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6 Geometry and space
Australian Curriculum links
Year 6
Proficiency strands
Problem-solving includes calculating angles
Reasoning includes investigating new situations using known properties of angles
Content descriptions
Measurement and geometry
Shape
Elaborations
Construct simple prisms and pyramids
(ACMMG140)
• considering the history and significance of pyramids from a range of cultural
perspectives including those structures found in China, Korea and Indonesia
• constructing prisms and pyramids from nets, and skeletal models
Year 7
Content descriptions
Measurement and Geometry
Shape
Elaborations
Draw different views of prisms and solids
formed from combinations of prisms
(ACMMG161)
• using aerial views of buildings and other 3-D structures to visualise the
structure of the building or prism
Geometric reasoning
Elaborations
Classify triangles according to their side and
angle properties and describe quadrilaterals
(ACMMG165)
• identifying side and angle properties of scalene, isosceles, right-angled and
obtuse angled triangles
• describing squares, rectangles, rhombuses, parallelograms, kites and
trapeziums
Demonstrate that the angle sum of a
triangle is 1808 and use this to find the
angle sum of a quadrilateral (ACMMG166)
• using concrete materials and digital technologies to investigate the angle sum
of a triangle and quadrilateral
Identify corresponding, alternate and
cointerior angles when two parallel straight
lines are crossed by a transversal
(ACMMG163)
• defining and classifying angles such as acute, right, obtuse, straight, reflex and
revolution, and pairs of angles such as complementary, supplementary,
adjacent and vertically opposite
• constructing parallel and perpendicular lines using their properties, a pair of
compasses and a ruler, and dynamic geometry software
Investigate conditions for two lines to be
parallel and solve simple numerical
problems using reasoning (ACMMG164)
• defining and identifying alternate, corresponding and allied angles and the
relationships between them for a pair of parallel lines cut by a transversal,
including using dynamic geometry software
Year 8
Proficiency strands
Reasoning includes using congruence to deduce properties of triangles
Content descriptions
Measurement and geometry
Geometric reasoning
Elaborations
Define congruence of plane shapes using
transformations (ACMMG200)
• understanding the properties that determine congruence of triangles and
recognising which transformations create congruent figures
• establishing that two figures are congruent if one shape lies exactly on top of
the other after one or more transformations (translation, reflection, rotation),
and recognising the equivalence of corresponding sides and angles
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MathsWorld 7 Australian Curriculum edition Teacher book
Develop the conditions for congruence of
triangles (ACMMG201)
• constructing triangles using the conditions for congruence
• solving problems using the properties of congruent figures, justifying reasoning
and making generalisations
• investigating the minimal conditions needed for the unique construction of
triangles, leading to the establishment of the conditions for congruence (SSS,
SAS, ASA and RHS), and demonstrating which conditions do not prescribe
congruence (ASS, AAA)
• plotting the vertices of two-dimensional shapes on the Cartesian plane,
translating, rotating or reflecting the shape and using coordinates to describe
the transformation
Establish properties of quadrilaterals using
congruent triangles and angle properties,
and solve related numerical problems using
reasoning (ACMMG202)
• establishing the properties of squares, rectangles, parallelograms, rhombuses,
trapeziums and kites
• identifying properties related to side lengths, parallelism, angles, diagonals and
symmetry
Source: Australian Curriculum
Pre-test
Most students should be able to recognise common geometric shapes by their appearance, even if they
are not aware of geometric properties at this stage. Students need to be able to recognise shapes when
they are represented in a non-standard position. For example, some students are unable to recognise a
square as a square if it is tilted. Similarly, some students think shapes J and K in question 8 are triangles
because they are ‘pointy’, rather than noticing that they each have more than three sides.
1
●
Look at this set of letters.
a Which of the letters have perpendicular lines?
b Which letters have parallel lines?
c Which letters have both parallel and perpendicular lines?
2
●
3
●
4
●
What word could be used to describe this set of lines?
Which of these things are usually vertical and which are usually horizontal?
a tabletop
b floor
c classroom walls
d ceiling
e whiteboard
Look again at the letters in question 1.
a Which of the letters contain right angles?
b In which of the letters can you see acute angles?
c In which of the letters can you see obtuse angles?
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For each of these angles
i name the angle using the given letters.
ii what is the size of each of these angles?
40
40
30
40
0 90 80 7
0
10 10
01
2
01
80 90 100 110
13
12
60
01
30
50
0
170 180
10
20
K
160
30
50
0
170 180
10
20
160
30
R
70
60
40
30
50
50
J
50
8
●
30
Q
L
Use your protractor to measure each of these angles and state whether the angle is an acute,
obtuse or reflex angle.
a
7
●
60
01
01
13
12
40
0 10 2
0
100 90 80 70
110
14
0
2
01
80 90 100 110
01
180 170 1
60 1
50
1
70
60
14
40
50
6
●
b
P
0 10 2
0
a
180 170 1
60 1
50
1
5
●
6
chapter
Geometry and space
c
b
How many degrees are in each of the following?
a a straight angle
b one revolution about a point
c a right angle.
Which of the following shapes are
a triangles?
b quadrilaterals?
A
B
C
D
E
F
G
H
I
J
K
L
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MathsWorld 7 Australian Curriculum edition Teacher book
9
●
10
●
11
●
Which of these shapes are squares?
A
B
C
D
Consider the shapes on the right.
a This triangle has three equal sides.
What do we call this type of triangle?
b This quadrilateral has both pairs
of opposite sides equal and all the
angles are right angles. What do we
call this type of quadrilateral?
Name these three-dimensional shapes.
a
b
c
6 cm
6 cm
12
●
6 cm
Which of these shapes are prisms?
A
B
C
D
Ans wer s
1
●
2
●
3
●
4
●
5
●
6
●
7
●
8
●
a E, F, H, L, T
b E, F, H, Z
c E, F, H
Parallel
Horizontal: tabletop, floor, ceiling
Vertical: classroom walls, whiteboard
a E, F, H, L, T
a i
b i
/PQR or /RQP
/JKL or /LKJ
b A, K, V, W, X, Y, Z
c A, K, X, Y
ii 80°
ii 145°
a 808, acute
b 3048, reflex
c 1058, obtuse
a 1808
b 3608
c 908
a C, F, L
b A, B, D, G, H, I, J
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9
●
10
●
11
●
12
●
6
chapter
Geometry and space
A, D
a equilateral triangle
b rectangle
a cube
b square pyramid
c hexagonal prism
B, C
Warm-up: parking problem
Chapter warm-ups are included in the teacher and student ebooks as separate worksheet for ease of
printing. This chapter warm-up links with the analysis task Parking problem at the end of the chapter,
where students investigate the advantages and disadvantages of each type of parking. The parking signs
below indicate whether parallel parking or angle parking applies. Parallel parking is where the cars
park end-to-end parallel to the kerb (the edge of the roadway), as shown in the photograph, and angle
parking is where the cars park at an angle to the kerb, usually 458, 608 or 908.
a Find out the type of parking––angle parking or parallel parking––that applies in the road outside
your school, in the shopping street of your town or suburb, or in the road where you live.
b Why do you think this type of parking has been used?
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MathsWorld 7 Australian Curriculum edition Teacher book
Teaching note: the van Hiele theory of geometry learning
When thinking about the teaching and learning of geometry, it is useful to gain an understanding of the
levels of student understanding identified by Pierre van Hiele and Dina van Hiele-Geldof. In the 1950s
this Dutch couple extensively explored children’s learning of geometry concepts. Russian mathematics
educators found their work of interest, but it was not until the 1980s that the van Hiele theory came to
the attention of American researchers.
The theory has now been the subject of research world-wide and various aspects of the theory have been
criticised. There has been a renumbering of the original levels, so that the van Hieles’ Level 0 is now
Level 1. John Pegg has also suggested the splitting of the current Level 2 (old Level 1) into two levels.
The van Hiele theory has also been criticised for its hierarchical nature of learning (that children do not
progress to a higher level until they have mastered the previous level) and research has now shown that
children can be at different levels for different geometry concepts. However, it does provide a useful
framework for thinking about the development of geometric understanding. For further reading see
Pegg, J (1995), ‘Learning and teaching geometry’, in: Grimison L & Pegg J, Teaching secondary school
mathematics: theory into practice. Sydney, Harcourt Brace, 1995.
In the currently used numbering, the van Hiele levels may be summarised as follows:
Level
Characteristic of the level
Example
Level 1
Recognising shapes by their visual
appearance alone
That shape is a square.
Level 2
Level 2a
Level 2b
Recognising shapes by their properties
Recognising one property
Recognising more than one property
Level 3
Recognising relationships between
properties
If all the angles of a square are right angles,
then the opposite sides must be parallel.
Level 4
Deductive reasoning and
understanding the minimum properties
that will identify a shape
A rhombus with one right angle is sufficient
to identify a shape as a square (why?);
students are able to complete proofs such as
the angle sum of a triangle.
The shape is a square because it has four
equal sides
The shape is a square because it has four
right angles, opposite sides are parallel, all
sides are equal.
Many students go through secondary school with a Level 1 understanding of geometry. The
MathsWorld Australian Curriculum books place a strong emphasis on developing students’ thinking and
understanding beyond Level 1.
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6.1 Lines, rays and segments
Teaching note: parallel lines
Many students will be familiar with parallel bars in the playground or in gymnastics. It is important for
students to recognise that the definition ‘Parallel lines are lines that will never meet even if they are
extended in either direction’ is incomplete. This is the case only if the lines are in the same plane. We can
also say that the perpendicular distance between parallel lines is the same at all points along the lines.
Question 13 in exercise 6.1 is worth doing as a class activity, with each
pair of students having a rectangular cardboard box, such as a shoe box
or cornflakes packet. This provides an excellent means of conveying the
notion that lines that never meet are not necessarily parallel. This idea
can also be developed by looking at lines in the classroom, for example a
vertical wall edge and its opposite horizontal ceiling-wall edge. However,
the hands-on approach of looking at edges and drawing lines on a box
helps students to see directly that there are line segments that will never
meet, even if extended, that are not parallel. We must specify then that
parallel lines are lines in the same plane that will never meet.
A
B
D
C
F
H
G
Introducing symbols
Symbols for parallel and perpendicular are introduced in drawings as well as
in written statements, for example AB || CD.
Extra example 1
A four-sided shape is shown on the right.
b
a Which sides of this shape appear to be parallel?
b Mark the diagram with symbols to show this.
a
c
d
Working
Reasoning
a Sides b and d appear to be parallel.
These two sides
b
are the same distance apart at all positions.
■
are in the same plane.
■
do not meet.
We use arrowheads to show that two lines or
line segments are parallel.
b
a
■
c
d
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Extra example 2
The diagram on the right shows two line segments, AB and CD. C
a What word can we use to describe AB and CD?
B
b Mark the diagram with a symbol to show this.
90°
A
D
Working
Reasoning
a AB and CD are perpendicular.
Lines are perpendicular if there is an angle of
908 between them.
b
The right angle symbol shows that the
segments are perpendicular.
C
B
90°
A
D
Teaching note: compass and ruler constructions
Compasses are not always appropriate in classrooms. The Math Open Reference Project website has
excellent animations of compass and ruler constructions that can be paused and replayed. Scrolling
down the page leads to a clearly set out proof of why each construction works.
www.mathopenref.com/tocs/constructionstoc.html
The following are direct links to constructing a line perpendicular to a given line passing through a
point on the line (as in example 3 in the student book) and passing through an external point (as in
example 4 in the student book).
www.mathopenref.com/constperplinepoint.html
www.mathopenref.com/constperpextpoint.html
exercise 6.1
Below are the answers to the questions in exercise 6.1 in the student book.
Ans wer s
1
●
Line
Ray
Segment
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2
●
3
●
4
●
6
There are parallel segments in the rows of bricks. The legs of the parallel bars are parallel to each
other.
a CD, EF
chapter
Geometry and space
6.1
b AB, CD, EF
A
C
B
D
5
●
P
R
S
Q
6
●
In striped material, the stripes are parallel to each other. In check material the horizontal stripes
are perpendicular to the vertical stripes.
7
●
F
A
8
●
9
●
10
●
C
B
D
E
Chris is building a house with rectangular rooms. He has checked that the floor is horizontal. The
walls must be vertical and perpendicular to the floor. The walls on opposite sides of the room must
be parallel to each other. Two walls meeting at a corner of the room must be perpendicular to each
other.
a perpendicular
b vertical
a
c parallel
d horizontal
b
c
11
●
a meet
c don’t meet, but not parallel
b don’t meet, but not parallel
d don’t meet and parallel
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12
●
a Bay Street, Bank Street or Junction Street
b Main Street, Tower Street or River Street
13
●
Banksia Street
Railway Avenue
Grant Street
exercise 6.1
2
●
3
●
Explain why you think this shape is called a parallelogram.
These two lines are parallel.
a Measure how far apart the lines are.
b Show on the diagram where you measured to
find how far apart the lines are.
Jess is making a corduroy teddy bear for her little brother. She notices that each of the paper
pattern pieces has a line labelled ‘Place on straight grain of fabric’. Using the word parallel, explain
in a sentence what you think ‘Place on straight grain of fabric’ means.
Paper
pattern
piece
Place on
straight grain
of fabric
1
●
additional questions
CUT 2
BODY
FRONT
Corduroy
fabric
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Ans wer s
1
●
2
●
6
chapter
Geometry and space
6.1
Both pairs of opposite sides of the shape are parallel.
a 2 cm
b
2 cm
3
●
Place the arrow on the pattern parallel to the direction of the lines in the fabric.
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6.2 Angles
Teaching note: dual concept of angle
It is important that students understand the dual nature of the concept angle,
■
a dynamic concept of an amount of turning, for example as a door opens,
■
and the more familiar static concept of an angle at a vertex.
Plastic geostrips or strips of card and paper fasteners are useful in developing the concept of an angle as
an amount of turning. Plastic geostrips are available from Modern Teaching Aids, www.teaching.com.au.
Another useful approach is to take students outside where there is room to spread out and ask them
to make angles with their arms. If they start by putting both arms out horizontally together to the left,
they can then move their right arm in a clockwise direction to make approximate representations of
given angles.
Teaching note: developing angle sense
Students enjoy playing a game of ‘Simon Says’. Instruct them to close their eyes, turn clockwise through
a right angle, turn anticlockwise through a straight angle, and so on.
Students can vary this by making the angles with outstretched arms rather than turning their body. By
concentrating so hard on the direction and type of angle, students make enough errors for this to be fun.
Teaching note: symbol sense
Students should be encouraged to use correct mathematical language. They will see that we use the
same symbol for perpendicular and a right angle because perpendicular means at right angles to.
Teaching note: estimating angles
As well as being able to measure angles accurately, students should also be able to estimate angle sizes.
A useful activity is for students to work in pairs, each carefully drawing a set of angles using pencil,
ruler and protractor. Each student keeps a record of the sizes of the angles they have constructed. The
angle sheets can then be swapped, with each student estimating the sizes of the angles drawn by the
other student, then comparing their estimates with the constructed angle sizes. This can also be used as
an exercise in accurate construction.
Teaching note: measuring angles
Three angle-measurers used by various trades people are shown on page 256 of the student book. There
may be students in your class whose parents use one of these in their regular work. Students could
think about situations where angles are important.
The Year 5 Australian Curriculum includes ‘measuring and constructing angles using both 1808 and
3608 protractors’. Measuring angles with each protractor is included here for those students who do not
already have this skill.
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6
Students should think about the type of angle before they start, so they can check that they have
used the correct scale on their protractors. The 3608 protractor is easier to use when working with
reflex angles.
chapter
Geometry and space
6.2
Extra example 3
Use a 3608 protractor to measure the following angles.
a
b
c
Working
Reasoning
a
Place the protractor with its centre at the
vertex of the angle, with the zero mark on one
arm of the angle. Measure around from zero
in a clockwise direction on the outer scale.
0
40 35
03
3
03
30
40
33
03
20
50
60
100 90 80 70
10
01
13
0
30
1
22
0
02
14
10
01
50
200
160
190
170
50
60 1
170 1
180
14
23
0
30
4
02
Check: the angle is an acute angle so its
measure must be between 08 and 908.
12
02
12
24
50 26
280 270 260 250
80 90 100 110
290
70
00
02
60
03
10
03
0 270 280 29
20
31
00
31
40
30
350 3
40
50
0
32
10 2
0
0
20
02
21
190 20
0
The angle measures 738.
b
0 270 280 29
26
250
40
02
0
22
50
14
0
10
60 1
340
350
10
170 1
Place the protractor with its centre at the
vertex of the angle, with the zero mark on
one arm of the angle. Measure around from
zero in an anticlockwise direction on the
inner scale.
0
0
02
0 190
0
22
21
31
03
00
60
30
290
70
280 270 260 250
02
14
40
24
80 90 100 110
50
01
30
03
20
33
10 2
0
350 3
40
Check: the angle is an obtuse angle so its
measure must be between 908 and 1808.
50
02
30
20
190 20
0
1
30
180
31
50
40
170
1
30
00
60
03
160
0 90 80 70
10 10
1
20
32
23
03
30
01
12
The angle measures 1248.
continued
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Extra example 3 continued
Reasoning
c
Place the protractor with its centre at the
vertex of the angle, with the zero mark on one
arm of the angle. Measure around from zero
in a clockwise direction on the outer scale
because it is the reflex angle that is required.
40
350
340
30
10 2
0 3
0
0
20
80 90 100 110
70
12
50
70
60
60
40
10
110
20
01
13
270 260 2
50 2
Check: the angle is a reflex angle so its
measure must be between 1808 and 3608.
0
01
23
30
22
02
01
50
10
200
160
60 1
50
14
0
20
170 1
190
170
23
40
350 3
40 3
30
3
10
20
280
290
00
03
31
260 270 280 290
250
30
40
03
2
0 90 80
0
1
30
14
3
3
20
50
0
Working
180
0 200
21
02
19
The angle measures 2438.
Extra example 4
Use a 3608 protractor to draw the following angles.
a 1958
b 988
Working
c 378
Reasoning
a
01
40
23
0
30
01
50
20
23
40
30
20
10
190 20
02
10
2
60 1
50
14
0
01
13
2
0
50
0
Rule a line segment for one arm of the angle.
Place the protractor with its centre at the
right-hand end of the line segment as shown.
Place a pencil mark at 1958. Join the pencil
mark to the right hand of the line segment to
make the other arm of the angle.
180
170 1
32
170
190
200
160
10
350 3
40 3
30
32
0
0 270 260 25
90 28
02
2
00
03
31
02
22
10 2
0
0 3
40 35 0
03
0
33
60
14
40
50
80 90 100 110
70
12
260 270 280 290
250
30
40
03
0 90 80 70
02
10
10 10
6
01
195°
continued
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6
Extra example 4 continued
6.2
Working
Reasoning
b
50
60
1
31
0
50
170 1
60
350
10
20
340
30
15
01
40
1
30
00
90 80 7
0
30
10
20
90 3
100
110
40
0
190
200
10
02
22
31
03
00
50
60
30
290
70
280 270 260 250
80 90
14
40
5
01
20
10 2
0 3
0
350 3
40 3
30
3
02
24
19
0
60 17 180 0 200 2
24
Rule a line segment and place the centre of the
protractor at the left-hand end of the segment,
with the protractor line exactly along the
segment. Measure around from 0 to 988 and
place a small mark. Remove the protractor and
join the mark to the left-hand end of the line
segment to form the angle.
03
22
0
23
260 270 280 2
32
0
02
01
chapter
Geometry and space
0
20
100 11
01
13
98°
c
03
40
31
13
0
23
0
50
350 3
40
10
350
20
30
40
13
01
23
20
02
40
110
250
100 90 80 70
260 270
50
60
3
20
10
2
14
0
33
20
50
40
03
Rule a line segment and place the centre of
the protractor at the right-hand end of the
segment, with the protractor line exactly along
the segment. Measure around from 0 to 378
and place a small mark. Remove the protractor
and join the mark to the right-hand end of the
line segment to form the angle.
02
60 1
190 20
170 1
180
0
190
170
200
160
10
33
03
02
20
02
22
30
20
0 270 260 25
90 28
2
00
01
10 2
0
60
14
40
50
80 90 100 110
70
1
0
00
280 29
03
31
37°
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MathsWorld 7 Australian Curriculum edition Teacher book
exercise 6.2
BLM
This exercise includes questions that reinforce the concept that we must be directly in front of an angle
to measure it, and that angles can be important for safety (for example in wheel chair ramps) and in
sport. A blackline master is included in the teacher and student ebooks providing a larger diagram for
question 13.
Question 13
Ans wer s
Below are the answers to the questions in exercise 6.2 in the student book.
1
●
2
●
3
●
4
●
5
●
6
●
7
●
8
●
9
●
10
●
11
●
a
c
e
g
i
/TOP (or /POT), acute, 508
/BUS (or /SUB), reflex, 2968
/HDG (or /GDH), obtuse, 1638
/MOP (or /POM), right, 908
/HFK (or /KFH), straight, 1808
b
d
f
h
/JCP (or /PCJ), obtuse, 1308
/PAT (or /TAP), reflex, 2708
/XYZ (or /ZYX), acute, 258
/KTP (or /PTK), obtuse, 1308
a obtuse angle
e obtuse angle
b acute angle
f reflex angle
c reflex angle
g revolution
d right angle
h straight
a line
e reflex angle
b perpendicular
f ray
c obtuse angle
g acute angle
d line segment
h parallel
a 668
e 758
b 3018
f 1538
c 1258
g 2908
d 2408
h 1058
a 708
f 958
b 2608
g 2058
a 338
b 408
c 1508
h 638
d 458
i 2858
e 1328
j 3458
a The photograph at the left, because it has been taken from a position more directly in front of
the beam.
b approximately 308
c 608
a A wheelchair ramp cannot be too steep otherwise the person will go down the slope too
quickly and will not be able to stop. The ski slope needs to be steep enough for the skier to
build up enough speed to keep going. The escalator needs to be steep enough so that it does
not take up too much horizontal space, but not so steep that people could fall over on it.
b i 38
ii 308
iii 308
c The photograph has been taken from a position side on to the escalator. If the angle is to be
measured accurately, the photograph must be taken from a position directly in front of the
escalator, at right angles to the tiled wall.
a 408
b 538; yes, the angle is greater than 508
a a = 49, b = 50, c = 51, d = 48, e = 41, f = 39, g = 36, h = 39
b answers will vary
a obtuse
b acute
35°
70°
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c reflex
d reflex
6
e acute
155°
chapter
Geometry and space
6.2
25°
80°
f acute
270°
g obtuse
h acute
313°
138°
12
●
13
●
Answers will match question 11.
a
b mid-on
A
61°
283°
B
264°
110°
138°
142°
153°
165°
194°
168°
180° 172°
175°
185°
exercise 6.2
1
●
additional questions
Match each of the terms listed on the left with their correct description from the list on the right.
Line segment
an angle of 908
Ray
lines that are at right angles to each other
Parallel lines
an angle of 1808
Perpendicular lines
an angle less than 908
Acute angle
part of a line with a definite starting point but no finishing point
Obtuse angle
an angle greater than 908 but less than 1808
Right angle
part of a line with a definite starting point and finishing point
Straight angle
an angle of 3608
Reflex angle
two lines which are always the same distance apart
Revolution
an angle greater than 1808 but less than 3608
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2
●
In
a
b
c
d
e
the following diagram name
a pair of parallel line segments.
a pair of perpendicular lines.
an acute angle.
an obtuse angle.
a right angle.
E
F
G
A
D
C
B
Ans wer s
1
●
2
●
a
b
c
d
e
Line segment
part of a line with a definite starting point and finishing point
Ray
part of a line with a definite starting point but no finishing point
Parallel lines
two lines which are always the same distance apart
Perpendicular lines
lines that are at right angles to each other
Acute Angle
an angle less than 908
Obtuse Angle
an angle greater than 908 but less than 1808
Right Angle
an angle of 908
Straight Angle
an angle of 1808
Reflex Angle
an angle greater than 1808 but less than 3608
Revolution
an angle of 3608
FD and GC
Answers will vary. Two answers are: FD and BE; GC and BE.
Answers will vary. Two answers are: /FGC; /DEF.
one of /AGC or /AFD
one of /EDF or /ECG
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6.3 Calculating angle sizes
Teaching note: using geometric language
Students should appreciate how appropriate geometric language (for example, ‘vertically opposite
angles’, ‘supplementary angles’) facilitates communication. They should be given the opportunity to
practise this language. This is one of the benefits of students using interactive geometry software such as
GeoGebra, particularly when they are working in pairs. Using the names of the various software tools
(for example, ‘perpendicular line’, ‘parallel line’, ‘midpoint’) reinforces correct language.
GeoGebra: complementary and supplementary angles
Complementary
and
supplementary
angles
The GeoGebra (and HTML) file Complementary and supplementary angles in the student and teacher
ebooks allows the size of angles to be changed to see pairs of angles that are complementary or
supplementary.
GeoGebra: vertically opposite angles
Vertically
opposite
angles
The interactive GeoGebra file, Vertically opposite
angles, is included in the teacher and student ebooks
in both GeoGebra and HTML formats. Students
will notice that any two adjacent angles make a
straight angle. The reasoning about why vertically
opposite angles are equal is based on the reasoning ‘if
a 1 b 5 180 and a 1 c 5 180, then b and c must be
equal’.
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Teaching note: when do we put a degrees symbol?
Pronumerals are introduced to refer to unknown angles as an alternative to using angle names such
as /ABC. It should be noted that in this chapter the pronumerals used for angle sizes stand for a
number—the number of degrees in the angle. The pronumerals already have a degrees sign after them
on the diagram, so the answers will be numeric only, without the degrees sign; for example, a 5 50, not
a 5 50°.
Teaching note: using algebra
Students could be introduced to equations informally as shown in extra example 5 part a and example
13 part b in the student book. Alternatively, they could use a visual or arithmetic approach and show
their working on a copy of the diagram.
Extra example 5
a Find the value of a.
b ABC is a straight line. Find the size of /ABD.
D
78°
a°
36°
A
B
C
Working
Reasoning
a a 1 36 = 90
a = 90 2 36
a = 54
The two adjacent angles add to 908.
b /ABD 1 /DBC
/ABD 1 788
/ABD
/ABD
=
=
=
=
1808
1808
1808 2 788
1028
ABC is a straight line. /ABD and /DBC are
supplementary angles.
Worksheet: pool angles
The worksheet Pool angles is included in the student and teacher ebooks.
Pool angles
This activity uses a pool table as its context. A diagram first shows how the ball bounces off the edge
of the table at the same angle as it strikes the table. Students are then asked to use their ruler and
protractor to draw the path of the ball after one, two, three and four bounces.
Pool angles
If you have ever played pool you will know the importance of angles.
When a ball hits the edge, it bounces away at the same angle, as shown.
The drawing below shows a pool table. A ball is hit in the direction
shown.
a Use your ruler and protractor to find where the ball is going to hit
after its first bounce. Carefully draw the path of the ball.
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6
b Using your ruler and protractor, carefully draw the path of the ball for its second, third and fourth
bounces. (Assume that the ball continues to bounce so that it forms equal angles with the edges of
the table each time.)
chapter
Geometry and space
6.3
55°
Ans wer s
a
b
55°
55°
exercise 6.3
Below are the answers to the questions in exercise 6.3 in the student book.
Ans wer s
1
●
2
●
3
●
4
●
5
●
a 1458
b 238
c 908
d 588
e 1718
f 428
a 218
b 1098
c 728
d 1608
e 1158
f 1258
a 768
b 658
c 948
d 1008
e 588
f 408
a
c
e
g
i
a = 38, b
a = 61, b
a = 82, b
a = 70, b
k 5 72
=
=
=
=
38, c = 114
110, c = 110
59, c = 39, d = 59
39, c = 71, d = 70
Angle Complement
b
d
f
h
d = 26, e = 116, f = 64
a = 48, b = 132, c = 42
a = 110, b = 70, c = 70
a = 36
Angle Complement
148
768
328
588
888
28
758
158
458
458
158
758
908
08
18
898
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6
●
7
●
8
●
9
●
10
●
11
●
12
●
13
●
14
●
15
●
16
●
17
●
a a = 50
b b = 10
Angle Supplement
c c = 32
208
908
908
238
1578
1758
58
918
898
788
1028
1008
808
18
1798
b b = 86
a 1078
a
d
g
j
e e = 64
f f = 24
d d = 45
e e = 65
f f = 90
Angle Supplement
1608
a a = 32
d d = 45
c c = 138
b ADB is a straight line, so the angles must add to 1808
e = 46
m = 23
a = 45, b = 45, c = 85
x = 285
a a = 60; b = 120
b
e
h
k
d = 70
h = 68
x = 66
x = 97
y = 107
w = 33
x = 40
x = 90
c
f
i
l
b c = 60; d = 120; e = 60; f = 120
a 308
c 608
e Queensberry Street
b 308; same size as angle at A
d 1208; supplementary angles
f Elizabeth Street, Swanston Street
a = 24
a /AOB or /AOC or straight angle /AOD
b /AOC
808
a 368
b Bec and Sarah’s total angle = 1088, Emma’s total angle = 1448
a Blue 908; Green 1208; Yellow 728
b Red 788
c
72°
120°
78°
90°
exercise 6.3
additional questions
Teaching note
In question 2, students may need help in reasoning that at 10 minutes past 5, the hour hand will have
moved towards 6. They have already calculated in part c that the hour hand moves 0.58 per minute, so
in 10 minutes, the hour hand will have moved through 10 3 0.5°, that is, 58. Hence, the required angle is
the angle between 2 and 5 (that is, 908) plus 58, making an angle of 958 between the two hands.
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1
●
2
●
a Through how many degrees does the minute hand of a clock
turn each minute?
b What is the size of the obtuse angle between the hour hand and the
minute hand of a clock at exactly 5 o’clock?
c Through how many degrees does the hour hand turn each minute?
d What is the angle between the hour hand and the minute hand at
exactly 10 minutes past 5 o’clock?
chapter
6
Geometry and space
6.3
11 12 1
12
2
9
3
8
4
7
6
5
Justify your reasoning for each of the following. For example, you could copy the diagram and
label the sizes of any angles you calculate along the way to finding the required angle.
a Calculate the size of ⬔MFX.
b Find the value of x.
M
Y
F
2x°
N
3
●
x°
55°
X
60°
G
When rays of light meet a flat shiny surface, such as a mirror, they are reflected at the same angle
as they meet the mirror. For example, the diagram below shows how the rays of light are reflected
if they meet the mirror at an angle of 758.
Ray of light
meeting the mirror
75°
Reflected
ray of light
75°
Mirror
A periscope is a device for seeing around corners. It consists of a bent tube with two mirrors
arranged so that the rays of light are bent by 908 by each mirror and that the light rays end up
parallel to their original direction.
Copy the drawing and show how you would place the two mirrors so that the rays of light follow
the path shown through the periscope.
Light
rays
Eye
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4
●
Calculate the size of the angles between the blades of this wind generator.
Ans wer s
1
●
2
●
3
●
a 68
b 1508
c 0.58
d 958
a 1458, /MFX = /NFY, as vertically opposite angles are equal.
b x = 40, 2x 1 x 1 60 5 180° as supplementary angles.
Mirror 1
Light
rays
Eye
Mirror 2
45°
4
●
120°
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6.4 Angles and parallel lines
GeoGebra: parallel lines
Parallel lines
A second interactive GeoGebra (and HTML) file Parallel lines is also included in the teacher and
student ebooks. This time the lines AB and CD have been constructed so that they are parallel.
Students can now confirm the angle relationships they observed in the Lines and transversals diagram
when AB and CD were dragged until they appeared to be parallel. It is useful for students to see both
interactive diagrams, as they then understand that any two lines may be cut by a transversal, but that
the special angle relationships of equal alternate and corresponding angles and supplementary cointerior
angles occur only when the lines are parallel. The converse is of course true, too: if alternate angles or
corresponding angles are equal or if cointerior angles are supplementary, then the lines must be parallel.
GeoGebra: lines and transversals
Lines and
transversals
The interactive GeoGebra file
Lines and transversals is included
in the student and teacher ebooks
in both GeoGebra and HTML
formats. Two lines and a transversal
can be dragged on the screen
to observe angle relationships.
Students will see that vertically
opposite angles are always equal.
In this interactive diagram, the
lines AB and CD have not been
constructed to be parallel. As the
two lines are dragged until they
appear parallel, students will notice
that other angle relationships
become apparent.
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Class activity: angles and parallel lines
Class activity
Angles and
parallel lines
The class activity Angles and parallel lines includes four GeoGebra screen images of a pair of lines
cut by a transversal. In the first two screens the lines are not parallel, but students can observe the
relationship between vertically opposite angles. In the second pair of screens, the lines have been
dragged until they are parallel. Students can now observe the additional angle relationships that occur
for parallel lines cut by a transversal.
Extra example 6
Find the values of the pronumerals.
74°
a°
b°
Working
Reasoning
a = 74 (Alternate angles)
The angles marked a8 and 748 are alternate angles. As
the lines are parallel, the angles are equal.
b = 106 (Corresponding angles)
The angles marked 748 and b8 are corresponding angles.
As the lines are parallel, the angles are equal.
Note also that the angles marked a8 and b8 are vertically
opposite angles so they are equal.
Extra example 7
Find the size of /PAB.
x°
126°
Working
Reasoning
x 1 126 5 180
x 5 180 2 126
x 5 54
The angles marked 1268 and x8 are allied angles
between parallel lines. Allied angles are supplementary,
that is, they add to 1808.
Teaching note: the converse is also true
Students have seen that when parallel lines are crossed by a transversal, alternate angles and
corresponding angles are equal and allied angles are supplementary. The converse of this is also true:
if alternate angles and corresponding angles are equal and allied angles are supplementary, then the
lines must be parallel. Extra example 8 relates to this property.
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6
Extra example 8
chapter
Geometry and space
6.4
Are line segments AB and CD parallel?
A
C
51°
50°
B
D
Working
Reasoning
The angles marked 508 and 518 are
alternate angles.
The alternate angles are not equal so line
segments AB and CD are not parallel.
Alternate angles formed by a transversal
cutting across two lines are equal if the lines
are parallel.
exercise 6.4
Ans wer s
Below are the answers to the questions in exercise 6.4 in the student book.
1
●
2
●
3
●
a
b
c
d
e
f
g
h
a 47
d d = 61, k = 119
g x = 101, y = 101
b 94
e 48
h a = 36, b = 144
c m = 152, n= 152
f 88
i a = 31, b = 149, c = 149
a 56
d 115
g 134
b 124
e e = 59, f = 59
h a = 57, b = 123, c = 57
c 143
f m = 42, n = 42
i w = 112, x = 68, y = 68, z = 112
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4
●
5
●
6
●
a
b
c
d
AB
AB
AB
AB
a
c
e
g
/ACH
/BCG
/MBA
/BCH
is
is
is
is
parallel to CD because allied angles add to 1808.
not parallel to CD because allied angles add to 1828.
parallel to CD because allied angles add to 1848.
parallel to CD because allied angles add to 1808.
b
d
f
h
/BCG
/BCH
/CBM
/ABN,/MBC,/BCH,/GCD
358. Harry’s line of sight and the sea are parallel, so d8 and 358 are alternate angles, which are
equal.
exercise 6.4
1
●
additional questions
Copy each of the following figures. On your drawing, label the sizes of all angles that you had to
find in order to determine the value of the pronumeral, then write the value of the pronumeral.
a
b
118°
m°
w°
36°
c
d
28°
53°
k°
y°
50°
37°
Ans wer s
1
●
a
b
c
d
m = 36
w = 62
y = 90
k = 78
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6.5 Triangles
Teaching note: triangles as rigid shapes
The student text at the beginning of this section shows a triangle made from plastic geostrips and
photographs of triangles used in constructions because of their rigidity. It is worth getting students to
experience the rigidity of a triangle compared with a quadrilateral (which can be pushed into different
shapes). Plastic geostrips or rolls of newspaper taped together can be used successfully for this. Plastic
geostrips are slightly ‘bendy’, so even though the triangles are rigid in the plane, the slightly flexible
plastic strips can be distorted by heavy-handed students who may then miss the point of the activity!
In this case, the rigidity of stiff rolls of newspaper may be better. Alternatively, if your school has a
dome kit, components of the kit could be used. Construction of the entire dome would then make a
worthwhile additional activity.
The rigidity of the triangle may be compared with quadrilaterals, which can be made rigid by joining a
diagonal (see beginning of section 6.6).
Diagonal
Teaching note: the angles in a triangle
Class activity
Sum of the angles
of a triangle
Tearing the corners from a triangle and arranging them to form a straight line is an excellent visual
reinforcement for the concept that the three angles of a triangle add to 180°. It is advisable that
students tear, rather than cut, the corners from their triangle. If they cut the corners, they may become
confused about which is the cut edge and which are the arms of the angle of the triangle. If they make
a second copy of their triangle, students can paste the triangle and the arranged pieces into their
mathematics books.
Students are sometimes not given the opportunity to progress from this teaching demonstration to
a reasoned mathematical argument. It is important that students do not think the corner tearing
demonstration actually proves the relationship. The following diagram (also included in the student
book, shows that if we construct a line parallel to one side of the triangle, the same arrangement of
the three angles to make a straight line can be used. This time, though, we are making use of alternate
angles between parallel lines cut by a transversal to prove the relationship. Proof in this sense is an
answer to the question ‘Why do the angles add to 180°?’
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GeoGebra: angle sum of a triangle
Angle sum of
a triangle
The interactive GeoGebra file, Angle sum of a triangle, is included in the student and teacher ebooks.
The vertices of the triangle can be dragged to see that the three angles always add to 180°. Again, this is
not a proof, but a very convincing demonstration. Clicking on the Proof checkbox displays the parallel
line and gives students the opportunity to reason why the relationship is true.
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6
GeoGebra: exterior angle of a triangle
Exterior angle
of a triangle
chapter
Geometry and space
6.5
The interactive GeoGebra file, Exterior angle of a triangle, is included in the student and teacher
ebooks. The vertices of the triangle can be dragged to see that the three angles always add to 180°.
After the students have had a chance to find the angle relationship for themselves, clicking on the
check box displays the relationship and students are asked to explain it. In the triangle shown, we
know that /ABC 1 /BCA 1 /CAB 5 180° (three angles of the triangle). But we also know that
/ABC 1 /ABD 5 180° (straight line). So /BCA 1 /CAB must equal /ABD.
Using pronumerals for the angles may make the explanation clearer for students.
a°
b°
d°
c°
a 1 b 1 c = 180
(angles of a triangle add to 1808)
d 1 c = 180
(adjacent angles that make a straight line add to 1808)
So a 1 b must equal d.
GeoGebra: types of triangles
Triangles
The interactive GeoGebra file, Triangles, is included in the student and teacher ebooks in both
GeoGebra and HTML formats. Three sliders allow the side lengths of the triangle to be changed.
Students can see the side and angle properties of the different types of triangles and can also see that
there are some sets of lengths for which it is not possible to make a triangle. The check box Show type
can be clicked off to give students the opportunity to name the triangle type themselves.
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Teaching note: can any three side lengths make a
triangle?
Triangles
The Triangles GeoGebra construction identifies side lengths that will not allow a triangle to be
constructed. Making triangles with geostrips as shown on page 293 in the student book reinforces students
understand why any three side lengths will not necessarily make a triangle. If pairs of students are given
a random set of three geostrips (with three paper fasteners) some will find they can make triangles and
others will not. This leads to a discussion of why some students were unable to make a triangle.
Class activity: constructing special triangles
The class activity Constructing special angles is available in the teacher ebook as a student worksheet.
Class activity
Constructing
special triangles
Required equipment: sharp pencil, ruler, pair of compasses
Construction 1
This construction encourages students to think about triangle properties. Because they have used the
same compass opening for two sides of their triangle, the triangle must be isosceles.
Construction 2
In this construction, the same compass opening is used for all three sides of the triangle so the triangle
must be equilateral. This is then linked with the angles which must, of course, be 608.
These constructions could also be completed using interactive geometry software such as Geogebra. The
circle tool then takes the place of compasses.
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Constructing special triangles
6
chapter
Geometry and space
6.5
Construction 1
a In your mathematics workbook, draw a line segment 10 cm long.
■ Label the ends A and B.
Using your ruler to measure, open your compass to 12 cm.
■
Place the compass point at A and draw an arc.
■
Place the compass point at B and draw another arc to intersect (cross) the first arc.
12
cm
■
A
A
B
A
B
10 cm
B
10 cm
10 cm
■
Label the intersection point C.
■
Join each end of AB to C to form a triangle. Label the lengths of sides AC and BC.
C
A
10 cm
B
b What type of triangle have you constructed?
c Measure the three angles of your triangle and label them on the triangle.
Construction 2
Start with a 10 cm segment as in construction 1.
■
Label the ends of the segment D and E.
■
Place the compass point at the D and open the compass so that the pencil is exactly at E.
Draw an arc.
D
10 cm
E
D
10 cm
E
D
10 cm
E
F
D
E
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■
Now do the opposite—place the point at E and place the pencil D. Draw another arc to intersect
the first arc. Label the intersection point F.
■
Join each end of DE to F to make a triangle.
d What sort of triangle is it? Can you explain why?
e Carefully measure each of the angles of your triangle. If you have constructed your triangle
accurately, each of the three angles should be 608. How close to 608 are your angles? Explain why
the angles are 608.
Answers
a No answer required.
b Isosceles triangle
c Students should find their angles close to 658, 658 and 508. (Note: by calculation, the angles are
approximately 65.388, 65.388 and 49.248.)
d The triangle is an equilateral triangle because the length of two sides is determined by arcs that are
equal in length to the 10 cm segment which forms the other side of the triangle.
e The three angles are equal and must add to 1808.
exercise 6.5
Ans wer s
Below are the answers to the questions in exercise 6.5 in the student book.
1
●
2
●
3
●
4
●
5
●
6
●
7
●
a ^ABC (or ^CAB or ^BCA)
c ^THA (or ^HAT or ^ATH)
b ^BOX (or ^OXB or ^XBO)
d ^ENP (or ^PEN or ^NPE)
equilateral triangle: angles are 608, sides are equal
a
c
e
g
i
a a = 63
g g = 50
a
d
g
j
b
d
f
h
right-angled scalene triangle
right-angled isosceles triangle
obtuse-angled isosceles triangle
obtuse-angled scalene triangle
acute-angled isosceles triangle
b b = 49
h h = 30
a = 32
x = 60, y = 60, z = 60
g = 60
d = 50, e = 80
a a = 65
g g = 70
b b = 77
h h = 110
a right-angled scalene triangle
c c = 76
i i = 61
b
e
h
k
acute-angled equilateral triangle
acute-angled isosceles triangle
equilateral triangle (acute-angled)
acute-angled scalene triangle
d d = 36
j j = 90
b = 64
e = 74
h = 106
m = 45
c c = 90
i m = 44
e e = 86
k k = 30
c
f
i
l
d d = 108
j j = 39
f f = 53
l l = 108
c = 70, d = 40
y = 45, z = 75
t = 30, u = 120
x = 60, y = 60
e e = 125
k a = 75
f f = 52
l x = 70
b
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8
●
a i
b i
C
8 cm
A
5 cm
d i
5 cm
6 cm
A
B
ii acute-angled triangle
iii equilateral triangle
e i
C
7 cm
5 cm
5 cm
f
i
C
5 cm
5 cm
A
3 cm
B
7 cm
ii acute-angled triangle
iii isosceles triangle
C
4 cm
B
10 cm
ii right-angled triangle
iii scalene triangle
C
A
8 cm
A
ii acute-angled triangle
iii scalene triangle
c i
6.5
C
6 cm
B
9 cm
6
chapter
Geometry and space
5 cm
A
8 cm
B
B
ii right-angled triangle
iii scalene triangle
ii obtuse-angled triangle
iii isosceles triangle
g i
h i
C
C
6 cm
12 cm
6 cm
5 cm
A
A
B
8 cm
B
13 cm
ii right-angled triangle
iii scalene triangle
9
●
10
●
a
c
e
g
no; 5 cm 1 3 cm , 9 cm
no; 2 cm 1 2 cm = 4 cm
no; 5 cm 1 4 cm = 9 cm
yes; 10 cm 1 3 cm . 11 cm
a
A
ii acute-angled triangle
iii isosceles triangle
b
d
f
h
yes; 5 cm 1 6 cm . 7.5 cm
yes; 3 cm 1 3 cm . 4 cm
no; 6 cm 1 8 cm , 16 cm
no; 7 cm 1 9 cm , 20 cm
b
A
7 cm
7 cm
B
40°
6 cm
35°
C
i acute-angled triangle
ii scalene triangle
B
7 cm
C
i acute-angled triangle
ii isosceles triangle
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c
d
A
A
4 cm
8 cm
B
C
6 cm
C
8 cm
i acute-angled triangle
ii isosceles triangle
e
45°
B
30°
i obtuse-angled triangle
ii scalene triangle
f
C
C
4 cm
A
8 cm
60°
B
9 cm
55°
A
i obtuse-angled triangle
ii scalene triangle
g
30°
30°
8 cm
M
F
5 cm
K
i obtuse-angled triangle
ii isosceles triangle
11
●
a
25°
B
50°
8 cm
i right-angled triangle
ii isosceles triangle
C
55°
40°
50°
6 cm
20°
6 cm
A
C
50°
B
i obtuse-angled triangle
ii scalene triangle
e
A
5 cm
B
i obtuse-angled triangle
ii isosceles triangle
C
9 cm
C
70°
40°
40°
i right-angled triangle\
ii scalene triangle
f
C
40°
C
i right-angled triangle
ii scalene triangle
d
A
B
A
B
i obtuse-angled triangle
ii scalene triangle
c
L
5 cm
b
A
B
i acute-angled triangle
ii scalene triangle
h
G
E
10 cm
A
20°
7 cm
B
i right-angled triangle
ii scalene triangle
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g
h
R
35°
P
55°
8 cm
i right-angled triangle
ii scalene triangle
12
●
40°
40°
Q
10 cm
i obtuse-angled triangle
ii isosceles triangle
base 1.73 m, slope 2 m
scale : 10 cm = 1 metre
slope
1 metre
slope measures 20 cm = 2 metres
base measures 17.3 cm = 1.73 metres
30°
base
exercise 6.5
1
●
6.5
R
P
Q
6
chapter
Geometry and space
additional questions
Find values of the pronumerals in each of the following diagrams.
a
b
c
8x°
64°
4x°
3x°
22°
34°
y°
w°
d
e
z°
157°
2
●
3
●
62°
130°
f
103°
r°
t°
x°
126°
This right-angled triangle has sides of length 3 cm and 4 cm as shown.
a Measure the length of the third side of the triangle.
3 cm
b For many centuries builders have known that a triangle with side
lengths in these particular proportions is a right-angled triangle, and
4 cm
they have used this fact to construct right angles when laying out the
foundations for buildings.
Suggest how you could use a ruler and a long piece of rope (for example, about 15 metres long)
to mark out a right angle on the ground. If possible, try out your suggestion.
Recent technology provides rugby players with instantaneous
information about their chances of kicking a goal when they
are at different distances from the goal. The player’s distance
from the goal, the angle of vision to the goal posts (shaded)
and the apparent width between the goals is displayed.
Calculate the angle of vision, g8, for the situation shown in
the diagram.
45°
19°
g°
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4
●
5
●
6
●
Consider two triangles with the dimensions given.
i Side lengths 7 cm, 5 cm, 4 cm
ii Side lengths 14 cm, 10 cm, 8 cm
a Use your ruler, pencil and compass to construct each of the triangles.
b Notice that the triangles have the same shape but one is larger than the other. Measure the
three angles in each triangle and write them on your constructions.
c Suggest another set of three side lengths that would give a triangle with the same shape as
these two triangles, but a different size.
Consider two triangles with the dimensions given.
i Side lengths 5 cm, 4 cm, 3 cm
ii Side lengths 10 cm, 8 cm, 6 cm
a Use your ruler, pencil and compass to construct each of the following triangles.
b What sort of triangles are they?
Harry and Kate are making a skateboard ramp. They want the ramp to make an angle of 30° with
the ground and they want the top of the ramp to be exactly one metre above the ground. Harry
and Kate have made a rough sketch of the side view of the ramp.
They now need an accurate drawing of the side of the ramp so that
slope
they know how long to cut the wood for the base and the slope.
1 metre
Choose a suitable scale (for example, 10 cm on your drawing
30°
represents 1 m on the actual skateboard ramp) and make an
base
accurate diagram using your ruler and protractor.
Measure the length of the base and the slope and work out the lengths Harry and Kate will need
to cut for each.
Ans wer s
1
●
2
●
3
●
4
●
a w = 120
b x = 12
c y = 67
d z = 46
e r = 112
t = 68
f x = 23
a 5 cm
b Knot or mark the rope into 3 m, 4 m and 5 m lengths and stretch into triangle shape.
g 5 26
a and b
i
102°
ii
102°
5 cm
34°
4 cm
44°
10 cm
8 cm
34°
44°
14 cm
7 cm
c Answers will vary. For example, 21 cm, 15 cm, 12 cm; 28 cm, 20 cm, 16 cm; 3.5 cm, 2.5 cm, 2 cm.
5
●
a i
ii
3 cm
90°
90°
4 cm
6 cm
8 cm
5 cm
6
●
b right-angled triangles
10 cm
Slope = 2 m
Base = 1.73 m
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6.6 Quadrilaterals
Teaching note: angles in quadrilaterals
Class activity
Sum of the angles
of a quadrilateral
As for the angles of a triangle, the corner-tearing shown at the beginning of this section in the student
book is a useful activity to reinforce the concept that the four angles of a quadrilateral add to 360°.
Again, the visual demonstration can be extended to a consideration of why this is so. Dividing the
quadrilateral into two triangles by drawing a diagonal readily shows that the 360° is made up of two
lots of 180°.
GeoGebra: angle sum of a quadrilateral
Angle sum of a
quadrilateral
The interactive GeoGebra (and HTML) file Angle sum of a quadrilateral provides another, perhaps
even more convincing, demonstration for the angle sum.
Extra example 9
Find the values of the pronumerals in this rhombus.
a°
b°
56°
c°
Working
Reasoning
a 1 56 = 180
a = 124
A rhombus is a special parallelogram.
Adjacent angles in all parallelograms are
supplementary (add to 1808).
b = 56
Opposite angles of parallelograms are equal.
c = 124
a = 124
Opposite angles of parallelograms are equal.
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Extra example 10
Identify each of these quadrilaterals. Draw a diagram for each to show the given information.
a The quadrilateral has four right angles. One pair of opposite sides has length 12 cm and other
pair of opposite sides has length 10 cm.
b The quadrilateral has one pair of opposite sides parallel and no equal sides.
Working
Reasoning
a The quadrilateral is a rectangle.
The first sentence tells us that the
quadrilateral is a rectangle (which includes
squares).
The second sentence tells us that it is a
rectangle but not a square.
12 cm
10 cm
10 cm
12 cm
b The quadrilateral is a trapezium.
Trapeziums are the only quadrilaterals with
just one pair of parallel sides.
Teaching note: class inclusion
Although we think of squares, rhombuses and rectangles as discrete shapes, they are all members of the
family or class of parallelograms. They all satisfy the definition of a parallelogram as a plane shape with
both pairs of opposite sides parallel.
In the same way we can consider squares as special cases of both rhombuses and rectangles as
■
■
a square is a special rectangle where all sides are equal.
a square is a special rhombus where all angles are right-angles.
GeoGebra:
parallelogram family
Parallelogram
family
The GeoGebra file Parallelogram
family includes sliders for side
lengths and angle. Students can
make the special shapes of rhombus,
rectangle and square simply
changing the side lengths so that
they are all equal, or by changing
the angles to 908. Clicking on the
Check box displays the special
parallelogram name.
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6
Class activity: how does a folding umbrella work?
chapter
Geometry and space
6.6
Required equipment: class set of umbrella ribs (three old umbrellas), GeoGebra file Umbrella
Class activity
Umbrella
How does a folding umbrella work?
If you have an old folding umbrella, it is worth pulling it apart to separate the eight ribs. Three old
umbrellas gives a class set! The construction of each rib is based on a parallelogram and students can
move the hinged parallelogram to see how the shape remains a parallelogram, even though the angles
change.
Umbrella
a What special quadrilateral is formed by the hinged shape in the umbrella rib?
b How is the hinged shape constructed so that this special quadrilateral is formed?
c What important function do you think this special quadrilateral plays in the opening and closing of
the umbrella?
Ans wer s
a parallelogram
b The ribs have been constructed so that both pairs of opposite sides are equal. This ensures that the
shape is a parallelogram and, therefore, that both pairs of opposite sides stay parallel.
Note that this is the converse of the statement that a plane shape with both pairs of opposite sides
parallel also has both pairs of sides equal.
c This ensures that the umbrella folds neatly, with the folded parts moving parallel to each other as
the umbrella is folded.
Class activity: hinged quadrilaterals
Required for this activity: GeoGebra or HTML files Toolbox and Car jack
Class activity
Hinged
quadrilaterals
Optional: several car jacks of the type shown below and an expanding box for tools, sewing or fishing.
Although rectangles are probably the most common quadrilateral (for example in books, tables, floors
and windows), the special properties of the rhombus make it extremely useful in the design of tools and
other items.
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Car jack
Car jack
Car jacks of this type are readily available from ‘op shops’ or car wreckers’ yards. Based on a rhombus,
the jack hinges neatly and compactly. But the particular rhombus property that is important for the
operation of the jack is that the diagonals of a rhombus are perpendicular. The screw thread represents
one of the diagonals of the rhombus. The design of the base of the jack ensures that this diagonal
remains horizontal. The interactive GeoGebra file Car jack demonstrates how the other (invisible)
diagonal remains perpendicular to the base, ensuring that the car will be lifted vertically, perpendicular
to the ground.
Expanding toolbox
Expanding
toolbox
The design of the expanding sewing box shown here is also found in tool boxes and fishing boxes. The
interactive GeoGebra file Expanding box shows how the trays stay parallel to the base as they are
pulled out. The brackets attached to the drawers form parallelograms that stay parallelograms because
their opposite sides are of fixed equal lengths. There may be a student in the class who could bring a
similar box to demonstrate how it works.
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6
Hinged quadrilaterals
chapter
Geometry and space
6.6
1 Car jack
a What is a car jack designed to do?
Open the GeoGebra file Car jack. Drag point P to simulate operating the car jack.
b What special quadrilateral is formed by the four equal sides of the quadrilateral?
c What property of this special quadrilateral is useful for storing the jack?
d The horizontal screw thread of the jack represents one of the diagonals of the quadrilateral. What do
you know about the two diagonals of this special quadrilateral?
e Why is this property useful for the way the jack is designed to work?
2 Toolbox
a What special quadrilateral shape is formed by the hinged brackets connecting the trays?
b How is the box constructed so that this special quadrilateral shape is formed?
c Can you explain why the trays stay parallel to each other (and to the base of the toolbox) as they
lift out?
Ans wer s
1 Car jack
a A car jack is designed to lift a car to change a wheel.
b rhombus
c The sides are equal, so it can be folded flat.
d The diagonals are at right angles (perpendicular).
e The care moves at right angles to the ground, that is, vertically upwards.
2 Toolbox
a A parallelogram (or in this case the parallelogram may be a rhombus)
b The important aspect of the design is that the bracket strips are connected to the trays so that a
parallelogram is formed.
c The design is using the property that a shape with both pairs of opposite sides equal will also have
both pairs of opposite sides parallel.
Teaching note: exercise 6.6 question 9
Class activity
Hinged
quadrilaterals
Scissor lift
Question 9 relates to the scissor lift which incorporates the rhombus properties of parallel and equal
sides for compactness of closing. Again, as in the case of the car jack, the perpendicular diagonals of
the rhombus ensure that the hinged vertices of the parallel bars move vertically, perpendicular to the
ground. This ensures that the work platform remains horizontal. Students could consider whether any
parallelogram would work, or whether it is important that the special case rhombus is used.
If possible, it is worth getting students to construct the scissor lift model from geostrips and paper
fasteners. They enjoy the tactile feel of opening and closing the model while observing how the sides
remain parallel and the pivot points move up and down perpendicular to the horizontal. One previously
disengaged student became totally involved with this activity. He had been sitting operating the
rhombus geostrip linkage for a short time then suddenly announced: ‘My dad drives one of those scissor
lifts’. This one experience seemed to generate an interest in mathematics for this student. Students also
enjoy operating the virtual scissor lift in the interactive GeoGebra file Scissor lift.
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Scissor lift
What property of
the rhombus makes
it useful in the
scissor lift?
Move point P
horizontally.
exercise 6.6
Ans wer s
Below are the answers to the questions in exercise 6.6 in the student book.
1
●
2
●
3
●
4
●
5
●
a a = 55
g g = 108
b b = 49
h h = 114
a square
d rectangle
g parallelogram
c c = 124
i i = 110
d d = 113
j j = 132
b rhombus
e parallelogram
h square
e e = 103
k k = 58
f f = 56
l l = 82
c trapezium
f kite
i rhombus
rectangle, trapezium, square
a
d
g
j
m
k = 93
d = 69
m = 110
x = 118, y = 62, z = 118
a = 121, b = 121
a
b
e
h
k
n
p = 90, q = 90, r = 90
e = 101, f = 50
u = 40, v = 140
p = 40, q = 140, r = 40
d = 70
c
f
i
l
o
t = 45
x = 62, y = 118
m = 72, n = 108
x = 95, y = 97
a = 125, b = 60
5.4 cm
114°
66°
4 cm
5.4 cm
66°
114°
5.4 cm
b Both pairs of opposite sides of a parallelogram are parallel and equal.
c The opposite angles of a parallelogram are equal. The adjacent angles of a parallelogram add
to 1808.
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6
●
7
●
6
a Yes, they all have both pairs of opposite sides parallel and equal.
b A rhombus is a parallelogram with four equal sides. A rectangle is a parallelogram with four
right angles. A square is a parallelogram, with four equal sides and four right angles.
chapter
Geometry and space
6.6
a, c
90°
3.3 cm
3.3 cm
102°
102°
4.2 cm
4.2 cm
66°
axis of symmetry
b The kite has two pairs of equal sides and one pair of equal angles.
8
●
a
b two
74°
114°
66°
9
●
10
●
11
●
12
●
13
●
14
●
105°
74°
75°
106°
106°
78°
102°
90°
90°
a rhombus
b The bars stay parallel to each other. Both pairs of opposite sides of the rhombus are parallel
and equal, so the bars close up neatly and compactly. The diagonals of each rhombus are at
right angles to each other so the rhombuses move up and down (or in and out) at right angles
to the base. This is particularly important in the case of the scissor lift to ensure that the work
platform remains parallel to the ground.
b /DCB 5 90°
/CBA 5 110°
/BAD 5 50°
a kite
a trapezium
b It is the best shape to fit a bicycle with the least amount of wasted space. The two trapeziums
fit beside each other to make a rectangle.
608 and 1208
h = 72, s = 74
a Examples are shown below.
i
ii
iii
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b No. The angles in a quadrilateral always add up to 3608, so if you had three right angles then
the fourth angle would also have to be a right angle.
15
●
16
●
a a = 29
b b = 120
c c = 123, d = 57
James is correct. Both a square and a rhombus have both pairs of opposite sides parallel and all
sides equal. They also both have opposite angles equal. The difference is that for a square all the
angles must be 908, but for the rhombus there is no such limitation.
exercise 6.6
1
●
2
●
additional questions
a Using your ruler and protractor, carefully draw a quadrilateral which has exactly
i four right angles.
ii two right angles.
iii one right angle.
b Is it possible to draw a quadrilateral that has exactly three right angles? Explain.
Find the value of the pronumerals in each of the following figures.
a
b
c
72°
a°
42°
23°
d
57°
67°
94°
3
●
d x = 106
92°
d°
65°
b°
c°
59°
77°
x°
When a square is folded in half from corner to corner
as shown, two triangles are formed.
What are the sizes of the three angles in each triangle?
Ans wer s
1
●
a i
ii
iii
b No, because three right angles add to 2708, leaving 908 for the fourth angle, so there would be
four right angles.
2
●
3
●
a a = 29
b b = 120
c c = 123, d = 57
d x = 106
458, 458, 908
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6.7 Representing three-dimensional
objects in two dimensions
Teaching note: oblique and isometric drawings
The advantages and disadvantages of oblique and isometric representations can be discussed with
students. The oblique drawing has the advantage that front and back faces are accurately depicted, but
the disadvantage is that the depth cannot be accurately shown. The isometric drawing has the advantage
that all three dimensions—length, width and depth—can be shown accurately, but the drawing appears
distorted to our eye because we are accustomed to a perspective view.
Extra example 11
Use 1 cm graph paper to construct an oblique drawing of a box with a front face of 9 cm by 7 cm.
Working
Reasoning
Draw a rectangle 9 cm by 7 cm, then draw line segments
at 458 to the horizontal from three vertices of the
rectangle. The length of these segments depends on how
deep you wish the box to appear.
Draw the remaining horizontal and vertical line
segments to complete the box.
Extra example 12
Use isometric grid paper to construct an isometric drawing of a box 20 mm high with a base
50 mm by 20 mm.
continued
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Extra example 12 continued
Working
Reasoning
Draw a vertical line segment 2 units long
to represent the height of the box. Draw
segments 5 and 2 units long to represent
two edges of the base of the box.
Draw the remaining line segments to complete
the box, making sure the vertical edges are all
2 units long. The box could have been drawn
the other way around, with the 5 unit segment
to the left and the 2 unit segment to the right.
Teaching note: plans and elevations
There is an excellent isometric drawing tool provided at the National Council of Teachers of
Mathematics website at www.nctm.org.
Extra example 13
Use this isometric drawing to draw the
a plan.
b front elevation.
c side elevation.
Front
continued
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6
Extra example 13 continued
chapter
Geometry and space
6.7
Working
Reasoning
a
The plan shows the ‘floor’ area that the shape
covers.
b
The front elevation shows what we would see
if we looked directly at the front of the shape
without seeing the depth. It appears as if the
blocks have been moved forward so that they
are all in the same plane.
c
The side elevation shows what we would see
if we looked directly at the side of the shape
without seeing the depth.
exercise 6.7
Ans wer s
Below are the answers to the questions in exercise 6.7 in the student book.
1
●
2
●
3
●
4
●
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5
●
a i
ii
iii
4 cm
5 cm
4 cm
4 cm
5 cm
4 cm
b i
ii
iii
2 cm
3 cm
2 cm
4 cm
3 cm
4 cm
c i
ii
iii
2 cm
2 cm
4 cm
5 cm
4 cm
5 cm
d i
ii
4 cm
3.5 cm
1.5 cm
6
●
iii
3.5 cm
1.5 cm
4 cm
a i
ii
iii
b i
ii
iii
c i
ii
iii
d i
ii
iii
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7
●
e i
ii
iii
f i
ii
iii
a i
ii
iii
2.5 cm
6.7
3 cm
3 cm
6 cm
6
chapter
Geometry and space
4 cm
6 cm
4 cm
b i
ii
iii
25 cm
25 cm
40 cm
50 cm
40 cm
50 cm
8
●
4m
5m
3.5 m
9
●
10
●
11
●
a
b
c
d
Answers will vary according to prisms used. Check with your teacher.
a plan
b side elevation
c isometric drawing
d oblique drawing
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exercise 6.7
additional questions
1
●
Make an isometric drawing of a box 2 units high, 4 units wide and 6 units long.
2
●
Make an oblique drawing of a box 2 units high, 4 units wide and 6 units long.
3
●
Use this isometric drawing to draw the
a plan.
b front elevation.
c side elevation.
Front
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Ans wer s
6
chapter
Geometry and space
6.7
1
●
2
●
3
●
a
b
plan
c
front elevation
side elevation
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Analysis tasks
Answers to the student book task, A parking problem, are included in this section together with
two additional analysis tasks: Polyominoes and Boom angles.
Student book: a parking problem
A parking problem links to the chapter warm-up worksheet Parallel and angle parking and looks at
how angles are important in safe street (kerbside) parking. Students will discover that the greater the
angle the cars make with the kerbside, the greater the number of cars that can fit. However, there
are many other issues to take into account, such as the safety of cars backing out in front of traffic.
Discussion of these issues could be at whole class level or students could work in groups.
BLM
A parking
problem
The teacher ebook includes a blackline master template (A parking problem) of 1:100 scale
parking space models of the 2.7 3 5.5 m spaces for angle parking and the slightly longer 2.7 3 6 m
spaces for parallel parking. It is easier if the two different sizes of parking space are photocopied
onto different coloured paper. These pieces could be saved and used again by other classes to save
class time—in this case, thin card may be preferable. The pieces could also be laminated for future
use. For the slightly longer parking spaces for parallel parking, each group should be given at least
10 pieces. (Actually, only six are needed, but we don’t want to give them the answer before they
start!). For the angle parking spaces, each group should have at least 30 pieces so that they can set
out the 908 and 608 parking at the same time.
A nss w e rs
An
a
Parallel parking: 6 cars
b Spaces can be shorter for 608 or 908 angle parking because there is space in the road behind the
parked cars, so cars do not need extra space for getting in or out of the parking space.
c
908 angle parking: 14 cars
d
608 angle parking: 12 cars
Note: Students may also like to see how many cars would fit for 458 angle parking.
458 angle parking: 9 cars
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e
Advantage
Disadvantage
Parallel parking
Parallel parking is suited to
narrower roads and is safer for
cars leaving the parking space.
Fewer cars will fit.
Angle parking
More cars can fit in a certain
length of roadway than for
parallel parking.
458 and 608 parking fit more
cars than parallel parking, but
require less road width than 908
parking.
Angle parking spaces extend
out further into the roadway,
and it is harder for drivers to
see traffic coming when they
are backing out of an angle
parking space.
6
chapter
Geometry and space
f Factors affecting choice of parking type include
■ width of the road
■ number of parking spaces needed
■ length of the available parking strip
■ amount of traffic on the road, which could make angle parking more dangerous
■ depth of gutters in many towns, where parallel parking would not allow car doors to be
opened on the passenger side
■ how busy the area is in terms of parking requirements.
Additional task: polyominoes
This analysis task develops spatial visualisation skills. The task starts with triominoes, shapes made
from three squares joined by sides, then leads on to tetrominoes and pentominoes. Students are
asked to work out how many different tetrominoes and pentominoes are possible, drawing each of
the possible arrangements of squares. They are then asked to use the shapes to fill certain regions.
The game of dominoes has rectangle shaped pieces made up of two squares
joined together.
We could join together different numbers of squares to make triominoes (three squares),
tetrominoes (four squares) and pentominoes (five squares).
If we have two squares there is only one way of putting them together, as shown above.
If we imagine that we can pick up the triominoes and turn them over (reflect them) and turn them
around (rotate them), there are two different ways of putting three squares together.
These two triominoes are the same
The two different triominoes
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a Fill this 6 3 3 rectangle using a combination of
the two different triominoes.
b There are five different tetrominoes. Draw the
five tetrominoes, being careful to check whether
some of your tetrominoes are actually the same.
c Fill this 8 3 3 rectangle using
i two different tetrominoes.
ii four different tetrominoes.
d There are 12 different pentominoes. Draw the
12 tetrominoes, again being careful to check
whether some are actually the same.
e Fill this 10 3 3 rectangle using
i two different pentominoes.
ii four different pentominoes.
f Fill this square with five different pentominoes.
An s w e rs
a
b
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c i
6
chapter
Geometry and space
ii
d
e i
ii
f
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Additional task: boom angles
This task uses the angles displayed on the boom of a crane together with the safe loads and
the horizontal reach of the crane. The task gives practice in drawing angles and in interpreting
information in a table.
A
Boom angles
The part of a crane that can be tilted at different
angles is called the boom. The angle, /ABC, that the boom makes
with the horizontal is called the boom angle.
a Use your protractor to find the boom angle for this crane.
Boom
B
C
°
90
0°
Cranes have an angle measurer resembling a 908 protractor on the
boom which tells the crane driver the angle the boom makes with
the horizontal.
80
°
70°
0°
° 2
60° 50
° 40° 30
10
°
The crane shown in the photographs below was working on a building construction site and was
lifting concrete walls into position. The boom is shown in two different positions.
Position 1
Position 2
b Measure the boom angle (marked) for the crane in the two different positions, and compare
your answers with the crane’s angle measurer in each position.
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c The boom is telescopic and its length can be changed, depending on
how far it must reach to set down its load. If the boom is fully extended
and the angle is too small, the crane may tip over with very heavy loads.
The capacity chart in the cabin tells the driver the boom angle that is
safe for a particular load, and how far the crane can reach horizontally
for that angle and that boom length.
The capacity chart on the right shows
the loads that can be safely lifted for
different boom angles when the boom
is fully extended to 31.5 m. It also
shows how far the crane can reach for
each angle.
For each of the two positions of the
crane in part b, use the capacity chart
to find the maximum load the crane
could safely lift.
d What load could the crane safely lift if
the boom angle is 408?
e For each of the two positions of the
crane in part b, use the capacity chart
to find how far the boom could reach
horizontally.
f Using your protractor and ruler, make
a careful drawing of the boom to show
the angle it would need to make with
the horizontal for a load of 8100 kg.
6
chapter
Geometry and space
Horizontal reach
Boom length 31.5 m
Boom angle
(8)
Load
(kg)
Horizontal reach
(m)
79.0
10 000
6
77.0
10 000
7
75.0
10 000
8
73.5
9000
9
71.5
8100
10
67.5
6550
12
63.5
5000
14
59.0
4000
16
55.0
3200
18
50.0
2500
20
45.5
1800
22
40.0
1300
24
33.5
900
26
25.5
650
28
An
A
n s w e rs
a 44°
b position 1 = 48°; position 2 = 80° (so the crane’s measurer is wrong in both positions: it says
53° instead of 48° in position 1 and 51° instead of 80° in position 2)
c position 1 = 2200 kg; position 2 = 10 000kg
d 1300 kg
e position 1 = 21 m; position 2 = 6 m
f
71.5°
Further investigations in chapter 15
The following investigations in chapter 15 are suitable as additional or alternative tasks for chapter 6.
■
■
Catching the sun’s heat
Star polygons
The investigation Catching the sun’s heat involves drawing angles accurately with a protractor and
interpreting given data to show the ideal tilt angles for solar panels in various places in Australia.
The investigation Star polygons builds on students’ familiarity with a 5-pointed star to look at other
star polygons. The task incorporates fractions as well as interesting geometry. A blackline master file
in the student and teacher ebooks provides the templates for the vertices.
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Review Geometry and space
Visual map suggestion
Lines,
segments
and rays
Perpendicular
At right
angles
Parallel
Same distance apart; in the
same plane and never meet
Acute
< 90˚
Right
90˚
Obtuse
⬎ 90˚
⬍ 180˚
Parallel lines
Alternate angles are equal
Corresponding angles are equal
Allied angles are supplementary
Straight
180˚
Revolution
360˚
Vertically opposite
angles are equal
Angles
Complementary angles
add to 90˚
Reflex
⬎ 180˚
⬍ 360˚
Supplementary angles
add to 180˚
Triangles
3 sides
Sum of angles = 180°
Sides
Scalene
Isosceles
Angles
Equilateral
Acuteangled
Obtuseangled
Rightangled
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6
chapter
Geometry and space
Quadrilaterals
4 sides
Sum of angles = 360°
Special
Quadrilaterals
Parallelograms
Both pairs of opposite sides
parallel and equal.
Adjacent angles supplementary.
Diagonals bisect each other.
Trapezia
1 pair of opposite
sides parallel.
2 pairs of
supplementary
allied angles.
Rectangles
All angles 90°.
Diagonals equal in length.
Squares
All angles 90° and
all sides equal.
Diagonals
equal and
intersect at
right angles.
Kites
Two pairs of
adjacent sides equal.
One pair of opposite
angles equal.
Diagonals
intersect at
right
angles.
Rhombuses
All sides equal.
Diagonals
intersect at
right angles.
Note that the boxes for rectangles, squares and rhombuses have been included in the box for
parallelograms to indicate that they all belong to the family of parallelograms. Similarly squares have
been shown overlapping the boxes for rectangles and rhombuses.
Revision answers
1
●
6
●
7
●
8
●
2
●
C
E
3
●
C
4
●
B
5
●
E
a The line PQ is parallel to the line RS.
b i AB is perpendicular to CD. ii EF is parallel to GH.
a 144°
a i
b i
b 215°
/MLG or /GLM
/PHC or /CHP
ii acute angle
ii reflex angle
iii 65°
iii 280°
9
●
D
A
C
B
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10
●
E
C
D
A
11
●
12
●
B
a complementary: 53° and 37°
b supplementary: 139° and 41°
a
b 45°
E
135°
C
13
●
14
●
15
●
16
●
17
●
18
●
19
●
D
a k 5 28
b /EXY 5 73°
a a 5 50 (cointerior angles in parallel lines add to 1808)
b b 5 146 (corresponding angles in parallel lines are equal)
a a 5 124
b b 5 22
a rectangle; a 5 30
d square; d 5 90
b parallelogram; b 5 98
e rhombus; e 5 105
c kite; c 5 115
f trapezium; f 5 90, g 5 130
a Yes, the sum of the two shortest sides is more than the third side, so the sides will meet.
b Yes, the sum of the two shortest sides is more than the third side, so the sides will meet.
c No, the sum of the two shortest sides is equal to the third side, so the sides will not meet.
a p 5 30
a
b 1508
A
5.5 cm
B
20
●
c a 5 80; b 5 100; c 5 43
c 4 pieces
d 908
b isosceles
c acute-angled
b isosceles
c right-angled
6 cm
C
6 cm
a
F
5 cm
D
5 cm
E
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21
●
a
b scalene
c obtuse-angled
L
30°
J
22
●
i
23
●
i
6
chapter
Geometry and space
40°
K
8 cm
ii
24
●
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Practice quiz answers
Practice quiz
Chapter 6
1
●
6
●
11
●
C
A
D
2
●
7
●
12
●
3
●
8
●
13
●
E
A
A
4
●
9
●
14
●
E
D
C
5
●
10
●
15
●
C
A
D
D
B
B
Chapter tests
Test A
Chapter test A
Chapter 6
Multiple-choice questions
1
●
2
●
3
●
4
●
5
●
Two lines which are at right angles to each other are
A adjacent.
B perpendicular.
C complementary.
D parallel.
E
E vertically opposite.
The size of /AFD is
A 378
B 538
C 1278
D 1438
E 2338
A
D
C
F
37°
B
Which one of the following is a pair of complementary angles?
A 818 and 98
B 908 and 1808
C 648 and 1168
D 778 and 238
E 608 and 208
Which one of the following statements is not correct?
A Two line segments that are parallel could not be perpendicular.
B Two lines that are in the same plane and do not meet must be parallel.
C Complementary angles add to 908.
D Vertically opposite angles are always equal.
E Adjacent angles are always supplementary.
Which of the following sets of three lengths could be the sides of a triangle?
A 5 cm, 8 cm, 15 cm
B 6.5 cm, 6.5 cm, 13 cm
C 8.1 cm, 9.8 cm, 17.9 cm
D 6.2 cm, 12.4 cm, 18.6 cm
E 11.5 cm, 8.1 cm, 16.7 cm
[5 3 2 = 10 marks]
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chapter
6
Geometry and space
Short-answer questions
6
●
7
●
Carefully draw diagrams using a ruler and a pencil, then add the correct symbols to show
a two intersecting line segments that are perpendicular to each other.
b two lines that are parallel to each other.
[1 1 1 = 2 marks]
Find the size of the following angle and state its type.
50
280 270 260 250
12
01
30
24
02
30
3
50
350 3
40 3
30
3
40
30
01
20
40
110
250
100 90 80 70
60
260 270 280 290
50
340
10
20
13
02
3
0
22
23
350
0
10
40
30
190 20
02
15
01
3
20
170
180
170 1
60
10
03
30
8
●
160
190
200
10
02
10 2
0
01
40
290
22
20
0
30
80 90 100 110
14
10
30
70
60
S
P
W
A
Y
Road
Philip Street, Hamilton Road, Bay Road and Market Street
meet at a junction as shown.
a What is the size of the acute angle that Philip Street
makes with Hamilton Road?
b Which two roads or streets form a pair of vertically
opposite angles?
c What is the size of the acute angle that Market Street
makes with Hamilton Road?
reet
76°
Ba
24°
t
ee
tr
tS
e
ark
M
43° St
Philip
yR
oad
9
●
[1 1 1 1 1 1 1 = 4 marks]
Hamilton
a Measure the size of /WAY.
b Name the angle that is the supplement of /WAY.
[1 1 1 1 1 = 3 marks]
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10
●
A
a
b
c
d
pair of parallel lines is cut by a transversal as shown.
Name a pair of alternate angles.
Name a pair of corresponding angles.
A
Name a pair of cointerior angles.
What is the size of /DGH?
C
H
143°
G
D
F
E
B
[1 1 1 1 1 1 1 = 4 marks]
11
●
12
●
A quadrilateral has two sides of length 7 cm and two sides of length 10 cm. Jason says that the
quadrilateral must be a rectangle. Draw diagrams to show that there are two other possibilities. In
each case name the special quadrilateral you have drawn.
[2 1 2 = 4 marks]
Find the value of the pronumerals in each of the following.
a
b
43°
68°
b°
3b°
a°
55°
c
d
b°
82°
48°
125°
d°
e
f
a°
65°
x°
145°
112°
40°
g
h
104°
80°
a°
98°
a°
[8 3 2 = 16 marks]
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6
chapter
Geometry and space
Extended-response questions
13
●
14
●
Follow these steps.
a Use your compass, ruler and pencil to construct ^ABC with the following sides: BC = 7 cm,
AB = 3.5 cm and CB = 5 cm.
b Classify ^ABC according to its side lengths.
c Classify ^ABC according to its angle sizes.
[2 1 1 1 1 = 4 marks]
Follow these steps.
a Starting with the edge shown, construct an isometric drawing of 5 cm cube. (Assume the
isometric grid is a 1 cm grid.)
b Draw a front elevation of this three-dimensional shape.
[2 1 1 = 3 marks]
[Total = 50 marks]
Te s t A ans wer s
1
●
6
●
7
●
8
●
9
●
10
●
2
●
B
a
3
●
D
A
4
●
D
5
●
E
b
2578, reflex
a 298
b /PAW
a 808
b Hamilton Road and Philip Street
a
b
c
d
c 578
/AFG and /DGF or /BFG and /CGF
/AFG and /CGH or /BFG and /DGH or /EFB and /FGD or /EFA and /FGC
/AFG and /CGF or /BFG and /DGF
378
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11
●
b parallelogram
kite
7
7
10
10
10
7
7
10
12
●
13
●
a b 5 28
e a 5 105
a
b a 5 82
f x 5 103
A
3.5 cm
B
14
●
c b 5 42
g a 5 76
d d 5 27
h a 5 82
b Scalene
c Obtuse-angled
5 cm
7 cm
a
C
b
Test B
Chapter test B
Chapter 6
Multiple-choice questions
1
●
Two angles that add to 908 are
A complementary.
B supplementary.
C acute.
D obtuse.
E reflex.
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2
●
3
●
4
●
5
●
The value of x is
A 108
B 508
C 1008
D 1408
E 2808
x°
6
chapter
Geometry and space
37°
43°
x°
Which of the following is a pair of supplementary angles?
A 1638 and 278
B 598 and 1218
C 418 and 498
D 2408 and 608
E 908 and 1808
Which one of the following statements is correct?
A Adjacent angles are always supplementary.
B Two line segments that intersect at 908 are parallel.
C Vertically opposite angles are always complementary.
D Two lines are parallel if they are in the same plane and they never meet.
E Two line segments are parallel if they are in the same plane and they never meet.
Which of the following could not be the sides of a triangle?
A 3 cm, 4 cm, 5 cm
B 8.2 cm, 9.8 cm, 17.9 cm
C 8.1 cm, 9.9 cm, 17.9 cm
D 5 m, 12 m, 16 m
E 5 m, 8 m, 15 m
[5 3 2 = 10 marks]
Short-answer questions
6
●
Name a pair of
a supplementary angles.
b complementary angles.
c vertically opposite angles.
F
E
A
O
B
C
D
[1 1 1 1 1 = 3 marks]
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a Find the size of the following angle.
0 270 280 29
50 26
02
40
21
170 1
60
350
190
200
340
15
01
30
350 3
40 3
30
3
22
00
60
30
290
70
280 270 260 250
80 90
02
24
20
100 11
01
1
40
03
50
30
1
40
10 2
0
10
02
0
20
31
190 20
0
0
1
10
20
180
31
50
30
170
1
30
00
60
40
160
03
0 90 80 70
10 10
1
20
03
02
23
32
20
4
02
50
7
●
0
13
b State its type.
8
●
[1 1 1 = 2 marks]
Measure the size of /POM.
P
Q
O
R
M
[1 mark]
9
●
Find the value of the pronumerals.
a
y°
x°
b
35°
a°
2a°
3a°
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c
6
chapter
Geometry and space
d
130°
x°
c°
115°
e
30°
f
100°
40°
y°
145°
d°
g
P
h
Q
100°
d°
x°
95°
48°
79°
S
60°
R
[8 3 2 = 16 marks]
Extended-response questions
10
●
11
●
12
●
Follow these steps.
a Use a compass and a ruler to construct a triangle with sides of length 5 cm, 5 cm and 7 cm.
b Classify the triangle according to its side lengths.
[2 1 1 = 3 marks]
Follow these steps.
a Use a pencil and a ruler to draw a diagram showing a line AB that is parallel to a line CD. Use
the correct symbols to show that the lines are parallel.
b On the same diagram, draw a line segment EF that cuts both AB and CD.
c Label the point M where EF cuts AB and the point N where EF cuts CD.
d Name a pair of alternate angles.
e Name a pair of corresponding angles.
f Name a pair of cointerior angles.
[2 1 1 1 1 1 1 1 1 1 1 = 7 marks]
Complete the following.
a State two properties that rectangles, rhombuses, parallelograms and kites have in common.
b Draw and name three different types of quadrilaterals with two sides of length 3 cm and two
sides of length 5 cm.
[2 1 3 = 5 marks]
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13
●
Follow the steps.
a Use the isometric grid paper below to make an isometric drawing of a box 3 units high, 4 units
wide and 6 units long. Start with the edge shown.
b Draw a front elevation of this 3-dimensional shape.
[2 1 1 = 3 marks]
[Total = 50 marks]
Te s t B ans wer s
1
●
6
●
7
●
8
●
9
●
A
2
●
3
●
C
B
4
●
D
5
●
E
a /FOA and /AOD or /FOB and /BOD or /FOC and /COD or /EOF and /FOC or
/EOA and /AOC or /EOB and /BOC or /FOE and /EOD or /EOD and /DOC
b /AOB and /BOC or /EOF and /FOA or /COD and /FOA
c /EOF and /COD
a 75°
b acute
1358
a x 5 145,y 5 35
e y 5 45
b a 5 60
f d 5 70
c c 5 130
g x 5 110
d x 5 35
h d 5 42
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10
●
a
6
chapter
Geometry and space
b isosceles
5 cm
5 cm
7 cm
11
●
a, b, c
E
M
A
B
d /AMN and /DNM or /BMN and /CNM
e /AME and /CNM or /BME and /DNM or
/AMN and /CNF or /BMN and /DNF
f /AMN and /CNM or /BMN and /DNM
N
C
D
F
12
●
a
four sides; angles add to 3608
3 cm
3 cm
b
3 cm
3 cm
5 cm
5 cm
rectangle
13
●
a
5 cm
5 cm
parallelogram
kite
b
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