

SOLUTION EXPLANATION
a a = 115 Alternate angles in parallel lines cut by a
alternate angles in parallel lines transversal are equal.

b a = 55 Corresponding angles in parallel lines are

corresponding angles in parallel lines equal.

c a = 180 − 110 Cointerior angles in parallel lines sum to 180°.

= 70
cointerior angles in parallel lines

Now you try

Find the value of a in these diagrams and give a reason for each answer.
a b c
a° 65°
50°
120° a°
a°

Example 8 Determining whether two lines are parallel

Giving reasons, state whether the two lines cut by the transversal are parallel.
a b
75°
78° 58°
122°

SOLUTION EXPLANATION
a not parallel Parallel lines cut by a transversal have equal
Alternate angles are not equal. alternate angles.

b parallel 122° + 58° = 180°

The cointerior angles sum to 180°. Cointerior angles inside parallel lines are
supplementary (i.e. sum to 180°).

Now you try

Giving reasons, state whether the two lines cut by the transversal are parallel.
a b

109° 60°
71°

55°

Exercise 7C
FLUENCY 1, 2, 4–6(1/2) 2, 4–6(1/2) 3, 4–6(1/2)

Example 6 1 Name the angle that is: A

a corresponding to ∠BGA
b alternate to ∠FGH B
F
c cointerior to ∠CHG G
d vertically opposite to ∠FGA. E C
H

Example 6 2 Name the angle that is: C

a corresponding to ∠ABE F
G
b alternate to ∠ABE E
H B
c cointerior to ∠ABE A
d vertically opposite to ∠ABE. D

Example 6 3 Name the angle that is: C D

a corresponding to ∠EBH
b alternate to ∠EBH E F
B
c cointerior to ∠EBH A
d vertically opposite to ∠EBH.
H G

Example 7 4 Find the value of a in these diagrams, giving a reason.

a b c
110°
a°
130° a° a°

70°

d e f 67°
a° a°
130°
a°
120°

g h i
a°
115° a°
a° 62° 100°

j k l
64° 116°
117° a°
a° a°

Example 7 5 Find the value of each pronumeral in the following diagrams.

a b c
a° 120° d°
70° b° c° b° c°
c° b°
a°
a° 82°
d e f
a° 119°
85°
b° c° a° a°
b° b°
c°

Example 8 6 Giving reasons, state whether the two lines cut by the transversal are parallel.
a 59° b c
81° 112°
58° 68°
81°

d e f
132°
132°
79° 78° 100°
60°

PROBLEM-SOLVING 7–8(1/2) 7–8(1/2), 9 8(1/2), 9, 10

7 Find the value of a in these diagrams.

a b c
35°
a° 41° a°
a°

70°

d e f 141°
60°
a°
a°
150°
a°

8 Find the value of a in these diagrams.

a b c
80° a°
115° a°

a°
62°

d e f

a° a° 57°

42°
a°
67°

g h i
a° a°

80° 130° 121°

a°

9 A transversal cuts a set of three parallel lines.

a How many angles are formed?
b How many angles of different sizes are formed if the transversal is not perpendicular to the three lines?

10 Two roads merge into a freeway at the same angle, as shown.

Find the size of the obtuse angle, a°, between the parallel roads
and the freeway. a°
60°

freeway

REASONING 11 11, 12 11–13

11 a This diagram includes two triangles with two sides that are parallel. 20°
Give a reason why: b°
i a = 20 ii b = 45. 45°
a°

b Now find the values of a and b in the diagrams below.

i 25° ii iii
b°
b° a°
a°
a° 50° 35° b°
41° 25°

35°

12 This shape is a parallelogram with two pairs of parallel sides.

a° c°
a Use the 60° angle to find the value of a and b.
60° b°
b Find the value of c.
c What do you notice about the angles inside a parallelogram?

13 Explain why these diagrams do not contain a pair of parallel lines.

a b c 130°
130°
40°
150°
140°

300°

ENRICHMENT: Adding parallel lines – – 14, 15

14 Consider this triangle and parallel lines.

a Giving a reason for your answer, name an angle equal to: A B C
i ∠ ABD ii ∠CBE.

D E
b What do you know about the three angles ∠ ABD, ∠DBE and ∠CBE?
c What do these results tell you about the three inside angles of the
triangle BDE? Is this true for any triangle? Try a new diagram to check.

15 Use the ideas explored in Question 14 to show that the angles inside a quadrilateral
(i.e. a four-sided shape) must sum to 360°. Use this diagram to help.

7D Solving compound problems with parallel

lines EXTENDING
LEARNING INTENTION
• To be able to combine facts involving parallel lines and other geometric properties to ﬁnd missing angles in
a diagram

Parallel lines are at the foundation of construction

in all its forms. Imagine the sorts of problems
engineers and builders would face if drawings and
constructions could not accurately use and apply
parallel lines. Angles formed by intersecting beams
would be difficult to calculate and could not be
transferred to other parts of the building.

A builder makes sure that the roof rafters are all parallel, the
ceiling joists are horizontal and parallel, and the wall studs
are perpendicular and parallel.

Lesson starter: Not so obvious

Some geometrical problems require a combination of two or more ideas
a°
before a solution can be found. This diagram includes an angle of size a°.

• Discuss if it is possible to find the value of a. 65°

• Describe the steps you would take to find the value of a. Discuss your
reasons for each step.

KEY IDEAS
■ Some geometrical problems involve more than one step. A E
Step 1: ∠ABC = 75° (corresponding angles on parallel lines)
Step 2: a = 360 − 75 (angles in a revolution sum to 360°)
= 285
B 75°

a° C D

BUILDING UNDERSTANDING
1 In these diagrams, first find the value of a and then find the value of b.
a b c
a° b° a° 74°
65°
b° 125° a°
b°

2 Name the angle in these diagrams (e.g. ∠ABC) that you would need to find first before finding
the value of a. Then find the value of a.
a E b C c A

B 60° 70°
A a°
a° B E 60° D
A
F G a°
70°
D
C D B
F

Example 9 Finding angles with two steps

Find the value of a in these diagrams.

a A B b D
170° F
A
a°
D

C 60° 70°
E a° C
B

SOLUTION EXPLANATION
a ∠BDE = 360° − 90° − 170° Angles in a revolution add to 360°.
= 100° ∠ABC corresponds with ∠BDE, and
∴ a = 100
BC and DE are parallel.

b ∠ABC = 180° − 70° ∠ABC and ∠BCD are cointerior

= 110° angles, with AB and DC parallel.
∴a = 110 − 60
∠ABC = 110° and a° + 60° = 110°
= 50

Now you try

Find the value of a in these diagrams.
a A G b C D F
H
50°
60° C a°
F
A B E G J
B 105°
a° I
E
D

Exercise 7D
FLUENCY 1, 2(1/2), 3, 4(1/2) 2, 4 2(1/2), 4(1/2)

Example 9a 1 Find the value of a in these diagrams.

a E b

C 65°
a°

D F
165°
a°
B
A
Example 9a 2 Find the value of a in these diagrams.
a 300° b A c A
E B a°
B a° 150°
B F
E F
a°
A C
D C E
D
C 65°

D
d D e F f
A G
a°
E B
130° 107° a°
C F E a°
H
B G D I
C
57°
A
H

Example 9b 3 Find the value of a in these diagrams.

a b B
C a°
55° 60°
30° 100°
a° A

Example 9b 4 a C b A c A
a°
D D
62° 30°
45°
38° B
a° 85° D
B
A 75° B a°
C
C
d A B e E D C f E
a° A
a° 80°
a°
45° D
40°
C D E A B

35°
C
B

PROBLEM-SOLVING 5 5 5(1/2), 6

5 Find the size of ∠ABC in these diagrams.

a A b c A C
A
60° B 110°
70° B
130°
C
75° B
130°
C
d A e f
25° B
B 60° B
35°
50° 30° C C
40°
C
A
A

6 Find the value of x in each of these diagrams.

a b c
140°
x° 100°
130° x°
110° x°
60°

280°

REASONING 7 7, 8(1/2) 8, 9

7 What is the minimum number of angles you need to know to find all the angles marked in these
diagrams?
a d° b d° c
c° e° d°
e° c°
b° f ° f°
a° g°
b° b°
h° a°
e° c° a°

8 In these diagrams, the pronumeral x represents a number and 2x means 2 × x. Find the value of x in
each case.
a b
120°
60°
(2x)°

(2x)°

c d

(x + 20)°
(3x)°
60° 50°

e f

(x − 10)°
70° 60°
(4x)° 80°

9 Find the value of a in these diagrams.

a b
(2a)°
(3a)°
a° a°
c

(5a)° 150°

ENRICHMENT: Adding parallel lines – – 10(1/2)

10 Adding extra parallel lines can help to solve more complex geometry problems.

You can see in this problem that the value of a° is the sum of two alternate angles after adding the extra
(dashed) parallel line.

40° 40°

40°
a° 70°

70° 70°

Find the value of a in these diagrams.

a b c
50° 50°
a°
80° a°
120°
a° 50°
60°

d e f a°
300°
a°
30° 20° 70°
140°
a° 280°

260°

7E Classifying and constructing triangles

LEARNING INTENTIONS
• To understand that triangles can be classiﬁed by their side lengths (scalene, isosceles, equilateral) or by their
interior angles (acute, right, obtuse)
• To know that in an isosceles triangle, the angles opposite the apex are equal and the two sides (legs)
adjacent to the apex are of equal length
• To be able to classify triangles based in side lengths or angles
• To be able to construct triangles using a protractor and ruler

The word ‘triangle’ (literally meaning ‘three

angles’) describes a shape with three sides.
The triangle is an important building block
in mathematical geometry. Similarly, it is
important in the practical world of building and
construction owing to the rigidity of its shape.

In this Melbourne sports stadium, engineers have used triangular

structural supports, including equilateral and isosceles triangles.
These triangles are symmetrical and evenly distribute the weight
of the structure.

Lesson starter: Stable shapes

Consider these constructions, which are made from straight pieces of steel and bolts.

Assume that the bolts are not tightened and that there is some looseness at the points where they are
joined.

• Which shape(s) do you think could lose their shape if a vertex is pushed?
• Which shape(s) will not lose their shape when pushed? Why?
• For the construction(s) that might lose their shape, what could be done to make them rigid?

KEY IDEAS
■ Triangles can be named using the vertex labels. C
triangle ABC or ΔABC

B
■ Triangles are classified by their side lengths. A

scalene isosceles equilateral

60°

3 different sides
3 different angles
60° 60°
2 equal sides
2 equal angles 3 equal sides
3 equal angles (60°)
■ Triangles are also classified by the size of their interior angles.

acute right obtuse

all acute angles one right angle one obtuse angle

■ The parts of an isosceles triangle are named as shown opposite. The apex
base angles are equal and two sides (called the legs) are of equal
length. The two sides of equal length are opposite the equal angles.
base legs
■ Sides of equal length are indicated by matching markings. angles
■ Rulers, protractors and arcs drawn using a pair of compasses
base
can help to construct triangles accurately.
right triangles isosceles triangles equilateral triangles

Three side lengths (e.g. constructing Two sides and the angle Two angles and a side
a triangle with side lengths 3 cm, between them (e.g. constructing (e.g. constructing a
5 cm, 6 cm) a triangle with side lengths triangle with angles
4 cm and 5 cm with a 40° angle 35° and 70° and a side
between them) length of 5 cm)

C
C
C
3 cm 5 cm 4 cm
35° 70°
A B
5 cm
40° B
A B A
6 cm 5 cm

BUILDING UNDERSTANDING
1 Describe or draw an example of each of the triangles given below. Refer back to the Key ideas
in this section to check that the features of each triangle are correct.
a scalene b isosceles c equilateral
d acute e right f obtuse
2 Answer these questions, using the point labels A, B and C for the given A
isosceles triangle.
a Which point is the apex?
b Which segment is the base?
c Which two segments are of equal length? C B
d Which two angles are the base angles?

Example 10 Classifying triangles

These triangles are drawn to scale. Classify them by:

i their side lengths (i.e. scalene, isosceles or equilateral)
ii their angles (i.e. acute, right or obtuse).
a b

SOLUTION EXPLANATION
a i isosceles Has 2 sides of equal length.
ii acute All angles are acute.

b i scalene Has 3 different side lengths.

ii obtuse Has 1 obtuse angle.

Now you try

These triangles are drawn to scale. Classify them by:
i their side lengths ii their angles.
a b

Example 11 Constructing triangles using a protractor and ruler

Construct a triangle ABC with AB = 5 cm, ∠ABC = 30° and ∠BAC = 45°.

SOLUTION EXPLANATION
First, measure and draw segment AB.
Then use a protractor to form the angle 30° at
point B.
A 30°
B
5 cm

Then use a protractor to form the angle 45° at

C point A.
Mark point C and join with A and with B.
45° 30°
A B
5 cm

Now you try

Construct a triangle ABC with AB = 4 cm, ∠ABC = 45° and ∠BAC = 60°.

Example 12 Constructing a triangle using a pair of compasses and ruler

Construct a triangle with side lengths 6 cm, 4 cm and 5 cm.

SOLUTION EXPLANATION
Use a ruler to draw a segment 6 cm in length.
6 cm

Continued on next page

Construct two arcs with radius 4 cm and 5 cm,

using each end of the segment as the centres.
5 cm 4 cm

6 cm

Mark the intersection point of the arcs and

draw the two remaining segments.
5 cm 4 cm

6 cm

Now you try

Construct a triangle with side lengths 5 cm, 6 cm and 7 cm.

Exercise 7E
FLUENCY 1–8 3–8 3–5, 7, 8

Example 10i 1 Classify each of these triangles according to their side lengths (i.e. scalene, isosceles or equilateral).
a b c

Example 10ii 2 These triangles are drawn to scale. Classify them according to their angles (i.e. acute, right or obtuse).
a b c

Example 10 3 These triangles are drawn to scale. Classify them by:

i their side lengths (i.e. scalene, isosceles or equilateral)
ii their angles (i.e. acute, right or obtuse).
a b c

Example 11 4 Use a protractor and ruler to construct the following triangles.

a triangle ABC with AB = 5 cm, ∠ABC = 40° and ∠BAC = 30°
b triangle DEF with DE = 6 cm, ∠DEF = 50° and ∠EDF = 25°
c triangle ABC with AB = 5 cm, ∠ABC = 35° and BC = 4 cm

Example 12 5 Construct a triangle with the given side lengths.

a 7 cm, 3 cm and 5 cm. b 8 cm, 5 cm and 6 cm.

6 Construct an isosceles triangle by following these steps. apex

a Draw a base segment of about 4 cm in length.
b Use a pair of compasses to construct two arcs of equal radius. (Try about 5 cm but
there is no need to be exact.)
c Join the intersection point of the arcs (apex) with each end of the base.
base
d Measure the length of the legs to check they are equal.
e Measure the two base angles to check they are equal.

7 Construct an equilateral triangle by following these steps.

a Draw a segment of about 4 cm in length.
b Use a pair of compasses to construct two arcs of equal radius.
Important: Ensure the arc radius is exactly the same as the length of the
segment in part a.
c Join the intersection point of the arcs with the segment at both ends.
d Measure the length of the three sides to check they are equal.
e Measure the three angles to check they are all equal and 60°.

8 Construct a right triangle by following these steps. C

a Draw a segment, AB, of about 4 cm in length.
b Extend the segment AB to form the ray AD. Make AD
about 2 cm in length.
c Construct a circle with centre A and radius AD. Also
mark point E. D A E B
d Draw two arcs with centres at D and E, as shown in the
diagram. Any radius will do as long as they are equal for
both arcs.
e Mark point C and join with A and B.

PROBLEM-SOLVING 9 9, 10 10, 11

9 Is it possible to draw any of the following? If yes, give an example.

a an acute triangle that is also scalene
b a right triangle that is also isosceles
c an equilateral triangle that is also obtuse
d a scalene triangle that is also right angled

10 Without using a protractor, accurately construct these triangles. Rulers can be used to set the pair of
compasses.
a triangle ABC with AB = 5.5 cm, BC = 4.5 cm and AC = 3.5 cm
b an isosceles triangle with base length 4 cm and legs 5 cm
c an equilateral triangle with side length 3.5 cm
d a right triangle with one side 4 cm and hypotenuse 5 cm
Essential Mathematics for the Victorian Curriculum ISBN 978-1-009-48064-2 © Greenwood et al. 2024 Cambridge University Press
Year 7 Photocopying is restricted under law and this material must not be transferred to another party.
442 Chapter 7 Geometry

11 Copy and complete the following table, making the height of each cell large enough to draw a triangle
in each cell. Draw an example of a triangle that fits the triangle type in both the row and column. Are
there any cells in the table for which it is impossible to draw a triangle?
Triangles Scalene Isosceles Equilateral
Acute
Right
Obtuse

REASONING 12 12, 13 13, 14

12 a Is it possible to divide every triangle into two right triangles using one line segment? Explore with
diagrams.
b Which type of triangle can always be divided into two identical right triangles?

13 Try drawing a triangle with side lengths 4 cm, 5 cm and 10 cm. Explain why this is impossible.

14 a Is the side opposite the largest angle in a triangle always the longest?
b Can you draw a triangle with two obtuse angles? Explain why or why not.

ENRICHMENT: Gothic arches – – 15

15 a The Gothic, or equilateral arch, is based on the equilateral triangle. Try to

construct one, using this diagram to help.

b The trefoil uses the midpoints of the sides of an equilateral triangle.

Try to construct one, using this diagram to help.

7F Classifying quadrilaterals and other polygons

LEARNING INTENTIONS
• To be able to determine if a polygon is convex or non-convex
• To be able to determine if a polygon is regular or irregular
• To be able to classify a polygon by the number of sides it has
• To know what a quadrilateral is
• To be able to classify a quadrilateral as a parallelogram, rectangle, rhombus, square, trapezium or kite based
on a diagram or a description

Polygons are closed plane shapes with straight

sides. Each side is a segment and joins with two
other sides at points called vertices. The number
of sides, angles and vertices are the same for each
type of polygon, and this number determines the
name of the polygon. The word polygon comes from
the Greek words poly meaning ‘many’ and gonia
meaning ‘angle’.

Quadrilaterals are polygons with four sides. There

are special types of quadrilaterals and these are
Architects designed these giant domes for plant greenhouses
identified by the number of equal side lengths and at the Eden Project, England. Steel hexagons, pentagons and
the number of pairs of parallel lines. triangles support plastic ‘pillows’ full of air that insulate plants
from the cold.

Lesson starter: Quadrilaterals that you know

You may already know the names and properties of some of the special quadrilaterals. Which ones do you
think have:

• 2 pairs of parallel sides?

• all sides of equal length?
• 2 pairs of sides of equal length?

Are there any types of quadrilaterals that you know which you have not yet listed?

KEY IDEAS
■ Polygons are closed plane figures with straight sides.
side
■ A vertex is the point at which two sides of a shape meet. convex vertex
(Vertices is the plural form of vertex.)
■ Convex polygons have all vertices pointing outward and all interior
(inside) angles smaller than 180°.
non-
■ Non-convex (or concave) polygons have at least one vertex pointing inward convex
and at least one interior angle larger than 180°.

■ Polygons are classified by the number of sides Number of sides Type

they have. 3 Triangle or trigon
■ Regular polygons have sides of equal length and 4 Quadrilateral or tetragon
angles of equal size. 5 Pentagon
• In a diagram, sides of equal length are shown 6 Hexagon
using markings (or dashes). 7 Heptagon or septagon
8 Octagon
regular irregular 9 Nonagon
pentagon pentagon 10 Decagon
11 Undecagon
12 Dodecagon

D quadrilateral ABCD

C
■ Polygons are usually named with capital letters for each A
vertex and in succession, clockwise or anticlockwise.
B
■ A diagonal is a segment that joins two vertices, dividing a
shape into two parts. diagonals

■ Special quadrilaterals
Square Rectangle

Rhombus Parallelogram

Kite Trapezium

■ Quadrilaterals with parallel sides contain two pairs of cointerior angles.

d°
c° c + d = 180
a°

b°
a + b = 180

BUILDING UNDERSTANDING
1 Consider these three polygons.
i ii iii

a The three shapes are examples of what type of polygon?

b Which shape(s) are convex and why?
c Which shape(s) are non-convex and why?
d State the missing words in this sentence. The third shape is called a __________
__________.

2 Draw an example of each of the quadrilaterals listed. Mark any sides of equal length with single
or double dashes, mark parallel lines with single or double arrows and mark equal angles using
the letters a° and b°. (Refer back to the Key ideas in this section should you need help.)
a square b rectangle c rhombus
d parallelogram e trapezium f kite
3 a Draw two examples of a non-convex quadrilateral.
b For each of your drawings, state how many interior angles are greater than 180°.

Example 13 Classifying polygons

a State the type of polygon and whether it is convex or non-convex.

b Is the polygon regular or irregular?

Continued on next page

SOLUTION EXPLANATION
a convex pentagon The polygon has 5 sides and all the vertices are
pointing outward.

b irregular The sides are not of equal length and the angles are
not equal.

Now you try

a State the type of polygon and whether it is convex or
non-convex.
b Is the polygon regular or irregular?

Example 14 Classifying quadrilaterals

State the type of each quadrilateral given below.

a b

SOLUTION EXPLANATION
a non-convex quadrilateral One interior angle is greater than 180°.

b trapezium There is one pair of parallel sides.

Now you try

State the type of each quadrilateral given.

a b

Exercise 7F
FLUENCY 1–6 2–8 2–6, 8

Example 13 1 a State the type of polygon shown below and whether it is convex or non-convex.
b Is the polygon shape regular or irregular?

2 How many sides do each of these polygon have?

a pentagon b triangle
c decagon d heptagon
e undecagon f quadrilateral
g nonagon h hexagon
i octagon j dodecagon

Example 13 3 a Which of the given shapes are convex?

b State the type of polygon by considering its number of sides.
i ii iii

iv v vi

Example 14a 4 Classify each of these quadrilaterals as either convex or non-convex.

a b c 175°

Example 14b 5 State the type of special quadrilateral given below.

a b c

d e f

Example 14b 6 List all the types of quadrilateral that have:

a 2 different pairs of sides of equal length
b 2 different pairs of opposite angles that are equal in size
c 2 different pairs of parallel lines
d only 1 pair of parallel lines
e only 1 pair of opposite angles that are equal in size.
7 State whether the following are polygons (P) or not polygons (N).
a circle b square c rectangle d oval
e cylinder f cube g line h segment
8 Use your knowledge of the properties of quadrilaterals to find the unknown angles and lengths in each
of these diagrams.
a 10 cm b bm 5m c
a°
100° a°
b°
a° 130°
b cm 50°

PROBLEM-SOLVING 9 9, 10 10, 11

9 Draw line segments to show how you would divide the given shapes into the shapes listed below.
a b c

two triangles
one rectangle and three triangles
two triangles
d e f

four triangles and two quadrilaterals one pentagon

one square and one heptagon

10 A diagonal between two vertices divides a polygon into two parts.

a What is the maximum (i.e. largest) number of diagonals that can be drawn for the following shapes
if the diagonals are not allowed to cross?
i convex pentagon
ii convex decagon
b What is the maximum number of diagonals that can be drawn for the following shapes if the
diagonals are allowed to cross?
i convex pentagon
ii convex decagon

11 Using the given measurements, accurately draw this equilateral triangle onto a piece of paper and cut it
into 4 pieces, as shown. Can you form a square with the four pieces?

3 cm
6 cm

6 cm

6 cm
3 cm

6 cm 6 cm

REASONING 12 12, 13 13, 14

12 State whether each of the following statements is true or false.

a A regular polygon will have equal interior (i.e. inside) angles.
b The sum of the angles inside a pentagon is the same as the sum of the angles inside a decagon.
c An irregular polygon must always be non-convex.
d Convex polygons are not always regular.

13 The diagonals of a quadrilateral are segments that join opposite vertices.

a List the quadrilaterals that have diagonals of equal length.
b List the quadrilaterals that have diagonals intersecting at 90°.

14 a Are squares a type of rectangle or are rectangles a type of square? Give an explanation.
b Are rhombuses a type of parallelogram? Explain.
c Is it possible to draw a non-convex trapezium?

ENRICHMENT: Construction challenge – – 15

15 Use a pair of compasses and a ruler to construct these figures. Use the diagrams as a guide, then
measure to check the properties of your construction.
a a rhombus with side length 5 cm

b a line parallel to segment AB and passing through point P

P P

A B A B

Each trapezium-shaped tabletop is a quadrilateral and two together

forms a hexagon. Could these two trapezium-shaped tables be joined
to make a parallelogram?

1 Consider the diagram below and answer the A E

7A
following.

Progress quiz
a Name the point where the line EH F
B
intersects KF. K G
b Name an angle which has its vertex at G.
c Name an angle adjacent to ∠FGH. I
d Name a set of three concurrent lines. C J
e Name an obtuse angle with its vertex at B
and use your protractor to measure the size D
of this angle. H

2 Find the value of each pronumeral below and give a reason for each answer.
7B
a b
x° x° 105°
62°

c d
x° x°
64°
157°

e f

x° 300° x°
47°
g h 140°
60°
x° 75°
x° x°

3 Name the angle that is: F

7C
a corresponding to ∠EGB D
b alternate to ∠AGH C H
B
c vertically opposite to ∠GHD A G
d cointerior to ∠CHG.
E

4 Find the value of a in these diagrams and give a reason for each answer.
7C
a b c

a° 116° 128° a°
76° a°

5 Giving reasons, state whether the two lines cut by the transversal are parallel.
7C
a b
Progress quiz

57°
111°
69°

56 °

6 Find the value of a in these diagrams.

7D
a b
Ext
54°

a°

a°
47°
145°

7 Classify this triangle by:

7E
a its side lengths
b its angles.

8 Construct an equilateral triangle with each side having a length of 6 cm.

9 State the special type of quadrilateral given in each diagram below.

7F
a b

c d
×

10 Name the type of shape stating whether it is concave or convex, regular or irregular.
7F
a b

c d

7G Angle sum of a triangle

LEARNING INTENTIONS
• To know that the sum of interior angles in a triangle is 180°
• To know what an exterior angle is
• To be able to ﬁnd exterior angles in triangles using supplementary angles
• To understand that the angle sum of a polygon can be determined by decomposing into triangles

The three interior angles of a triangle have a very important property. No matter the shape of the triangle,
the three angles always add to the same total.

The Millau Viaduct in France is the tallest bridge structure

in the world; its tallest pylon is 343 m. The different
triangles formed by the bridge’s cables all have the same
angle sum.

Lesson starter: A visual perspective on the angle sum

Use a ruler to draw any triangle. Cut out the triangle and tear off the three corners. Then place the three
corners together.

a°
b°
c° a° c°
b°

What do you notice and what does this tell you about the three angles in the triangle? Compare your
results with those of others. Does this work for other triangles?

KEY IDEAS
■ The angle sum of the interior angles of a triangle is 180°.
b°
a° c°

a + b + c = 180

■ If one side of a triangle is extended, an exterior angle is A

formed. In the diagram shown opposite, ∠DBC is the exterior
angle. The angle ∠DBC is supplementary to ∠ABC (i.e. adds B a°
to 180°).
D C
(180 − a)°
■ The angle sum S of a polygon with n sides can be determined by decomposing into triangles.
Quadrilateral (n = 4) Pentagon (n = 5)

Angle sum = 3 × 180° = 540°

Angle sum = 2 × 180° = 360°

BUILDING UNDERSTANDING
1 a Use a protractor to measure the three angles in this
triangle.
b Add your three angles. What do you notice?

2 For the triangle opposite, give reasons why:

a°
a a must equal 20 160°
b b must equal 60. b°
100°
3 What is the size of each angle in an equilateral triangle?
4 For the isosceles triangle opposite, give a reason why:
a a = 70 b°
b b = 40.

70° a°

Example 15 Finding an angle in a triangle

Find the value of a in these triangles.

a a° b
60°
95° a°
70°

SOLUTION EXPLANATION
a a = 180 − (60 + 95) The sum of angles in a triangle is 180.
= 180 − 155 Add the two known angles.
= 25 Find the difference between 180 and 155.

b a = 180 − (70 + 70) The two angles opposite the sides of equal
= 180 − 140 length (i.e. the base angles) in an isosceles
= 40 triangle are equal in size.
Add the two equal angles.
Find the difference between 140 and 180.

Now you try

Find the value of a in these triangles.
a b
a°

20°
20°
135°
a°

Example 16 Finding an exterior angle

Find the size of the exterior angle (x°) in this diagram.

a° x°
62°

Continued on next page

SOLUTION EXPLANATION
a = 180 − (90 + 62) The angle sum for a triangle is 180°.
= 180 − 152 Add the two known angles.
= 28 a is the difference between 180 and 152.

x = 180 − 28 Angles of size x° and a° are supplementary

= 152 (i.e. they add to 180°).
∴ The size of the exterior angle is 152°. x is the difference between 180 and 28.

Now you try

Find the size of the exterior angle (x°) in this diagram.

20°

132° x°
a°

Exercise 7G
FLUENCY 1, 2(1/2), 3, 4–5(1/2) 2–5(1/2) 2(1/2), 4–5(1/2)

Example 15a 1 Find the value of a in these triangles.

a b
30° a°
40°
110°
110°
a°

Example 15a 2 Find the value of a in each of these triangles.

a b c
a°
a°
20°
130° 35°
40° 80° a°

d e 35° f 15°
a° 120°
25° a° a°

20°

Example 15b 3 Find the value of a in each of these isosceles triangles.

a b
30°
a°
a°
72°

Example 15b 4 Find the value of a in each of these isosceles triangles.

a b c
a° 80°
74°
a°
65°
a°

d e f
a°
30°
70°
110°
a°
a°

Example 15 5 The triangles below have exterior angles. Find the value of x. For parts b to f, you will need to first
calculate the value of a.
a b c
60° x°
150°
x°
80°

a° a°
x° 150°

d e f
60° a°
100° x° x°
x° 82°
a° a°
70°
40° 60°

PROBLEM-SOLVING 6(1/2) 6–7(1/2) 6–7(1/2), 8

6 Find the value of a in each of these triangles.

a b
a°

110° a°

c d
100°
a° 42°

35°
a°

e f
a° (2a)°
56°
a°
40°

7 Each of these diagrams has parallel lines. Find the value of a.

a b c
a°

80°
40° 70°
35°
a°
50°
a°

d e f
a° 15°
20°
a°
35° a° 100°

30°

8 A plane flies horizontally 200 m above the

ground. It detects two beacons on the ground.
Some angles are known, and these are shown
in the diagram. Find the angle marked a°
between the line of sight to the two beacons. 120°
a° 200 m
140°

beacons

REASONING 9 9, 10 10, 11

9 In the Key Ideas, we can see how polygons can be decomposed into triangles.
For example: A quadrilateral can be decomposed into two triangles without any intersecting diagonals
so the angle sum of a quadrilateral equals 2 × 180° = 360°.
a By decomposing into triangles, without any intersecting diagonals, how many triangles are formed
in these polygons?
i pentagon ii hexagon iii heptagon
b Using your results from part a, determine the angle sum of the following polygons.
i pentagon ii hexagon iii heptagon

10 Determine the angle sum of the following polygons.

a octagon b nonagon c decagon

11 If S is the angle sum of a polygon with n sides, find a rule linking S with n.

ENRICHMENT: Exploring triangle theorems – – 12–14

12 a Find the sum 75° + 80°.

b Find the value of a in the diagram opposite. 75°
c What do you notice about the answers to parts a and b? a°
80°
d Do you think this would be true for other triangles with different
angles? Explore.

13 This diagram includes two parallel lines. a° b°

a The angles marked a° are always equal. From the list c°
(corresponding, alternate, cointerior, vertically opposite), give a
reason why. a° b°
b Give a reason why the angles marked b° are always equal.
c At the top of the diagram, angles a°, b° and c° lie on a straight line. What does this tell you about
the three angles a°, b° and c° in the triangle?

14 Complete these proofs. Give reasons for each step where brackets are shown.
a The angle sum in a triangle is 180°.
C
D E
∠DCA = a° (Alternate to ∠BAC and DE is parallel to AB.) c°
∠ECB = _______ (_____________________)
∠DCA + ∠ACB + ∠ECB = _______ (_____________________) a° b°
∴ a + b + c = _______________ A B
b The exterior angle outside a triangle is equal to the sum of the
two interior opposite angles.

b°

a° c°
A C D

7H Symmetry
LEARNING INTENTIONS
• To understand what a line of symmetry is
• To be able to determine the order of line symmetry for a shape
• To understand what rotational symmetry is
• To be able to determine the order of rotational symmetry for a shape

You see many symmetrical geometrical shapes in nature.

The starfish and sunflower are two examples. Shapes
such as these may have two types of symmetry: line and
rotational.

Lesson starter: Working with symmetry

Starﬁsh and sunﬂowers are
On a piece of paper, draw a square (with side lengths of both symmetrical, but in different ways.
about 10 cm) and a rectangle (with length of about
15 cm and width of about 10 cm), then cut them out.

• How many ways can you fold each shape in

half so that the two halves match exactly? The
number of creases formed will be the number of
lines of symmetry.
• Now locate the centre of each shape and
place a sharp pencil on this point. Rotate
the shape 360°. How many times does the
shape make an exact copy of itself in its original
position? This number describes the rotational
symmetry.

KEY IDEAS
■ An axis or line of symmetry divides a shape into two equal
parts. It acts as a mirror line, with each half of the shape
being a reflection of the other.
• An isosceles triangle has one line (axis) of symmetry.
• A rectangle has two lines (axes) of symmetry.

■ The order of rotation is the number of times a shape makes an exact copy 2
of itself (in its original position) after rotating 360°. 1
• We say that there is no rotational symmetry if the order of rotational
symmetry is equal to 1.
3

BUILDING UNDERSTANDING
1 How many ways could you fold each of these shapes in half so that the two halves match
exactly? (To help you solve this problem, try cutting out the shapes and folding them.)
a square b rectangle c equilateral triangle
d isosceles triangle e rhombus f parallelogram
2 For the shapes listed in Question 1 , imagine rotating them 360° about their centre. 1
How many times do you make an exact copy of the shape in its original position?

Example 17 Finding the symmetry of shapes

Give the number of lines of symmetry and the order of rotational symmetry for each of these
shapes.
a rectangle b regular pentagon

SOLUTION EXPLANATION
a 2 lines of symmetry

rotational symmetry:
order 2

b 5 lines of symmetry

rotational symmetry:
order 5

Now you try

Give the number of lines of symmetry and the order of rotational symmetry for each of these
shapes.
a isosceles trapezium b equilateral triangle

× ×

Exercise 7H
FLUENCY 1–6 2–7 2–4, 6, 7

Example 17 1 Give the number of lines of symmetry and the order of rotational symmetry for
this regular hexagon.

Example 17 2 Give the number of lines of symmetry and the order of rotational symmetry for
each shape.
a b

c d

e f

3 Name a type of triangle that has the following properties.

a 3 lines of symmetry and order of rotational symmetry 3
b 1 line of symmetry and no rotational symmetry
c no line or rotational symmetry

4 List the special quadrilaterals that have these properties.

a lines of symmetry:
i 1 ii 2 iii 3 iv 4
b rotational symmetry of order:
i 1 ii 2 iii 3 iv 4

5 State the number of lines of symmetry and the order of rotational symmetry for each of the following.
a b

c d

6 Of the capital letters of the alphabet shown in the font here, state which have:
a 1 line of symmetry A B C D E F G H I J K L M
b 2 lines of symmetry N O P Q R S T U V W X Y Z
c rotational symmetry of order 2.
7 Complete the other half of these shapes for the given axis of symmetry.
a b c

PROBLEM-SOLVING 8 8, 9 8, 9

8 Draw the following shapes, if possible.

a a quadrilateral with no lines of symmetry
b a hexagon with one line of symmetry
c a shape with line symmetry of order 7 and rotational symmetry of order 7
d a diagram with no line of symmetry but rotational symmetry of order 3
e a diagram with line of symmetry of order 1 and no rotational symmetry

9 These diagrams are made up of more than one shape. State the order of line symmetry and of rotational
symmetry.
a b c

REASONING 10 10 10, 11

10 Many people think a rectangle has four lines of symmetry, including the
diagonals.

a Complete the other half of this diagram to show that this is not true.

b Using the same method as that used in part a, show that the
try
diagonals of a parallelogram are not lines of symmetry. mme
of sy
line

11 A trapezium has one pair of parallel lines.

a State whether trapeziums always have:
i line symmetry
ii rotational symmetry.
b What type of trapezium will have one line of symmetry?

ENRICHMENT: Symmetry in three dimensions – – 12

12 Some solid objects also have symmetry. Rather than line symmetry,
they have plane symmetry. This cube shows one plane of symmetry,
but there are more that could be drawn.

State the number of planes of symmetry for each of these solids.

a cube b rectangular prism c right square pyramid

d right triangular prism e cylinder f sphere

7I Reflection and rotation

LEARNING INTENTIONS
• To understand that a shape can be reﬂected or rotated to give an image
• To be able to draw the result of a point or shape being reﬂected in a mirror line
• To be able to draw the result of a point or shape being rotated about a point

Reﬂection and rotation are two types of

transformations that involve a change in position of
the points on an object. If a shape is reflected in a
mirror line or rotated about a point, the size of the
shape is unchanged. Hence, the transformations
reflection and rotation are said to be isometric.

Lesson starter: Draw the image

Here is a shape on a grid. When designing ‘The Mirrored Staircase’ in this picture,
the architect has drawn one staircase and then reﬂected its
• Draw the image (result) after reflecting the shape design across the vertical line that is the axis of symmetry.
in the mirror line A.
• Draw the image (result) after reflecting the shape
in the mirror line B. mirror
• Draw the image after rotating the shape about point O by 180°. line A
• Draw the image after rotating the shape about point O by 90°
clockwise.
• Draw the image after rotating the shape about point O by 90°
anticlockwise. mirror
O line B
Discuss what method you used to draw each image and the
relationship between the position of the shape and its image after
each transformation.

KEY IDEAS
■ Reﬂection and rotation are isometric transformations that give an
image of an object or shape without changing its shape and size.
■ The image of point A is denoted A′. A B B′ A′
E F F′ E′
■ A reflection involves a mirror line, as shown in the diagram
image
opposite. D C C′ D′

mirror
line

■ A rotation involves a centre point of rotation (C) and an angle of A

rotation, as shown.
• A pair of compasses can be used to draw each circle, to help find
the position of image points. C D
B
90°
D′ image A′

B′
rotation 90° clockwise
about C

BUILDING UNDERSTANDING
1 Use the grid to reflect each shape in the given mirror line.
a b

c d

e f

y
2 Give the coordinates of the image point A′ after the point A(2, 0) is
2
rotated about point C(0, 0) by the following angles.
1
a 180° clockwise C A x
b 180° anticlockwise −2 −1
−1 1 2
c 90° clockwise −2
d 90° anticlockwise
3 a Are the size and shape of an object changed after a reflection?
b Are the size and shape of an object changed after a rotation?

Example 18 Drawing reflections

Draw the reflected image of this shape and give the y

coordinates of A′, B′, C′ and D′. The y-axis is the mirror
4
line.
3 mirror line
2 ( y-axis)
1
x
−4 −3 −2 −1 O 1 2 3 4
D −1
A
−2
B
C −3
−4

SOLUTION EXPLANATION
y Reflect each vertex A, B, C and
D about the mirror line. The line
4
segment from each point to its image
3 mirror line should be at 90° to the mirror line.
2 ( y-axis)
1
x
−4 −3 −2 −1 O 1 2 3 4
−1 A′
D A image D′
−2 B′
B
C −3 C′
−4

A′ = (1, − 1) , B′ = (1, − 2) , C′ = (3, − 3) , D′ = (3, − 1)

Now you try

Draw the reflected image of this shape and give the coordinates
of A′, B′, C′ and D′. The y-axis is the mirror line. y
B
4
C 3
2
1
D A
x
1 1O
−4 −3 −2 −− 1 2 3 4
−2
−3
−4

Example 19 Drawing rotations

Draw the image of this shape and give the coordinates y

of A′, B′ and D′ after carrying out the following rotations.
4
a 90° anticlockwise about C
3
b 180° about C 2
D
1
C A B x
−3 −2 −1
−1 1 2 3
−2
−3
−4

SOLUTION EXPLANATION
a y Rotate each point on a circular arc around
point C by 90° anticlockwise.
4 Join the three image points (A′, B′ and D′) with
3
line segments to form the result.
2 B′ D
1 A′
D′ C A B x
−3 −2 −1
−1 1 2 3
−2
−3
−4

A′ = (0, 1), B′ = (0, 2), D′ = (−2, 1)

b y Rotate each point on a circular arc around

point C by 180° in either direction.
4 Join the three image points (A′, B′ and D′) with
3
D line segments to form the result.
2
1
B′ A′ C A B x
−3 −2 −1
−1 1 2 3
−2
D′−3
−4

A′ = (−1, 0), B′ = (−2, 0), D′ = (−1, −2)

Now you try

Draw the image of this shape and give the coordinates of y
A′, B′ and D′ after carrying out the following rotations. 4 D
a 90° anticlockwise about C 3
b 180° about C 2
1
C A B
x
−5 −4 −3 −2 −−
11 1 2 3 4
−2
−3
−4

Exercise 7I
FLUENCY 1, 2(1/2), 3, 4 2–3(1/2), 4, 5 2–3(1/2), 4, 5

Example 18 1 Draw the reflected image of this shape and give the coordinates of A′, B′, C′ and D′. The y-axis is the
mirror line.

5
4
D A
3
2
1
C B
x
1 1O
−5 −4 −3 −2 −− 1 2 3 4 5
−2
−3
mirror line
−4
(y-axis)
−5

Example 18 2 Draw the image of each shape in the mirror line and give the coordinates of A′, B′, C′ and D′. Note
that the y-axis is the mirror line for parts a to c, whereas the x-axis is the mirror line for parts d to f.
a y b y
B4 4 A
3 3
C 2 2 D
1 1 B
D A C
x x
−4 −3 −2 −1O 1 2 3 4 −4 −3 −2 −1 O 1 2 3 4
−1 −1
−2 −2
−3 mirror line −3 mirror line
−4 −4

c y d y

4 4 B
3 mirror line 3
2 2 C
1 1
x A D
x
−4 −3 −2 −1 O 1 2 3 4 −4 −3 −2 −1 O
−1 −1 1 2 3 4
−2 A −2
−3 D −3 mirror line
−4 −4
B C
e y f y
A B
4 4
3 3
D
2 mirror line 2 mirror line
1 1
C x
x
−4 −3 −2 −1 O 1 2 3 4 −4 −3 −2 −1 O 1 2 3 4
−1 −1
A D−2 −2
−3 −3
B
−4 −4
C

3 Give the new coordinates of the image point A′ after point A has y
been rotated around point C(0, 0) by:
a 180° clockwise 4
A
b 90° clockwise 3
2
c 90° anticlockwise
1
d 270° clockwise C x
e 360° anticlockwise −4 −3 −2 −1
−1 1 2 3 4
f 180° anticlockwise. −2
−3
−4

Example 19 4 Draw the image of this shape and give the coordinates of A′, B′ y
and D′ after the following rotations.
a 90° anticlockwise about C 4
3
b 180° about C
2
c 90° clockwise about C
1
D A C x
−4 −3 −2 −1
−1 1 2 3 4
−2
B
−3
−4

Example 19 5 Draw the image of this shape and give the coordinates of A′, B′ and D′ y
after the following rotations.
4
a 90° anticlockwise about C B
3
b 180° about C 2
c 90° clockwise about C A
1 D
C x
−3 −2 −1
−1 1 2 3
−2
−3
−4

PROBLEM-SOLVING 6, 7 6–8 6, 8, 9

6 The mirror lines on these grids are at a 45° angle. Draw the reflected image.
a b c

d e f

7 On the Cartesian plane, the point A(− 2, 5) is reflected in the x-axis and this image point is then
reflected in the y-axis. What are the coordinates of the final image?

8 A point, B(2, 3), is rotated about the point C(1, 1). State the coordinates y
of the image point B′ for the following rotations. 3 B
a 180° 2
b 90° clockwise 1 C
c 90° anticlockwise x
−3 −2 −1 O 1 2 3
−1
−2
−3

9 For each shape given, by how many degrees has it been rotated and in which direction?
a b c

C
C
C

REASONING 10 10, 11 11, 12

10 a By repeatedly reflecting a shape over a moving

mirror line, patterns can be formed. This right-angled
triangle for example is reflected 4 times by placing
the mirror line vertically and on the right side each
time. Create a pattern using these starting shapes by
repeatedly placing the vertical mirror line on the
right side.
i ii

iii Create your own using reflection.

b By repeatedly rotating a shape about a point, patterns can be formed. This
diagram shows a semicircle rotated by 90° three times about the given
point. Create a pattern using three 90° rotations about the given point.
i ii

iii Create your own using rotation.

c See if you can combine reflections and rotations to create more complex patterns.

11 Write the missing number in these sentences.

a Rotating a point 90° clockwise is the same as rotating a point ________ anticlockwise.
b Rotating a point 38° anticlockwise is the same as rotating a point ________ clockwise.
c A point is rotated 370° clockwise. This is the same as rotating the point ________ clockwise.

12 A point S has coordinates (− 2, 5).

a Find the coordinates of the image point S′ after a rotation 180° about C(0, 0).
b Find the coordinates of the image point S′ after a reflection in the x-axis followed by a reflection in
the y-axis.
c What do you notice about the image points in parts a and b?
d Test your observation on the point T(− 4, − 1) by repeating parts a and b.

ENRICHMENT: Interactive geometry software exploration – – 13, 14

13 Explore reflecting shapes dynamically, using interactive geometry software.

a On a grid, create any shape using the polygon tool.
b Construct a mirror line.
c Use the reflection tool to create the reflected image about your mirror line.
d Drag the vertices of your original shape and observe the changes in the image. Also try dragging
the mirror line.

14 Explore rotating shapes dynamically, using interactive geometry software.

a On a grid, create any shape using the polygon tool.
b Construct a centre of rotation point and a rotating angle (or number).
c Use the rotation tool to create the rotated image that has your nominated centre of rotation and
angle.
d Drag the vertices of your original shape and observe the changes in the image. Also try changing
the angle of rotation.

The following problems will investigate practical situations drawing upon knowledge and skills developed
throughout the chapter. In attempting to solve these problems, aim to identify the key information, use
Applications and problem-solving

diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.

Roof trusses
1 When building a house, the frame to hold the roof up is constructed of roof trusses. Roof trusses come
in different designs and define the pitch, or angle, of the external roof and the internal ceiling. The
image here shows a roof with W trusses.
The standard W truss provides an external pitch for the roof and a flat internal ceiling for the plaster.

Standard W truss

A builder is interested in how the lengths and angles work together for a standard W roof truss and
the overall height of the truss above the ceiling which depends on the pitch angle.
a Why do you think this design of truss is called a W truss?
b How many segments of timber are required to construct one W truss?
c Using the guidelines below, construct an accurate scale of a W truss in your workbook with a roof
pitch of 30°.
i Draw a horizontal base beam of 12 cm.
ii Divide the base beam into five equal segments.
iii Measure 30° angles and draw the two sloping roof beams.
iv Divide each sloping roof beam into three equal segments.
v Draw the internal support beams by connecting lines between the equal segment markings on
the roof and base beams.
d On your accurate diagram, measure and label each of the internal angles formed by the support
beams.
e From your answers in part d, label any parallel support beams.
f From your accurate diagram, what is the vertical height from the top of the roof to the ceiling (base
beam)?
g Investigate the angle (pitch) of either a roof at school or a roof at home. If possible, take a photo of
the roof trusses and measure the relevant angles.

The perfect path for a hole-in-one

2 Visualising angles is a key to successfully playing wall
mini-golf. When a golf ball bounces off a straight wall incoming outgoing
we can assume that the angle at which it hits the wall angle centre angle
(incoming angle) is the same as the angle at which it angle
leaves the wall (outgoing angle).

You are interested in drawing a path where incoming angles equal outgoing angles so that you can
score a hole-in-one.

Applications and problem-solving

A sample mini-golf hole is shown here.
A two-bounce pathway hole
a Your first try. You may need to copy the diagram onto
barrier
paper first. Select a point on the starting line where
you choose to tee off from and choose an angle to hit
the ball at. Trace this path, ensuring each outgoing
angle equals each incoming angle and see how close barrier
you end up to going in the hole.
b Label each of your angles along your path.
c Can you draw an accurate path for a hole-in-one?
A three-bounce pathway
d Can you design a three-bounce path for a hole-in-
one? Label and measure your angles. start line (tee)
Design your own You can place the ball
e Design your own mini-golf hole that is possible for anywhere along this line.
someone to score a hole-in-one through bouncing off
walls or barriers. Show the correct pathway.
(Hint: You may wish to draw the correct pathway first and then place in the barriers to make it
challenging for players.)

Tessellating bathroom tiles

3 A tessellation is defined as an arrangement of shapes closely fitted together in a repeated pattern
without gaps or overlapping.
Elise is interested in what simple shapes tesselate and how to tile a bathroom using regular and
irregular shapes.
a Name the only three regular polygons that tessellate by themselves.
b Why can’t a regular pentagon or a regular octagon tessellate by themselves?
c Create your own tessellation using a combination of regular triangles, squares and hexagons.
Elise is designing her new bathroom and wishes to have a floor tile pattern involving just
pentagons. Elise’s builder says that it is impossible to tessellate the bathroom floor with regular
pentagons, but he is confident that he
can tessellate the floor with irregular
pentagons if he can find the right tiles.
d Suggest what the internal angles of an
irregular pentagon tile should be to ensure
they tessellate with one another.
e Draw an irregular pentagon that would
tessellate with itself.
f Create a bathroom floor pattern to show
Elise how irregular pentagon tiles can
tessellate.

7J Translation
LEARNING INTENTIONS
• To understand that a shape can be translated left, right, up or down
• To be able to draw the result of a point or shape being translated in a given direction
• To be able to describe a translation given an original point/shape and an image point/shape

Translation involves a shift in an object left, right, up or down.

The orientation of a shape is unchanged. Translation is another
isometric transformation because the size and shape of the
image is unchanged.

Lesson starter: Describing a translation

Consider this shape ABCD and its image A′B′C′D′.
In a dragster race along 300 m of straight
track, the main body of the car is translated
A B down the track in a single direction.

A′ B′
D C

D′ C′

• Use the words left, right, up or down to describe how the shape ABCD, shown opposite, could be
translated (shifted) to its image.
• Can you think of a second combination of translations that give the same image?
• How would you describe the reverse translation?

KEY IDEAS
■ Translation is an isometric transformation involving a shift left, right, up or down.

■ Describing a translation involves saying how many units a shape is shifted left, right, up and/or
down.

BUILDING UNDERSTANDING
1 Point A has coordinates (3, 2). State the coordinates of the y
image point A′ when point A is translated in each of the following 5
ways. 4
a 1 unit right 3
b 2 units left 2 A
c 3 units up 1
d 1 unit down x
O 1 2 3 4 5
e 1 unit left and 2 units up
f 3 units left and 1 unit down
g 2 units right and 1 unit down
h 0 units left and 2 units down
2 A point is translated to its image. State the missing word (i.e. left, right, up or down) for each
sentence.
a (1, 1) is translated _______ to the point (1, 3).
b (5, 4) is translated _______ to the point (1, 4).
c (7, 2) is translated _______ to the point (7, 0).
d (3, 0) is translated _______ to the point (3, 1).
e (5, 1) is translated _______ to the point (4, 1).
f (2, 3) is translated _______ to the point (1, 3).
g (0, 2) is translated _______ to the point (5, 2).
h (7, 6) is translated _______ to the point (11, 6).
3 The point (7, 4) is translated to the point (0, 1).
a How far left has the point been translated?
b How far down has the point been translated?
c If the point (0, 1) is translated to (7, 4):
i How far right has the point been translated?
ii How far up has the point been translated?

Example 20 Translating shapes

Draw the image of the triangle ABC after a translation 2 units to the right and
A
3 units down.

B C

Continued on next page

SOLUTION EXPLANATION

A Shift every vertex 2 units to the right and

3 units down. Then join the vertices to form the
C image.
B
A′

B′ C′

Now you try

Draw the image of the triangle ABC after a translation
B
1 unit left and 3 units down.

A C

Example 21 Describing translations

A point B(5, − 2) is translated to B′(− 1, 2). Describe the translation.

SOLUTION EXPLANATION
Translation is 6 units left and 4 units up. y

3
B′ 2
1
x
−2 −1 O 1 2 3 4 5
−1
−2
−3 B

Now you try

A point B(− 2, 5) is translated to B′(4, − 3). Describe the translation.

Exercise 7J
FLUENCY 1, 2, 3–4(1/2) 2–4(1/2) 2(1/2), 3–4(1/3)

Example 20 1 Draw the image of the triangle ABC after a translation 3 units to the left and
A
2 units down. C

Example 20 2 Draw the image of these shapes after each translation.

a 3 units left and 1 unit up b 1 unit right and 2 units up

c 3 units right and 2 units down d 4 units left and 2 units down

3 Point A has coordinates (− 2, 3). Write the coordinates of y

the image point A′ when point A is translated in each of the
following ways. 5
4
a 3 units right A
3
b 2 units left
2
c 2 units down 1
d 5 units down x
e 2 units up 1 1O
−5 −4 −3 −2 −− 1 2 3 4 5
f 10 units right −2
g 3 units right and 1 unit up −3
h 4 units right and 2 units down −4
i 5 units right and 6 units down −5
j 1 unit left and 2 units down
k 3 units left and 1 unit up
l 2 units left and 5 units down

Example 21 4 Describe the translation when each point is translated to its image. Give your answer similar to these
examples: ‘4 units right’ or ‘2 units left and 3 units up’.
a A(1, 3) is translated to A′(1, 6). b B(4, 7) is translated to B′(4, 0).
c C(− 1, 3) is translated to C′(− 1, − 1). d D(− 2, 8) is translated to D′(− 2, 10).
e E(4, 3) is translated to E′(− 1, 3). f F(2, − 4) is translated to F′(4, − 4).
g G(0, 0) is translated to G′(− 1, 4). h H(− 1, − 1) is translated to H′(2, 5).
i I(− 3, 8) is translated to I′(0, 4). j J(2, − 5) is translated to J′(− 1, 6).
k K(− 10, 2) is translated to K′(2, − 1). l L(6, 10) is translated to L′(− 4, − 3).

PROBLEM-SOLVING 5 5, 6 6, 7

5 A point, A, is translated to its image, A′. Describe the translation that takes A′ to A (i.e. the reverse
translation).
a A(2, 3) and A′(4, 1)
b B(0, 4) and B′(4, 0)
c C(0, − 3) and C′(− 1, 2)
d D(4, 6) and D′(− 2, 8)

6 If only horizontal or vertical translations of distance 1 are allowed, how many different paths are there
from points A to B on each grid below? No point can be visited more than once.
a B b B

A
A

7 Starting at (0, 0) on the Cartesian plane, how many different points can you move to if a maximum of
3 units in total can be translated in any of the four directions of left, right, up or down with all
translations being whole numbers? Do not count the point (0, 0).

REASONING 8 8 8, 9

8 A shape is translated to its image. Explain why the shape’s size and orientation is unchanged.

9 A combination of translations can be replaced with one single translation. For example, if (1, 1) is
translated 3 units right and 2 units down, followed by a translation of 6 units left and 5 units up, then
the final image point (− 2, 4) could be obtained with the single translation 3 units left and 3 units up.
Describe the single translation that replaces these combinations of translations.
a (1, 1) is translated 2 units left and 1 unit up, followed by a translation of 5 units right and 2 units
down.
b (6, − 2) is translated 3 units right and 3 units up, followed by a translation of 2 units left and 1 unit
down.
c (− 1, 4) is translated 4 units right and 6 units down, followed by a translation of 6 units left and
2 units up.
d (− 3, 4) is translated 4 units left and 4 units down, followed by a translation of 10 units right and
11 units up.

ENRICHMENT: Combined transformations – – 10

10 Write the coordinates of the image point after each sequence of transformations. For each part, apply
the next transformation to the image of the previous transformation.
a (2, 3)
• reflection in the x-axis
• reflection in the y-axis
• translation 2 units left and 2 units up
b (−1, 6)
• translation 5 units right and 3 units down
• reflection in the y-axis
• reflection in the x-axis
c (−4, 2)
• rotation 180° about (0, 0)
• reflection in the y-axis
• translation 3 units left and 4 units up
d (−3, −7)
• rotation 90° clockwise about (0, 0)
• reflection in the x-axis
• translation 6 units left and 2 units down
e (−4, 8)
• rotation 90° anticlockwise about (0, 0)
• translation 4 units right and 6 units up
• reflection in the x- and the y-axis

Construction material moved onto a building by a modern crane has undergone a

combination of transformations.

7K Drawing solids
LEARNING INTENTIONS
• To be able to draw pyramids, cylinders and cones
• To be able to use square or isometric dot paper to accurately draw solids

Three-dimensional solids can be represented as a drawing

on a two-dimensional surface (e.g. paper or computer
screen), provided some basic rules are followed.

Architects create 3D models of building plans by

hand or with computer software.

Lesson starter: Can you draw a cube?

Try to draw a cube. Here are two bad examples.

• What is wrong with these drawings?

• What basic rules do you need to follow when drawing a cube?

KEY IDEAS
■ Draw cubes and rectangular prisms by keeping:
• parallel sides pointing in the same direction
• parallel sides the same length.
■ Draw pyramids by joining the apex with the vertices on the base.

triangular pyramid (tetrahedron) square pyramid

apex apex

square base
triangular base
■ Draw cylinders and cones by starting with an oval shape.

cylinder cone

■ Square and isometric dot paper can help to accurately draw solids. Drawings made on
isometric dot paper clearly show the cubes that make up the solid.
square dot paper isometric dot paper

BUILDING UNDERSTANDING
1 Copy these diagrams and add lines to complete the solid. Use dashed line for invisible sides.
a cube b cylinder c square pyramid

2 Cubes are stacked to form these solids. How many cubes are there in each solid?
a b c

Example 22 Drawing solids

Draw these solids.

a a cone on plain paper
b this solid on isometric dot paper

SOLUTION EXPLANATION
a Draw an oval shape for the base and the apex
point. Dot any line or curve which may be
invisible on the solid.
Join the apex to the sides of the base.

b Rotate the solid slightly and draw each cube

starting at the front and working back.

Now you try

Draw these solids.
a a square-based pyramid on plain paper
b this solid on isometric dot paper

Exercise 7K
FLUENCY 1–4 2–4, 5(1/2) 2, 4, 5(1/2)

Example 22a 1 Draw a square-based prism on plain paper.

Example 22a 2 On plain paper, draw an example of these common solids.

a cube
b tetrahedron
c cylinder
d cone
e square-based pyramid
f rectangular prism

Example 22b 3 Copy these solids onto square dot paper.

a b

Example 22b 4 Draw these solids onto isometric dot paper.

a b c

5 Here is a cylinder with its top view (circle) and side view (rectangle).

top top side

side

Draw the shapes which are the top view and side view of these solids.
a cube b square prism c cone

d square pyramid e octahedron f sphere

( 2 sphere)
g square pyramid on cube h hemisphere _ 1 on i cone on hemisphere
square prism

PROBLEM-SOLVING 6 6, 7 7, 8

6 Here is the top (plan or bird’s eye) view of a stack of 5 cubes. How many different
stacks of 5 cubes could this represent?

7 Here is the top (plan) view of a stack of 7 cubes. How many different stacks
of 7 cubes could this represent?

8 Draw these solids, making sure that:

i each vertex can be seen clearly
ii dashed lines are used for invisible sides.
a tetrahedron (solid with 4 faces)
b octahedron (solid with 8 faces)
c pentagonal pyramid (pyramid with pentagonal base)

REASONING 9 9 9, 10

9 Andrea draws two solids as shown. Aiden says that they are drawings of exactly the same solid. Is
Aiden correct? Give reasons.
and

10 Match the solids a, b, c and d with an identical solid chosen from A, B, C and D.
a b c d

A B C D

ENRICHMENT: Three viewpoints – – 11

11 This diagram shows the front and left sides of a solid.

left front
a Draw the front, left and top views of these solids.
i ii

b Draw a solid that has these views.

i front left top

ii front left top

7L Nets of solids
LEARNING INTENTIONS
• To understand that a net is a two-dimensional representation of a solid’s faces
• To know what a polyhedron is
• To know what the five Platonic solids are
• To be able to draw a net of simple solids

The ancient Greek philosophers studied the properties

of polyhedra and how these could be used to explain
the natural world. Plato (427–347 bce) reasoned that
the building blocks of all three-dimensional objects
were regular polyhedra which have faces that are
identical in size and shape. There are five regular
polyhedra, called the Platonic solids after Plato,
which were thought to represent fire, earth, air, water
and the universe or cosmos.

Platonic solids form excellent dice; their symmetry

and identical faces provide fair and random results.
Platonic dice have 4, 6, 8, 12 or 20 faces and are used in
role-playing games such as ‘Dungeons and Dragons’.
Lesson starter: Net of a cube
Here is one Platonic solid, the regular hexahedron or cube,
and its net.

If the faces of the solid are unfolded to form a net, you can
clearly see the 6 faces.

Can you draw a different net of a cube? How do you know it will fold to
form a cube? Compare this with other nets in your class.

KEY IDEAS
■ A net of a solid is an unfolded two-dimensional square pyramid cylinder
representation of all the faces. Here are two
examples.

■ A polyhedron (plural: polyhedra) is a solid with flat faces.

• They can be named by their number of faces, e.g. tetrahedron (4 faces),
hexahedron (6 faces).

■ The five Platonic solids are regular polyhedra each with identical regular faces and the same
number of faces meeting at each vertex.
• regular tetrahedron (4 equilateral triangular faces)

• regular hexahedron or cube (6 square faces)

• regular octahedron (8 equilateral triangular faces)

• regular dodecahedron (12 regular pentagonal faces)

• regular icosahedron (20 equilateral triangular faces)

BUILDING UNDERSTANDING
1 State the missing words in these sentences.
a A regular polygon will have _________________ length sides.
b All the faces on regular polyhedra are __________________ polygons.
c The ________________ solids is the name given to the 5 regular polyhedra.

2 Which of the following nets would not fold up to form a cube?

A B C

3 Name the type of shapes that form the faces of these Platonic solids.
a tetrahedron b hexahedron c octahedron
d dodecahedron e icosahedron
4 Name the solids that have the following nets.
a b c

Example 23 Drawing nets

Draw a net for these solids.

a rectangular prism b regular tetrahedron

SOLUTION EXPLANATION
a This is one possible net for the rectangular
prism, but others are possible.

Continued on next page

b Each triangle is equilateral. Each outer triangle

folds up to meet centrally above the centre
triangle.

Now you try

Draw a net for these solids.
a triangular prism b square-based pyramid

Exercise 7L
FLUENCY 1–5 2–6 3(1/2), 4–6

Example 23a 1 Draw a net for this rectangular prism.

Example 23b 2 Draw a net for this pyramid.

Example 23 3 Draw one possible net for these solids.

a b c

d e f

4 Which Platonic solid(s) fit these descriptions? There may be more than one.
a Its faces are equilateral triangles.
b It has 20 faces.
c It has 6 vertices.
d It is a pyramid.
e It has 12 sides.
f It has sides which meet at right angles (not necessarily all sides).

5 Here are nets for the five Platonic solids. Name the Platonic solid that matches each one.
a b

c d

6 How many faces meet at each vertex for these Platonic solids?
a tetrahedron b hexahedron c octahedron
d dodecahedron e icosahedron

PROBLEM-SOLVING 7 7 7, 8

7 Try drawing a net for a cone. Check by drawing a net and cutting it out to see if it works. Here are two
cones to try.
a b

8 How many different nets are there for these solids? Do not count rotations or reflections of the
same net.
a regular tetrahedron b cube

REASONING 9 9 9, 10

9 Imagine gluing two tetrahedrons together by joining two faces as shown, to form a
new solid.
a How many faces are there on this new solid?
b Are all the faces identical?
c Why do you think this new solid is not a Platonic solid.
(Hint: Look at the number of faces meeting at each vertex.)

10 Decide if it is possible to draw a net for a sphere.

ENRICHMENT: Number of cubes – – 11

11 Consider a number of 1 cm cubes stacked together to form a larger cube.

This one, for example, contains 3 × 3 × 3 = 27 cubes.
a For the solid shown:
i how many 1 cm cubes are completely inside the solid with no faces on
the outside?
ii how many 1 cm cubes have at least one face on the outside?
b Copy and complete this table.
n (side length) 1 2 3 4 5
n3 (number of 1 cm cubes) 1 8
Number of inside cubes 0
Number of outside cubes 1

c For a cube stack of side length n cm, n ⩾ 2, find the rule for:
i the number of cubes in total
ii the number of inside cubes
iii the number of outside cubes.

BMX ramp

Modelling
Marion is designing a BMX ramp and wishes to use three equal length pieces of steel for each side of the
ramp. Her design is shown in this diagram with the three pieces of equal length steel shown as the line
segments AB, BC and CD, shown in green.

A B D
The points A, B and D are on a straight line and represent the base of the ramp. The line segment AC
represents the ramp slope.

Present a report for the following tasks and ensure that you show clear mathematical working and
explanations where appropriate.

Preliminary task
a Copy the diagram above, putting dashes on AB, BC and CD to indicate they are the same length.
b What type of triangle is ΔABC? Give a reason.
c If ∠BAC = 15°, use your diagram to find the size of:
i ∠ACB ii ∠ABC.
d If ∠ABC = 140°, determine the size of all the other angles you can find in the diagram.

Modelling task
Formulate a The problem is to find the steepest ramp that Marion can build using the three equal length pieces
of steel. Write down all the relevant information that will help solve this problem, including any
diagrams as appropriate.
Solve b Starting with ∠BAC = 25°, determine the following angles giving reasons for each calculation.
Illustrate with a diagram.
i ∠ACB ii ∠ABC iii ∠CBD iv ∠BDC v ∠BCD
c Determine the angle ∠BDC for:
i ∠BAC = 30° ii ∠BAC = 40°.
d Describe the problem that occurs when using the steel to make a ramp with ∠BAC = 45° and
illustrate with a diagram.
Evaluate and e Is it possible for Marion to use an angle ∠BAC greater than 45°? Explain why or why not.
verify f If the angle ∠BAC must be a whole number of degrees, determine the slope angle for the steepest
ramp possible.
Communicate g Summarise your results and describe any key findings.

Extension questions
Now Marion has five pieces of equal length steel (AB, BC, CD, DE and EF) and she uses them to make a
Modelling

ramp, AE, in the following way.

D F
A B

a Copy out the diagram and find all the angles if ∠BAC = 15°.
b Determine the steepest possible slope angle if all angles in the diagram must be a whole number of
degrees.
c Compare your answer to the angle found in the case when she used three pieces of equal
length steel.

Classifying shapes

Technology and computational thinking

Key technology: Dynamic geometry
Classification systems are used all around the world to help
sort information and make it easier for us to find what we are
looking for. Similarly, shapes are also classified into subgroups
according to their properties, including the number of parallel
sides, side lengths and angles.

1 Getting started
Let’s start by classifying triangles by using this flowchart.
Start

Are there Yes

3 equal Output equilateral
sides?
Output right isosceles
Output obtuse isosceles
No
Yes
Yes

Are there Yes Is there No Is there

2 equal a right an obtuse
sides? angle? angle?

No No

Output acute isosceles

Is there Yes
a right Output right scalene
angle?

Is there Yes
an obtuse Output obtuse scalene
angle?

Output acute scalene End

Work through the flowchart for each of these triangles and check that the algorithm classifies each
triangle correctly.
Technology and computational thinking

a b c

d e

2 Applying an algorithm
The special quadrilaterals that we consider here are: parallelogram, rectangle, rhombus, square, trapezium
and kite.
a Use the definitions in this chapter to think about what shared properties they have. Note the
following definitions:
• Parallelogram: A quadrilateral with two pairs of parallel sides
• Rectangle: A parallelogram with all angles 90 degrees
• Rhombus: A parallelogram with all sides equal
• Square: A rhombus with all angles 90 degrees OR a Rectangle with all sides equal
• Trapezium: A quadrilateral with one pair of parallel sides
• Kite: A quadrilateral with two pairs of adjacent equal sides
b Draw a flowchart similar to the one in part 1 for triangles, that helps to classify quadrilaterals. Test
your algorithm using a range of special quadrilaterals.

3 Using technology
Construct these shapes using dynamic geometry. The construction for an
isosceles triangle is shown in the diagram.
a isosceles triangle
b equilateral triangle
c right-angled triangle

4 Extension
a Construct as many of the special quadrilaterals as
you can using dynamic geometry. The construction
for a rectangle is shown here. Note that the
perpendicular line tool is used in this construction to
save having to construct multiple perpendicular lines
using circles.
b Test that your construction is correct by dragging one
of the initial points. When dragging, the properties of
the shape should be retained.

The perfect billiard ball path

Investigation
When a billiard ball bounces off a straight wall (with no side
spin), we can assume that the angle at which it hits the wall
(incoming angle) is the same as the angle at which it leaves
the wall (outgoing angle). This is similar to how light reflects
off a mirror.

Single bounce
Use a ruler and protractor to draw a diagram for each part wall
and then answer the questions. incoming outgoing
a Find the outgoing angle if: angle centre angle
i the incoming angle is 30° angle
ii the centre angle is 104°.
b What geometrical reason did you use to calculate
the answer to part a ii above?

Two bounces
Two bounces of a billiard ball on a rectangular table are shown here. a°
30°
a Find the values of angles a, b, c, d and e, in that order. Give a c°
b°
reason for each.
e°
b What can be said about the incoming angle on the first d°
bounce and the outgoing angle on the second bounce? Give
reasons for your answer.
c Accurately draw the path of two bounces using:
i an initial incoming bounce of 20° ii an initial incoming bounce of 55°.

More than two bounces

a Draw paths of billiard balls for more than two bounces starting at the
midpoint of one side of a rectangular shape, using the starting incoming
angles below.
i 45° ii 30°
b Repeat part a but use different starting positions. Show accurate diagrams,
using the same starting incoming angle but different starting positions.
c Summarise your findings of this investigation in a report that clearly explains what you have found.
Show clear diagrams for each part of your report.

1 Rearrange six matchsticks to make up

Up for a challenge? If you get stuck
four equilateral triangles. on a question, check out the ‘Working
Problems and challenges

with unfamiliar problems’ poster at

the end of the book to help you.

2 How many equilateral triangles of any size are

in this diagram?

3 What is the angle between the hour hand and minute hand of a clock
at 9 : 35 a.m.?

4 Two circles are the same size. The shaded circle rolls around the other
circle. How many degrees will it turn before returning to its starting
position?

5 A polygon’s vertices are joined by diagonals. How many diagonals

can be drawn in each of these polygons?
a decagon (10 sides)
b 50-gon

6 This solid is made by stacking 1 cm cubes. How many cubes

are used?

Measuring angles Angles

acute 0°− 90°

Chapter summary
right 90°
obtuse 90°− 180°
straight 180°
° 360°
reﬂex 180°− Angles at a point
revolution 360°
b°
a°
c° d°

Geometrical objects Complementary

a + b = 90
A
Supplementary
E
D c + d = 180
C
Vertically opposite
B a=c
∠ABC F Revolution
ray BD a + b + 90 + c + d = 360
line EF
segment AB
collinear points B, C, D Angles and
vertex B parallel lines

Parallel lines

d°
a° c ° tran
sve
b° rsa
l

a = b (corresponding)
a = d (alternate)
a + c = 180 (cointerior)

If a = 120, then b = 120,

d = 120 and c = 60.

Constructions
Two sides and the
Three side lengths
angle between them
C C
3 cm 5 cm 4 cm Compound problems
40° B with parallel lines (Ext)
A 6 cm B A 5 cm
A
Two angles and a side 30°
B
C
C 60°

A 35° 70° B ∠ABC = 30° + 60°= 90°

5 cm

Polygons Type Angle sum

Chapter summary

a°
regular convex irregular scalene isosceles equilateral 110°
octagon non-convex pentagon
60° 30°

acute right obtuse a = 180 – (110 + 30)

= 180 – 140
= 40

Solids
rectangular
cylinder
prism
Exterior angle
Triangles
c°
a°
b°
70°
If a = 70
b = 180 – (70 + 70)
Platonic solids = 40
c = 180 – 40
Regular polyhedron
= 140
tretrahedron (4)
hexahedron (6)
octahedron (8) Polygons, solids
dodecahedron (12) and transformations
icosahedron (20)

Quadrilaterals
Nets
rectangular
cylinder
prism

Symmetry
Special quadrilaterals
5 lines of symmetry parallelogram
rotational symmetry − rectangle

of order 5 − rhombus
− square
trapezium
regular
kite
pentagon
Transformations

Rotation Translation
y y
A (−2, 3) A′ (2, 3) A
3 B
180° 3
2 rotation 2
1 1 D C
x x A′ B′
−3 −2 −1O 1 2 3 −3 −2 −1 O 1 2 3
−1 −1
mirror 90° D′
−2 −2 triangle C′
line clockwise
(y-axis) −3 rotation −3 2 units right and
3 units down

Chapter checklist with success criteria

Chapter checklist
A printable version of this checklist is available in the Interactive Textbook ✔
1. I can name lines, rays and segments.
7A
e.g. Name this line segment.
A B
2. I can name angles.
7A
e.g. Name the marked angle.
P

Q
R
3. I can classify an angle based on its size.
7A
e.g. Classify 134° as an acute angle, a right angle, an obtuse angle, a straight angle, a reflex
angle or a revolution.
4. I can measure the size of angles with a protractor.
7A
e.g. Use a protractor to measure the angle ∠EFG.
G

E
F
5. I can draw angles of a given size using a protractor.
7A
e.g. Use a protractor to draw an angle of size 260°.
6. I can find angles as a point using complementary or supplementary angles.
7B
e.g. Find the value of a in these diagrams.
a b
65°
130° a°
a°

7. I can find the size of angles without a protractor using other angles at a point.
7B
e.g. Find the value of a without a protractor.
a°
120°

8. I can name angles in relation to other angles involving a transversal.

7C
e.g. Name the angle that is (a) alternate to ∠ABF, and (b) cointerior to ∠ABF.
A H

G
B F
C

D E
Essential Mathematics for the Victorian Curriculum ISBN 978-1-009-48064-2 © Greenwood et al. 2024 Cambridge University Press
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502
Chapter checklist Chapter 7 Geometry

✔
9. I can find the size of unknown angles in parallel lines.
7C
e.g. Find the value of a, giving a reason for your answer.

a°
110°

10. I can determine whether two lines are parallel given a transversal.
7C
e.g. State whether the two lines cut by this transversal are parallel.

58°
122°

11. I can solve problems involving parallel lines and angles at a point.
7D Ext
e.g. Find the value of a in this diagram.
D
A

60° 70°
a°
C
B

12. I can classify a triangle as scalene, isosceles or equilateral.

7E
e.g. Classify this triangle based on the side lengths.

13. I can classify a triangle as acute, right or obtuse.

7E
e.g. Classify this triangle based on the angles.

14. I can construct a triangle with given lengths and angles.

7E
e.g. Construct a triangle ABC with AB = 5cm, ∠ABC = 30° and ∠BAC = 45°.

15. I can construct a triangle using a ruler and pair of compasses.

7E
e.g. Construct a triangle with side lengths 6 cm, 4 cm and 5 cm.

16. I can name polygons based on the number of sides.

7F
e.g. State the name for a polygon with five sides.

17. I can classify polygons as convex/non-convex and regular/irregular.

7F
e.g. State whether this shape is convex or non-convex, and whether it is regular or irregular.

Chapter checklist
✔
18. I can classify quadrilaterals.
7F
e.g. Determine whether the quadrilateral shown is convex or non-convex and what type(s) of
special quadrilateral it is.

19. I can use the angle sum of a triangle to find an unknown angle.
7G
e.g. Find the value of a in this triangle.

20° 120°
a°

20. I can find an unknown angle within an isosceles triangle.

7G
e.g. Find the value of a in this diagram.

a°
70°

21. I can find exterior angles for a triangle.

7G
e.g. Find the value of x in this diagram.
62° a°
x°

22. I can determine the line and rotational symmetry of a shape.

7H
e.g. Give the order of line symmetry and of rotational symmetry for a rectangle.

7I 23. I can find the result of a reflection of a point or shape in the coordinate plane.
e.g. The shape ABCD is reflected in the y-axis. State the coordinates of A′, B′, C′ and D′ and
connect them to draw the image.
y

4
3 mirror line
2 (y-axis)
1
x
−4 −3 −2 −1 O 1 2 3 4
D −1
A
−2
B
C −3

7I 24. I can find the result of a rotation of a point or shape in the coordinate plane.
e.g. The triangle ABD is rotated 90° anticlockwise about C. State the coordinates of A′ , B′ and
D′ and hence draw the image.
y

4
3
D
2
1
C A B x
−3 −2 −1
−1 1 2 3
−2
−3
−4
25. I can draw the result of a translation.
7J
e.g. Draw the image of the triangle ABC after a translation 2 units to the right and 3 units down.
A

B C

26. I can describe a translation.

7J
e.g. A point B(5, − 2) is translated to B′(− 1, 2). Describe the translation.

27. I can draw simple solids.

7K
e.g. Draw a cone.

28. I can draw solids on isometric dot paper.

7K
e.g. Draw this solid on isometric dot paper.

29. I can draw a net for a solid.

7L
e.g. Draw a net for a rectangular prism and for a regular tetrahedron.

Short-answer questions

Chapter review
1 Name each of these objects.
7A
a D b A c P

C B
O
d e f
C T

S
A

2 For the angles shown, state the type of angle and measure its size using a protractor.
7A
a b

3 Find the angle between the hour and minute hands on

7A
a clock at the following times. Answer with an acute or
obtuse angle.
a 6:00 a.m.
b 9:00 p.m.
c 3:00 p.m.
d 5:00 a.m.

4 Without using a protractor, find the value of a in these diagrams.

7B
a b c
Chapter review

a°
70° 130°
145°
a°
a°
d e f
a°

a° 75°
41° a°
52°

g h i
(a + 30)°
(2a)°
a° a° a°
(2a)°

5 Using the pronumerals a, b, c or d given in the diagram, write b°

7C
down a pair of angles that are:
a vertically opposite
a°
b cointerior
c alternate
c°
d corresponding d°
e supplementary but not cointerior.

6 For each of the following, state whether the two lines cut by the transversal are parallel. Give
7C
reasons for each answer.
a b c
65°

92° 60°
65°
89°
130°

7 Find the value of a in these diagrams.

7D
a b c
Ext
85° a°
80° a°
59° a°
70°

d e f
a°

Chapter review
70°
a° 32°
140°
a°
150°

8 Use a protractor and ruler to construct these triangles.

7E
a triangle ABC with AB = 4 cm, ∠CAB = 25° and ∠ABC = 45°
b triangle ABC with AB = 5 cm, ∠BAC = 50° and AC = 5 cm

9 Use a protractor, pair of compasses and a ruler to construct these triangles.

7E
a triangle ABC with AB = 5 cm, BC = 6 cm and AC = 3 cm
b triangle ABC with AB = 6 cm, BC = 4 cm and AC = 5 cm

10 How many sides do these polygons have?

7F
a pentagon b heptagon c undecagon

11 A diagonal inside a polygon joins two vertices. Find how many

7F
diagonals can be drawn inside a quadrilateral if the shape is:
a convex b non-convex.

12 Name each of these quadrilaterals.

7F
a b c

13 Find the value of a in each of these shapes.

7G
a b 42° c
65°
80°
a° a°
70° a°

d e f
40°
20°
a°
a°
a°
g h i
a°

110° a° 15°
75° 25°
a°

14 Name the order of line and rotational symmetry for each of these diagrams.
7H
a b c
Chapter review

15 Write the coordinates of A′, B′ and C′ when this shape is y

7I
reflected in the following mirror lines.
4
a the y-axis 3
b the x-axis 2
1
x
−4 −3 −2 −1 O 1 2 3 4
−1 C
A
−2
−3 B
−4

16 Points A(0, 4), B(2, 0) and D(3, 3) are shown here. Write y
7I
down the coordinates of the image points A′, B′ and D′ 4 A
after each of the following rotations. D
3
a 180° about C(0, 0) 2
b 90° clockwise about C(0, 0) 1
C B
c 90° anticlockwise about C(0, 0) x
−4 −3 −2 −1
−1 1 2 3 4
−2
−3
−4

17 Write the coordinates of the vertices A′, B′ and C′ after y

7J
each of these translations.
4
a 4 units right and 2 units up 3
b 1 unit left and 4 units up 2
1
x
−4 −3 −2 −1 O 1 2 3 4
C −1
−2
−3
A B
−4

18 Draw a side view, top view and net for each of these solids.
7K
a b

Multiple-choice questions

Chapter review
1 Three points are collinear if:
7A
A they are at right angles.
B they form a 60° angle.
C they all lie in a straight line.
D they are all at the same point.
E they form an arc on a circle.

2 The angle shown here can be named: P

7A
A ∠QRP C ∠QPR E ∠PQP
B ∠PQR D ∠QRR

Q R
3 Complementary angles:
7B
A sum to 180° B sum to 270° C sum to 360°
D sum to 90° E sum to 45°

4 A reflex angle is:

7A
A 90° B 180° C between 180° and 360°
D between 0° and 90° E between 90° and 180°

5 What is the size of the angle measured by the

7A 80 90 100 11
70
protractor? 0
60 110 100 90 80 70 120
0 60 13
50 0 12
50 0
A 15° C 105° E 195° 1 3
150 40

14
0

0
40
14
60 30

15
B 30° D 165°

30
0 1
20

60 17
20
180 170 1
0 10

10 0
6 The angle a minute hand on a clock turns in 20

0 180
7A
minutes is:
A 72° B 36° C 18° D 144° E 120°

7 If a transversal cuts two parallel lines, then:

7C
A cointerior angles are equal.
B alternate angles are supplementary.
C corresponding angles are equal.
D vertically opposite angles are supplementary.
E supplementary angles add to 90°.

8 The three types of triangles all classified by their interior angles are:
7E
A acute, isosceles and scalene.
B acute, right and obtuse.
C scalene, isosceles and equilateral.
D right, obtuse and scalene.
E acute, equilateral and right.

9 A non-convex polygon has:

7F
A all interior angles of 90°.
Chapter review

B six sides.
C all interior angles less than 180°.
D all interior angles greater than 180°.
E at least one interior angle greater than 180°.

10 The quadrilateral that has 2 pairs of sides of equal length and 1 pair of angles of equal size is
7F
called a:
A kite B trapezium C rhombus D triangle E square

11 A rhombus has line symmetry of order:

7H
A 0 B 1 C 2 D 3 E 4

12 The point T(− 3, 4) is reflected in the x-axis; hence, the image point T′ has coordinates:
7I
A (3, 4) B (− 3, 4) C (0, 4) D (3, − 4) E (− 3, − 4)

13 The translation that takes A(2, − 3) to A′(− 1, 1) could be described as:

7J
A 3 units left.
B 4 units up.
C 3 units left and 4 units up.
D 1 unit right and 2 units down.
E 1 unit left and 2 units down.

Extended-response questions
1 A factory roof is made up of three sloping sections. The E F G
sloping sections are all parallel and the upright supports are
at 90° to the horizontal, as shown. Each roof section makes A D
a 32° angle (or pitch) with the horizontal. B C
factory
a State the size of each of these angles.
i ∠EAB
ii ∠GCD
iii ∠ABF
iv ∠EBF
b Complete these sentences.
i ∠BAE is ____________________________ to ∠CBF.
ii ∠FBC is ____________________________ to ∠GCB.
iii ∠BCG is ____________________________ to ∠GCD.
c Solar panels are to be placed on the sloping roofs and it is decided that the angle to the
horizontal is to be reduced by 11°. Find the size of these new angles.
i ∠FBC
ii ∠FBA
iii ∠FCG

2 Two cables support a vertical tower, as shown in the

diagram opposite, and the angle marked a° is the angle D

Chapter review
between the two cables.
a Find ∠BDC. a°
b Find ∠ADC.
c Find the value of a.
d If ∠DAB is changed to 30° and ∠DBC is changed 25° 60°
to 65°, will the value of a stay the same? If not,
A B C
what will be the new value of a?

3 Shown is a drawing of a simple house on a Cartesian y

plane.
4
Draw the image of the house after these transformations.
3
a translation 5 units left and 4 units down
2
b reflection in the x-axis 1
c rotation 90° anticlockwise about C(0, 0) x
−4 −3 −2 −1 O 1 2 3 4
−1
−2
−3
−4

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