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6Measurement andgeometry
Geometry‘Geometry’ comes from the Greek word geometria, whichmeans ‘land measuring’. The principles and ideas ofgeometry are evident everywhere – in road signs, buildings,bridges and patterns for tiles and wallpaper. Many examplesof geometric patterns can be seen in nature, such as thehexagonal cells on a honeycomb built by bees to storehoney.
n Chapter outlineProficiency strands
6-01 Triangle geometry U F R C6-02 Quadrilateral geometry U F R C6-03 Angle sum of a polygon U F R C6-04 Exterior angle sum of a
convex polygon U F R C
nWordbankangle sum The total of the sizes of the angles in a shape,such as a triangle
bisect To cut in half
convex polygon A polygon whose vertices all pointoutwards
diagonal An interval joining two non-adjacent vertices ofa shape
exterior angle of a triangle An ‘outside’ angle of a triangleformed after extending one of the sides of the triangle
polygon A plane shape with straight sides
regular polygon A polygon with all angles equal and allsides equal, such as a square
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
n In this chapter you will:• classify triangles and quadrilaterals according to their side and angle properties, and solve
related numerical problems using reasoning• solve problems involving the angle sum of a triangle and quadrilateral and the exterior angle of
a triangle• solve problems involving the angle sum of a polygon and the exterior angle sum of a convex
polygon
SkillCheck
1 Classify each angle as acute, obtuse, reflex, right, straight or a revolution.
a b c d
e f g h
2 Find the value of each pronumeral, giving reasons.
a b c
r°
r°w° y°
t°
k°
k°
t°
a°
a°
c°140°
85°
38°
80°
30°
110°
d e f
Worksheet
StartUp assignment 6
MAT09MGWK10065
Skillsheet
Types of angles
MAT09MGSS10018
Puzzle sheet
Angles
MAT09MGPS00050
Skillsheet
Angles andparallel lines
MAT09MGSS10021
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
k l
125°
a°
b°
k°
j76°
d° c°
b°
i
115°
n°m°
h45°
h°
k°
58°
x°y°
n° m°
g47°
b°c°
a°
3 Test whether each labelled pair of lines are parallel. Give reasons for each answer.
J
A C
B D
K
I
L
F
E
55°60°
78°
75°
112°
78°
a b c
G
H
6-01 Triangle geometry
Angle sum of a triangle
Summary
The angle sum of a triangle is 180�.a°
b°c°
a þ b þ c ¼ 180
Homework sheet
Geometry 1
MAT09MGHS10023
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
Proof:
UP
V
R
Q b°
a°
c°
Consider any triangle PQR with angles a�, b� and c�.Construct a line through P parallel to QR.
[ \UPQ ¼ b� (alternate angles, UV || QR)and \VPR ¼ c� (alternate angles, UV || QR)But \UPQ þ \QPR þ \VPR ¼ 180� (angles on astraight line)
[ a� þ b� þ c� ¼ 180�[ The sum of the angles of a triangle is 180�.
Example 1
Find the value of each pronumeral, giving reasons.
k°
55°
42° m°
52°
RQ
ba P
Solutiona k þ 55þ 42 ¼ 180
k ¼ 180� 55� 42
¼ 83
(angle sum of a triangle)
b \R ¼ \Q ¼ m� (nPQR is isosceles)
mþ mþ 52 ¼ 180
2mþ 52 ¼ 180
2m ¼ 128
m ¼ 64
(angle sum of a triangle)
The exterior angle of a triangle
Summary
The exterior angle of a triangle is equal to the sum of the twointerior opposite angles.
z ¼ x þ yx°
y°
z°
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
Example 2
Find the value of k in each diagram.
k°
59°
a b
42°110°
50°
k°
Solutiona k ¼ 42þ 59 ðexterior angle of triangleÞ
¼ 101
b k þ 50 ¼ 110 ðexterior angle of triangleÞk ¼ 60
Example 3
Find the size of the marked angle \BDF.
55°75°
D
F
EB
A
C
Solution\DBC ¼ \ABE ¼ 75� ðvertically opposite anglesÞ\BDF ¼ 75� þ 55� ðexterior angle of 4BCDÞ
¼ 130�
Exercise 6-01 Triangle geometry1 Find the value of each pronumeral.
70°110°37°
65°
45°
38°
42°53°
81° 67°
48°
60°
h°
w°
m°
k°x°
a°
a b c
d e f
See Example 1
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
56°
55°
40°
15°
y°
k°d°
m° m°
m°
h°
r°
g h i
j k l
2 What is the size of each angle in an equilateral triangle? Select A, B, C or D.
A 30� B 45� C 60� D 90�3 One angle of an isosceles triangle is equal to 80�. Which two of the following could
be the sizes of the other angles? Select two of A, B, C or D.
A 80� and 20� B 40� and 60� C 40� and 40� D 50� and 50�4 Find the value of m.
m°
m°
m°60°
55°
39°
73°40°
125°
m°m°
m°64°135°
50°
132°
a b c
d e f
Worked solutions
Triangle geometry
MAT09MGWS10029
See Example 2
9780170193085214
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
5 What is the size of the unknown angles in this triangle?Select A, B, C or D.
? ?
A 30 � B 45 �
C 60 � D 90 �
6 One angle of an obtuse-angled triangle is 50�. Which of the following could be the sizes of theother angles? Select A, B, C or D.
A 80� and 50� B 100� and 30� C 65� and 65� D 60� and 70�7 Find the size of \CBF in each diagram.
80°
45°
D E
A
C
BF
110°
120°
B
AE
C
F
53°
60°
B
C F
34°
E
FCD
B
E
CB
AF
a
d e f
b c
60°50°
A
D CE
FB
8 An isosceles triangle has a side length of 6 cm and one of its angles equal to 40�. Draw allpossible shapes of the triangle.
9 The diagram shows the shape of a roof truss.Find the value of each pronumeral. 50°
a° b°c°
10 Copy this diagram and mark all angles equal tothe angle marked d�.
d°
See Example 3
Worked solutions
Triangle geometry
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11 Find the value of each pronumeral.
3g°
3m°
2m°
2h°
h°
(2t + 1)°
30°
71°
50°
112°
a b c
d e f
70°
y°
2x°
Technology Sketching parallelograms and
rectanglesIn this activity you will use GeoGebra to construct two quadrilaterals.Constructing a parallelogram
1 To construct a parallelogram, use Interval between two points and construct two sides ofany length (as shown below).
2 Label the points (right-click on each point and Show label).
Skillsheet
Starting GeoGebra
MAT09MGSS10019
Skillsheet
Starting Geometer’sSketchpad
MAT09MGSS10020
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
3 Select Parallel line from the fourth drop-down icon menu and choose side BC and point A.
4 Repeat step 2 with side AB and point C. Now select Intersect Two Objects from thesecond drop-down icon menu and the lines through points A and C. Label the point D.
5 Use Angle to measure the size of \DAB and \ABC. Manipulate the vertices to change thesize of the parallelogram.
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Constructing a rectangle.
1 Make sure the axes and grid are removed by right-clicking and disabling them.
2 Click
Enter a point and 6 (cm). Label the interval endpoints A and B.
3 Click
Select point A and 6 cm interval. Repeat for point B.
4 Click New point to insert a point anywhere on the line under point B. Label it point C.
5 Construct a perpendicular line through C.
6 Click Intersect two objects and create a point of intersection at the perpendicular throughA and the perpendicular through C. Label the point of intersection, D.
7 Select the Move tool and click point C. Manipulate the line through CD.
8 Click Distance or length and choose points B and C. Use the Move tool to manipulatepoint C until BC ¼ 3.6 cm. Click Distance or length and points A and B to show the lengthof AB.
9 Now draw at least two other types of quadrilaterals.
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
6-02 Quadrilateral geometryA quadrilateral is any shape with four straight sides. A quadrilateral may be either convex ornon-convex (concave).
Convex quadrilateral Non-convex quadrilateral
• All vertices (corners) point outwards.• All diagonals lie within the shape.• All angles are less than 180�.
• One vertex points inwards.• One diagonal lies outside the shape.• One angle is more than 180� (reflex angle).
There are six special types of quadrilaterals.
Trapezium Parallelogram Rectangle
One pair of parallel sides Two pairs of parallel sides Four right angles
Rhombus Square Kite
Four equal sides Four equal sides and fourright angles
Two pairs of equal adjacent sides
Angle sum of a quadrilateral
Summary
The angle sum of a quadrilateral is 360�.a þ b þ c þ d ¼ 360
a°b°
c°d°
This property is true for both convex and non-convexquadrilaterals.
Worksheet
Classifyingquadrilaterals
MAT09MGWS00066
Worksheet
Properties of trianglesand quadrilaterals
MAT09MGWS10066
Worksheet
Naming quadrilaterals
MAT09MGWS10067
Worksheet
Deductive geometry
MAT09MGWS10068
Skillsheet
Naming shapes
MAT09MGSS10022
Homework sheet
Geometry 2
MAT09MGHS10024
‘Adjacent’ means ‘next to each other’.
Technology
GeoGebra: Angle sumof a quadrilateral
MAT09MGTC00010
Animated example
Angles and shapes
MAT09MGAE00011
Worksheet
Diagonal properties ofquadrilaterals
MAT09MGWS00067
Worksheet
Shapes and anglesreview
MAT09MGWS00068
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
Properties of quadrilaterals
Summary
Trapezium • One pair of opposite sides parallel• No axes of symmetry
Kite • Two pairs of equal adjacent sides• One pair of opposite angles equal• One axis of symmetry• Diagonals intersect at right angles
Parallelogram • Opposite sides are parallel and equal• Opposite angles are equal• No axes of symmetry• Diagonals bisect each other
Rhombus • All sides are equal• Two axes of symmetry• A special type of parallelogram• Diagonals bisect each other at right angles• Diagonals bisect the angles of the
rhombus
Rectangle • All angles are 90� (right angles)• Two axes of symmetry• A special type of parallelogram• Diagonals are equal (in length)• Diagonals bisect each other
Square • All sides are equal• All angles are 90� (right angles)• Four axes of symmetry• A special type of rhombus and rectangle• Diagonals are equal• Diagonals bisect each other at right angles• Diagonals bisect the angles of the square
9780170193085220
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
Example 4
Find the value of each pronumeral.
m°
125°a b
93°
78°
B C
EDA
d°
75°
Solutiona mþ 78þ 125þ 93 ¼ 360 (angle sum of a quadrilateral)
m ¼ 360� 78� 125� 93
¼ 64
b \CDA ¼ 180� � 75� (angles on a straight line)
¼ 105�
d ¼ 105 (opposite angles of a parallelogram)
Example 5
Find the size of \BED.
E
D
CBA
70° 110°
85°
Solution\ABE ¼ 70� (equal angles in isosceles nABE)
\EBC ¼ 180� � 70� (angles on a straight line)
¼ 110�
\BED ¼ 360� � 85� � 110� � 110� (angle sum of quadrilateral BCDE)
¼ 55�
Puzzle sheet
Mixed angle problems
MAT09MGPS00053
Puzzle sheet
Parallel lines
MAT09MGPS00051
Puzzle sheet
Triangles andquadrilaterals
MAT09MGPS00052
Can you see another methodfor finding d?
Can you see another methodfor finding \BED?
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Exercise 6-02 Quadrilateral geometry1 Find the value of m in each diagram.
m°
m° m°
m°m°
130°87°
69°
108°71°
85°
130°
110°
140°
55°
25°
84°30°100°110°
2m° 3m°
a b c
d e f
2 Copy and complete the table below.
Property Trapezium Kite Parallelogram Rhombus Rectangle SquareOne pair ofopposite sidesparallelOpposite sidesparallelOpposite sidesequal
�
All sides equalTwo pairs ofadjacent sidesequalDiagonals equal �Diagonals bisecteach otherDiagonals meet atright anglesDiagonals bisectthe angles of theshapeOpposite anglesequalOne pair ofopposite anglesequalAll angles 90�Axes of symmetry 0
See Example 4
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
3 Refer to the table you completed in question 2, and name all quadrilaterals that have:
a no axes of symmetry b one pair of parallel sidesc four equal sides d equal diagonalse opposite sides equal f four axes of symmetryg all adjacent sides equal h one axis of symmetryi opposite sides parallel j all angles measuring 90�k two axes of symmetry l diagonals which bisect each otherm opposite angles equal n diagonals meeting at 90�
4 Each triangle in this diagram is an equilateral triangle.a Name the different types of quadrilaterals you can find
in the diagram.
b How many of each type of quadrilateral are there in thediagram?
c How many triangles are there? (It may help to copy thediagram so that you can draw on it using coloured pencils.)
5 Which statements are always true?A A rhombus is a parallelogram.
B The diagonals of a parallelogram meet at right angles.
C A square is a rhombus.
D A parallelogram is a quadrilateral with a pair of opposite sides equal and parallel.
E A square is a rectangle.
F The diagonals of an isosceles trapezium bisect each other.
G The opposite angles of a rhombus are equal.
H The diagonals of a rhombus are equal and bisect each other at right angles.
I A rectangle is a parallelogram.
6 Name each quadrilateral using the properties marked on it.
b c
e f
a
d
g h i
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
7 Find the value of each pronumeral, giving reasons.
k°
a°p°
w°t°
n°
70°
125°
67°
35°
68°
70°
2a° 5c°
a b c
d e f
g hi
r° 110°
8 Find the size of \PQR in each diagram, giving reasons.
PQ V
WRR
W PU
T Q
T
R
QT
RPDC
Q
R
PTS
P Q
R
P
Q72° 80° 74°
110°
30°
60°
105°
110°
55°35°
a b c
d e f
See Example 5
Worked solutions
Quadrilateral geometry
MAT09MGWS10030
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
9 Use geometrical instruments or geometry software such as GeoGebra to construct eachquadrilateral described.a Parallelogram TUVW with sides TU ¼ 30 mm, UV ¼ 60 mm and \TUV ¼ 113�.
b Rhombus with side length 5 cm.
c Convex quadrilateral WXYZ with WX ¼ 65 mm, WZ ¼ 40 mm, \ZWX ¼ 54�,\WZY ¼ 114� and XY ¼ 24 mm.
d Non-convex quadrilateral with one side 4.5 cm and one angle 200�.
e 6 cm
65° 105°
50°
ED
G
F
8 cm
Just for the record Geometry in art
Geometry has been used in artwork forcenturies.Tesselations can be used to create artwork, suchas the regular hexagons used in the tessellationshown below.
Indian and Islamic artworks show incredibleintricacy, detail and colour within thegeometric images used.
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
Fractals use geometric formulas to create infinitely-occurring images, so they are usually created usingcomputer software. A famous fractal is the Mandelbrotset, shown here.
1 Design a tessellation or find examples of geometryin art.
2 Investigate fractals such the Mandelbrot set and theKoch snowflake, and their history.
Investigation: Angle sum of a polygon
To find the angle sum of a pentagon ABCDE (5 sides),follow this reasoning.
B
C
DE
A
From one vertex of the pentagon, draw all the diagonals.The diagonals from vertex A have divided the pentagoninto three triangles.
) Angle sum of a pentagon ¼ 3 3 angle sum of the 3 triangles
¼ 3 3 180�
¼ 540�
1 a Draw a hexagon (6 sides) and from one vertex draw all the diagonals.b How many diagonals are there?c How many triangles did you form?d Hence find the angle sum of a hexagon.
2 Repeat the procedure to find the angle sum of:a an octagon (8 sides) b a decagon (10 sides)
3 Copy and complete this table.
Polygon Number of sides Number of triangles Angle sumtriangle 3 1 180�quadrilateral 4 2pentagon 5hexagonoctagondecagon
4 Copy and complete this pattern: The number of triangles formed is always two ________than the number of __________ of the polygon.
5 a Using your own words, describe the rule for finding the angle sum of a polygon.b What is the angle sum of a polygon with 20 sides?c For a polygon with n sides, write a formula for the sum of its angles. Discuss your
result with other students.
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
6-03 Angle sum of a polygonA polygon is a general name for any shape with straight sides. The word is derived from theGreek, meaning ‘many angles’. Shapes with curved sides, such as circles, ellipses and semicircles,are not polygons.A polygon may be either convex or non-convex (concave).
Convex polygon Non-convex polygons
In a convex polygon, all vertices point outwards, alldiagonals lie within the shape and all angles are lessthan 180�.In a non-convex polygon, some vertices point inwards,some diagonals lie outside the shape and some anglesare more than 180� (reflex angles).A polygon’s name is determined by the number of sidesit has. The images below show the Pentagon buildingin the USA, a 50p coin from the UK, and a stop sign.
Name Number of sidesPentagon 5Hexagon 6Heptagon 7Octagon 8Nonagon 9Decagon 10Undecagon 11Dodecagon 12
6 a Draw a non-convex octagon.b Divide the polygon into triangles as shown.c How many triangles have been formed?d Find the angle sum of this non-convex octagon.e Does your rule for the angle sum of a polygon also
apply to the non-convex polygon?
NSW
Worksheet
Angle sum of a polygon
MAT09MGWK10069
Worksheet
Find the unknown angle
MAT09MGWK10070
Technology worksheet
Angle sum of a polygon
MAT09MGCT10002
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
Summary
The angle sum of a polygon with n sides is given by the formula A ¼ 180(n � 2)�.This formula applies to both convex and non-convex polygons.
Example 6
Find the angle sum of a nonagon (9 sides).
SolutionAngle sum ¼ 180ð9� 2Þ�
¼ ð180 3 7Þ�
¼ 1260�
Example 7
Find the number of sides of a polygon that has an angle sum of 720�.
Solution180ðn� 2Þ ¼ 720
180n� 360 ¼ 720
180n ¼ 1080
n ¼ 1080180
¼ 6
[ The polygon has 6 sides (hexagon).
Regular polygonsA regular polygon has all angles equal and all sides equal. Forexample, a regular pentagon has 5 equal sides and 5 equalangles. A square is a regular polygon but a rhombus is not.
Summary
The size of each angle in a regular polygon with n sides ¼ Angle sumNo. of sides
¼ 180 n� 2ð Þ�n
Worksheet
Equal angles
MAT09MGWK00063
9780170193085228
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
Example 8
Find the size of one angle in a regular hexagon.
SolutionA hexagon has six sides (n ¼ 6).
Size of one angle ¼ 180 6� 2ð Þ�
6
¼ 180 3 4ð Þ�
6¼ 120�
Each angle in a regular hexagon is 120�.
Exercise 6-03 Angle sum of a polygon1 How many sides has:
a a hexagon? b a quadrilateral? c a nonagon?d a decagon? e a heptagon? f a pentagon?g a dodecagon? h an octagon? i an undecagon?
2 Name each polygon.
a
d
b
e
g
c
f
h
3 Which polygons from question 2 are:
a convex? b regular?
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
4 Which shape is a regular octagon? Select A, B, C or D.
A B C D
5 What is the more common name for:
a a regular triangle? b a regular quadrilateral?
6 Find the angle sum of a polygon with:
a 5 sides b 8 sides c 15 sides d 12 sidese 7 sides f 10 sides g 20 sides h 11 sides
7 Find the value of each pronumeral.
cba
fed
g
h i
a°
120° 140°
130°
110°
100°145°
160°
b°
b°130°
90°
d°
d° d°
d°d°
c°
c°120°
120°
120°
120°
140°
140°
g°
96°
116°
50°
138°
x°
x°
144° 120°
96°
e° e°
e°
e°
e°e°
e°
e°
m°
y°
48°
n°
See Example 6
Worked solutions
Angle sum of a polygon
MAT09MGWS10031
9780170193085230
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
8 Find the number of sides of a polygon that has an angle sum of:
a 2160� b 1620� c 3960� d 2700�9 Which polygon has an angle sum of 1440�? Select A, B, C or D.
A pentagon B decagon C nonagon D octagon
10 Find the size of one angle in a regular
a octagon b decagon c dodecagon d hexagon
11 The angle sum of a regular polygon is 6840�.a How many sides does the polygon have?
b Find the size of each angle.
12 How many sides does a regular polygon have if each of its angles is:
a 165�? b 170�? c 144�?
Investigation: Exterior angle sum of a convex polygon
1 a Draw a triangle and extend each side to show the exteriorangles of the triangle as shown.Exterior angles are x�, y� and z�.Interior angles are a�, b� and c�.
b Use a protractor to measure angles x�, y� and z�.c Find the sum of the exterior angles.d Looking at the diagram, what must be the value of
a� þ x� þ b� þ y� þ c� þ z�?e But what do we know about the value of a� þ b� þ c�?f Therefore, what must be the value of x� þ y� þ z�?
a°
c° b°y°
z°
x°
2 Repeat this procedure for the exterior angles of a convex quadrilateral. What is the sumof the exterior angles of a convex quadrilateral?
3 Repeat the procedure for a convex pentagon and a convex hexagon. What do younotice about the sum of the exterior angles of those polygons?
4 a Draw any convex polygon and extend the sides. Label thevertices of your polygon A, B, C, etc.
b Start at A and move around the polygon, turning in thedirection indicated at each vertex.
c Continue until you return to A, facing the same way youstarted. What must be the sum of the turns in any roundtrip of a convex polygon?
d Test whether this rule works for a non-convex polygon.A
BD
E
C
See Example 7
See Example 8
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
6-04Exterior angle sum of a convexpolygon
Summary
The sum of the exterior angles of a convex polygon is 360�.
A
BD
E
C
Example 9
For a regular hexagon, find the size of:
a each exterior angle b each (interior) angle.
Solutiona Sum of exterior angles ¼ 360�
One exterior angle ¼ 360�4 6
¼ 60�
b Each angle ¼ 180� � 60� ðangles on a straight lineÞ¼ 120�
OR: Each angle ¼ 180 6� 2ð Þ�
6¼ 120�
NSW
9780170193085232
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
Example 10
Find the number of sides of a regular polygon if:
a each exterior angle is 12� b each (interior) angle is 160�.
Solutiona Sum of exterior angles ¼ 360�
Number of exterior angles ¼ 360 4 12
¼ 30
[ The regular polygon has 30 sides.
b Exterior angle ¼ 180� � 160� ðangles on a straight lineÞ¼ 20�
Sum of exterior angles ¼ 360�
Number of exterior angles ¼ 360 4 20
¼ 18[ The regular polygon has 18 sides.
OR: 180 n� 2ð Þ�
n¼ 160�
180ðn� 2Þ ¼ 160n
180n� 360 ¼ 160n
20n� 360 ¼ 0
20n ¼ 360
n ¼ 36020
¼ 18
[ The regular polygon has 18 sides.
Exercise 6-04 Exterior angle sum of a convex polygon1 Find the size of each exterior angle of a regular:
a octagon b decagon c 15-sided polygon d nonagon
2 Find the size of each angle of a regular:
a pentagon b dodecagon c nonagon d 16-sided polygon
3 Find the number of sides of a regular polygon if each exterior angle is:
a 24� b 36� c 40� d 10� e 18� f 60�4 Find the number of sides of a regular polygon if each angle is:
a 150� b 175� c 162� d 140� e 108� f 176�
See Example 9
See Example 10
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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
Mental skills 6 Maths without calculators
Dividing decimalsTo divide one decimal by another, first move the decimal points in both decimals the samenumber of places to the right so that the second decimal is a whole number.
1 Study each example.
a 0.24 ÷ 0.06 = 24 ÷ 6 = 4
c 0.006 ÷ 0.3 = 0.06 ÷ 3 = 0.002
e 1.6 ÷ 0.4 = 16 ÷ 4 = 4
b 0.45 ÷ 0.5 = 4.5 ÷ 5 = 0.9
d 27 ÷ 0.9 = 270 ÷ 9 = 30
f 5.6 ÷ 0.07 = 560 ÷ 7 = 802 Now evaluate each quotient.
a 0.25 4 0.5 b 63 4 0.7 c 3.2 4 0.4 d 0.18 4 0.2e 2.7 4 0.03 f 0.042 4 0.06 g 4 4 0.5 h 1.2 4 0.04i 0.072 4 0.9 j 0.35 4 0.1 k 0.28 4 0.07 l 0.033 4 0.11
3 Study each example.
Given that 112 4 14 ¼ 8, evaluate each expression.
a 112 ÷ 1.4 = 112.0 ÷ 1.4 = 1120 ÷ 14 = 112 × 10 ÷ 14 = 112 ÷ 14 × 10 = 8 × 10 = 80Estimate: 112 ÷ 1.4 ≈112 ÷ 1 = 112
b 0.112 ÷ 0.14 = 0.112 ÷ 0.14 = 11.2 ÷ 14 = 112 ÷ 10 ÷ 14 = 112 ÷ 14 ÷ 10 = 8 ÷ 10 = 0.8Estimate: 0.112 ÷ 0.14 ≈ 0.1 ÷ 0.1 = 1
c 1120 ÷ 1.4 = 11 200 ÷ 14 = 112 × 100 ÷ 14 = 112 ÷ 14 × 100 = 8 × 100 = 800Estimate: 1120 ÷ 1.4 ≈ 1120 ÷ 1 = 1120
d 1.12 ÷ 14 = 112 ÷ 100 ÷ 14 = 112 ÷ 14 ÷ 100 = 8 ÷ 100 = 0.08 Estimate: 1.12 ÷ 14 ≈ 1.12 ÷ 10 = 0.112
4 Now given that 368 4 23 ¼ 16, evaluate each quotient.
a 36.8 4 2.3 b 368 4 2.3 c 3.68 4 2.3 d 0.368 4 0.23e 36.8 4 23 f 3.68 4 0.23 g 36.8 4 0.23 h 0.368 4 2.3i 0.368 4 23 j 3.68 4 0.023 k 3.68 4 23 l 0.368 4 230
9780170193085234
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Geometry
Power plus
1 How many diagonals has:
a a quadrilateral? b an octagon? c a dodecagon?
2 a By drawing polygons with 3 to 10 sides and counting diagonals, copy and completethis table.
No. of sides, n 3 4 5 6 7 8 9 10No. of diagonals, d
b Find a formula for the number of diagonals, d, in a polygon with n sides.
3 Name all quadrilaterals whose diagonals:
a bisect each other at right angles b bisect each otherc intersect at right angles d have equal lengthe bisect the angles of the quadrilateral f are equal and bisect each other
4 Find the value of each pronumeral, giving reasons.
113°
40°
113°
83°
40°
118°
62°65°
54°
34°
117°
25°
h°k°
m°p°
w°
e°
x°f °
y°
n°
t°
v°
a°
d°
a b c
d e f
g h
36°
87°
30°
c°
i
9780170193085 235
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9
Chapter 6 review
n Language of maths
adjacent angle sum bisect co-interior
convex corresponding equilateral exterior angle
interior isosceles kite parallel
parallelogram polygon quadrilateral rectangle
regular polygon rhombus right angle square
supplementary trapezium vertex vertically opposite
1 What shape has two pairs of equal adjacent sides?
2 Name one property of a convex quadrilateral.
3 What type of angle is created when one side of a triangle is extended?
4 What is the sum of the exterior angles of any polygon?
5 What is a regular polygon? Are all regular polygons also convex?
6 Copy and complete: The exterior angle of a triangle is equal to the ______ of the_______ __________ ___________ angles.
n Topic overview• Write three ideas from this topic that were new to you.• Summarise what you know about the angle sum of a triangle, quadrilateral and polygon.• Name the three types of triangles, choose one and list all of its properties.• Name the six special quadrilaterals, choose one and list all of its properties.
Copy (or print) and complete this mind map of the topic, adding detail to its branches and usingpictures, symbols and colour where needed. Ask your teacher to check your work.
Polygons
GEOMETRY
Trianglegeometry
Quadrilateralgeometry
Puzzle sheet
Geometry crossword
MAT09MGPS10071
Quiz
Shapes and angles
MAT09MGQZ00011
Worksheet
Geometry summaryposter
MAT09MGWK10072
Worksheet
Mind map: Geometry(Advanced)
MAT09MGWK10074
9780170193085236
1 Find the value of each pronumeral, giving reasons.
a b c
d e f
91°42°
52°
127°
72°
35°3m°
2m°
3x°
k° e°
b°
t°
53°
87°y°
114°
57°
45°
y°
y°
g h i
5m°
2 What are the sizes of the angles in:
a an equilateral triangle? b a right-angled isosceles triangle?
3 Find the value of each pronumeral, giving reasons.
70°
98°
82°112°
120°
2e°
3e°
56°
110°
35°
g°
m°
y°
x° y°
m°
x°a b c
e fd
4 Name all quadrilaterals that have:
a both pairs of opposite sides parallel b two equal diagonalsc all sides equal d diagonals that bisect each other.
5 Draw each pentagon.
a a regular pentagon b a non-regular pentagon c a non-convex pentagon
6 a Show that the angle sum of a decagon is 1440�.b Find the size of one angle in a regular nonagon.
7 a Find the size of each exterior angle in a regular hexagon.b If each angle in a regular polygon is 150�, how many sides does it have?
See Exercise 6-01
See Exercise 6-01
See Exercise 6-02
See Exercise 6-02
See Exercise 6-03
See Exercise 6-03
See Exercise 6-04
9780170193085 237
Chapter 6 revision
