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4Measurement and geometry
GeometryGeometry is everywhere. Angles, parallel lines, triangles andquadrilaterals can be found all around us, in our homes, on transport,in construction, art and nature. This scene from Munich airport inGermany shows the importance of angles, lines and shapesin architecture and design.
n Chapter outlineProficiency strands
4-01 Angle geometry U F PS R C4-02 Angles on parallel lines U F PS R C4-03 Line and rotational
symmetry U F C4-04 Classifying triangles U F R C4-05 Classifying
quadrilaterals U F R C4-06 Properties of
quadrilaterals U F R C4-07 Angle sums of triangles
and quadrilaterals U F PS R4-08 Extension: Angle sum of
a polygon U F PS R
n Wordbankangle sum The total of the sizes of the angles in a shape,such as a triangle
bisect To cut in half
convex quadrilateral A quadrilateral whose vertices allpoint outwards.
diagonal An interval joining two non-adjacent vertices of ashape
exterior angle of a triangle An ‘outside’ angle of a triangleformed after extending one of the sides of the triangle
scalene triangle A triangle with no equal sides
supplementary angles Two angles that add to 180�
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n In this chapter you will:• investigate angles on a straight line, angles at a point and vertically opposite angles and use
results to find unknown angles• identify corresponding, alternate and co-interior angles when two parallel lines are crossed by a
transversal, and the relationships between them• investigate conditions for two lines to be parallel and solve simple numerical problems using
reasoning• identify line and rotational symmetries• classify triangles according to their side and angle properties• distinguish between convex and non-convex quadrilaterals• describe squares, rectangles, rhombuses, parallelograms, kites and trapeziums• apply the angle sum of a triangle and quadrilateral and that any exterior angle of a triangle
equals the sum of the two interior opposite angles
SkillCheck
1 Draw two different examples of each type of angle.
a acute angle b right angle c obtuse angle d reflex angle
2 Classify each angle size below as being acute, obtuse, right, reflex, straight or a revolution.
a 25� b 100� c 300� d 128�e 90� f 360� g 286� h 180�
3 Name each type of angle(s) marked.
a b c
4 Find the value of x in each equation.
a x þ 30 ¼ 90 b x þ 57 ¼ 180 c x þ 121 þ 77 ¼ 360
5 Copy each shape, name it and draw all axes of symmetry.
a b
Worksheet
StartUp assignment 4
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Skillsheet
Types of angles
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Geometry
dc
6 Write the order of rotational symmetry of each shape in question 5.
7 Which quadrilateral has all four sides equal and all four angles equal? Select the correctanswer A, B, C or D.
A parallelogram B rhombus C rectangle D square
8 Draw a scalene triangle.
9 a Draw a parallelogram and its diagonals.b Are the lengths of the diagonals you drew in part a equal?
4-01 Angle geometry
Classifying angles
Right angle Straight angle Revolution90� (quarter-turn) 180� (half-turn) 360� (complete turn)
Acute angle Obtuse angle Reflex angleLess than 90� Between 90� and 180� Between 180� and 360�
Skillsheet
Starting GeoGebra
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Skillsheet
Starting Geometer’sSketchpad
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Puzzle sheet
Angles: A dog day
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Worksheet
Straight angles, rightangles and revolutions
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Angle facts
Adjacent angles Complementary angles Angles in a right angleAngles next to each other,sharing a common arm.
\ABD and \DBC areadjacent.
B
A
C
D
Angles that have a sum of 90�,for example, 35� and 55�.
Are complementary.a þ b ¼ 90
a°b°
Supplementary anglesAngles that have a sum of180�, for example, 140�and 40�.
Angles on a straight line Angles at a point Vertically opposite anglesAre supplementary.x þ y ¼ 180
x° y°
(In a revolution)Add up to 360�.a þ b þ c þ d ¼ 360
a°b°
c°d°
Are equal.w ¼ y and x ¼ z
w°x°
y°
z°
Example 1
Find the value of each pronumeral, giving reasons.
cba x°70°
55°b°
a°y°72°
134°
Solutiona xþ 70 ¼ 90 ðAngles in a right angle.Þ
x ¼ 90� 70
¼ 20
The reason is written insidebrackets.
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b a ¼ 55 (Vertically opposite angles.)
bþ 55 ¼ 180
b ¼ 180� 55
¼ 125
(Angles on a straight line.)
c yþ 72þ 134þ 90 ¼ 360
yþ 296 ¼ 360
y ¼ 360� 296
¼ 64
(Angles at a point.)
Exercise 4-01 Angle geometry1 Classify each type of angle(s).
cba
fed
ihg
2 For each angle size, find the complementary angle.
a 16� b 33� c 71� d 2�
3 For each angle size, find the supplementary angle.
a 25� b 149� c 107� d 48�
4 What is the sum of the angles at a point? Select the correct answer A, B, C or D.
A 90� B 180� C 270� D 360�
5 Find the value of each pronumeral, giving reasons.
cba
fed
x°20°
a°
68° a°b°110°
y° 75°a°
161°x°
148°95°
See Example 1
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ihg
lkj
onm
rqp
a°
b° c°
128° x°160°
35°
x°
a°b°
c°45°
k°
32° n°110°
w°112°
92°
q°118° y°
320°
p°p°
28°p°
65°42°
a°
6 Find the value of each pronumeral, giving reasons.
cba
m°
60°65°
m°
a°a°
80°
fed
hg
x° 100° a°
b° c°
35°
65°a°
a° a°
45°
p°
p°
m° m°m°
Worked solutions
Exercise 4-01
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7 ABC is a straight line and EB bisects \DBF.Which of the following is the size of \EBC?
Select the correct answer A, B, C or D.
A 95� B 110�C 85� D 140�
30°40°
A
B
C
D
E
F
4-02 Angles on parallel linesWhen parallel lines are crossed by another line (called a transversal), special pairs of angles areformed.
Corresponding angles Alternate angles Co-interior anglesCorresponding angles onparallel lines are equal.
Alternate angles on parallellines are equal.
Co-interior angles onparallel lines aresupplementary(add to 180�).
• Corresponding angles are in ‘matching’ positions on the same side of the transversal:‘corresponding’ means ‘matching’
• Alternate angles are between the parallel lines on opposite sides of the transversal: ‘alternate’means ‘going back and forth in turns’
• Co-interior angles are between the parallel lines on the same side of the transversal: ‘co-interior’means ‘together inside’
Worksheet
Finding the unknownangle 1
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Skillsheet
Angles and parallellines
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Homework sheet
Angle geometry
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Worksheet
Angles in parallel lines
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Puzzle sheet
Angles in parallel lines
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Example 2
Find the value of each pronumeral, giving reasons.
ba
62°
y°
110°a°
Solutiona y ¼ 62 (Alternate angles on parallel lines)
b aþ 110 ¼ 180
a ¼ 180� 110
¼ 70
(Co-interior angles on parallel lines)
Example 3
Prove that the lines AB and CD are parallel.
Solution\AEF and \EFD are alternate angles.
\AEF ¼ \EFD ¼ 76�[ AB || CD (Alternate angles are equal)
Exercise 4-02 Angles on parallel lines1 Is each marked pair of angles corresponding, alternate or co-interior?
a b c
d e f
76°
76°
E
F
B
DC
A
‘[ AB || CD’ means ‘Thereforeline AB is parallel to line CD’
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2 Which angle is corresponding to the angle marked g�?Select A, B, C or D.
f °A D
B C
e° h°g°
3 For the diagram in question 2, name an angle that is:a co-interior to D
b alternate to the angle marked e�c equal to C
d corresponding to B
e supplementary to the angle marked e�.
4 Find the value of the pronumeral in each diagram, giving reasons for your answers.
p°
91°
d°
135°a° 61°
m°
120°
a°
70°
m°68°
a b c
d e f
5 Find the value of each pronumeral, giving reasons.
61°a°
b°
110°
c°d°
115°
b°a°
50° 60°
h°g°m°
80°p°
q°89°
y°
x°
w°y°
72°
a°55° 45°
n°60° 50°
a b c
d e f
g h i
See Example 2
Worked solutions
Exercise 4-02
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6 What is the value of y in this diagram?Select the correct answer A, B, C or D.
A 85 B 40C 35 D 65
40°105°
y°
7 For each diagram, decide whether AB is parallel to CD. If it is, then prove it.
48°48°
E
F
B
D
A
C 100°100°
E
F
B D
A C
120°
58°
E
F
B
D
A
C
115°65°E
F
B D
A Ca b c d
4-03 Line and rotational symmetryA shape is symmetrical if it looks the same after it has undergone a change of position or amovement. The two types of symmetry are line symmetry and rotational symmetry.When a shape with line symmetry is folded along a line, called an axis of symmetry, the two halvesof the shape fit exactly on top of each other. One half is the reflection or mirror-image of the otherhalf.
axis of symmetry
This shape has one axis of symmetry.
When a shape with rotational symmetry is rotated (spun) about a point, called the centre ofsymmetry, it fits exactly on itself at least once before one full revolution (360�). The number oftimes the shape fits on itself in one revolution is called its order of rotational symmetry.
centre of symmetry
This shape has rotational symmetry of order 4.
See Example 3
Worksheet
Symmetry
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Skillsheet
Line and rotationalsymmetry
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Exercise 4-03 Line and rotational symmetry1 Copy each shape and mark its axes of symmetry.
a b c
d e f
2 Count the number of axes of symmetry of each shape.
a b c d
3 Decide whether each shape has rotational symmetry. If it does, state the order of rotationalsymmetry.
a b c d
4 a Copy the capital letters below that have line symmetry and draw their axes of symmetry.
D H I K M N O R S V X Zb Copy the capital letters above that have rotational symmetry and mark their centre of
symmetry.
5 Draw a shape or pattern that has:
a one axis of symmetry b two axes of symmetry c four axes of symmetry
6 Draw a shape or pattern that has rotational symmetry:
a of order 2 b of order 4
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7 For each shape:i find how many axes of symmetry it has
ii state whether it has rotational symmetry and if it does, state the order
a b c
d e f
8 What shape has:a an infinite number of axes of symmetry?
b an infinite order of rotational symmetry?
4-04 Classifying trianglesTriangles can be classified in two ways: by their sides or by their angles.
Classifying by sides
Equilateral triangle Isosceles triangle Scalene triangleThree equal sides(Also three equal angles,each 60�)
60°60°
60°
Two equal sides(Also two equal angles,opposite the equal sides)
No equal sides(Also no equal angles)
Classifying by angles
Acute-angled triangle Obtuse-angled triangle Right-angled triangleThree acute angles (lessthan 90�)
One obtuse angle (between90� and 180�)
One right angle (90�)
Worksheet
Properties of triangles
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Worksheet
Triangle geometry
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Skillsheet
Naming shapes
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Homework sheet
Symmetry and triangles
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Puzzle sheet
Classifying triangles
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Exercise 4-04 Classifying triangles1 Classify each triangle according to its sides and angles.
a b c d
e
i
f
j
g
k
h
l
2 Sketch a triangle that is:
a right-angled and isosceles b equilateralc scalene and obtuse-angled d acute-angled and scalenee right-angled and scalene f acute-angled and isosceles
3 Which of the following describes this triangle when we classify itby sides and angles? Select the correct answer A, B, C or D.
A isosceles and obtuse-angled B isosceles and acute-angledC scalene and obtuse-angled D scalene and acute-angled
4 Is it possible to draw an obtuse-angled equilateral triangle? Justify your answer.
5 Which triangles in question 1 have:
a line symmetry? b rotational symmetry?
6 Find the value of each pronumeral, giving a reason.
ba
d
c
x°
42°
38° 38°
7 cm
y cm
x°
4.8 m
ml mm
fe60°
60°
60°
17.2 m
a m
b m
r°
15°10 cm
10 c
m
35°
35°
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7 Is it possible to draw a triangle with two obtuse angles? Why?
8 Which triangle is both obtuse-angled and scalene? Select A, B, C or D.
A B C D
9 Copy and complete this table.
TriangleNumber of axes ofsymmetry
Order of rotationalsymmetry
Equilateral triangleIsosceles triangleScalene triangle
Investigation: The perpendicular bisector in anisosceles trianglenABC is an isosceles triangle with AC ¼ AB.It has one axis of symmetry, AD.
A
CD
B
1 Why is CD ¼ DB?2 Why is \ADC ¼ \ADB?3 What is the size of \ADC and \ADB?4 AD bisects side CB. What does ‘bisect’ mean?5 AD ’ CB. What does ‘’’ mean?6 ‘In an isosceles triangle, the axis of symmetry is the
perpendicular bisector of the uneven side.’ Explainwhat this means in your own words.
Mental skills 4A Maths without calculators
Converting fractions and decimals to percentages
To convert a fraction or decimal into a percentage, multiply it by 100%.
1 Study each example.
a 25¼ 2
53 100% ¼ 2
513 10020% ¼ 2 3 20% ¼ 40%
b 2440¼ 24
403 100% ¼ 243
4053 100% ¼ 3
513 10020% ¼ 3 3 20% ¼ 60%
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Geometry
Just for the record It’s all Greek or Latin to me!Many of our words in geometry come from Greek or Latin. Latin was the language of theancient Roman Empire.
Word Origin MeaningEquilateral Latin: aequus latus Equal sidesEquiangular Latin: aequus angulus Equal cornersIsosceles Greek: isos skelos Equal legsScalene Greek: skalenos Uneven legAcute Latin: acutus SharpObtuse Latin: obtusus Dull or bluntReflex Latin: reflexus Bent backTriangle Latin: tri angulus Three cornersRectangle Latin: rectus angulus Right cornersQuadrilateral Latin: quadri latus Four sidesPolygon Greek: poly gonon Many anglesDiagonal Greek: dia gonios From angle to angleTrapezium/Trapezoid Latin/Greek: trapeza Small table
Explain what this sentence means, and illustrate with a diagram: ‘A rhombus is equilateralbut not equiangular’.
2 Now convert each fraction to a percentage.
a 710
b 3350
c 2760
d 2225
e 2432
f 3040
g 6075
h 45
i 1120
j 2880
k 1550
l 1620
m 5460
n 1840
o 1325
3 Study each example.
a 0:41 ¼ 0:41 3 100%
¼ 0:41
¼ 41%
b 0:08 ¼ 0:08 3 100%
¼ 0:08
¼ 8%c 0:9 ¼ 0:9 3 100%
¼ 0:90
¼ 90%
d 0:375 ¼ 0:375 3 100%
¼ 0:375
¼ 37:5%
4 Now convert each decimal to a percentage.
a 0.25 b 0.68 c 0.17 d 0.6 e 0.1f 0.333 g 0.59 h 0.702 i 0.84 j 0.7k 0.428 l 0.055 m 0.91 n 0.7825 o 0.314
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4-05 Classifying quadrilateralsA quadrilateral is any shape with four straight sides. A quadrilateral may be convex or non-convex.
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 anglesRhombus Square Kite
Four equal sides Four equal sides and fourright angles
Two pairs of equal adjacent sides
Exercise 4-05 Classifying quadrilaterals1 Is each quadrilateral convex or non-convex?
a b c
Worksheet
Properties ofquadrilaterals
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Worksheet
Classifyingquadrilaterals
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Worksheet
Always, sometimes,never true
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Puzzle sheet
What shape am I?
MAT08MGPS10013
Video tutorial
Classifyingquadrilaterals
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Puzzle sheet
Singing in the car
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Worksheet
Classifying trianglesand quadrilaterals
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‘Adjacent’ means ‘next to eachother’.
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d e f
2 Name each quadrilateral.
a b c d
3 Name all the quadrilaterals that have:
a four right angles b exactly one pair of parallel sidesc four equal sides d opposite sides equale opposite sides parallel f two pairs of equal adjacent sides
4 Which quadrilateral is also called a ‘diamond’?
5 Copy and complete this table.
QuadrilateralNumber of axesof symmetry
Order of rotationalsymmetry
RectangleParallelogramTrapeziumRhombusSquareKite
6 In the diagram on the right, what type of quadrilateral is:
a ACDF? b FBCD?
A
BF
E CD
7 A parallelogram is any quadrilateral with both pairs of opposite sides parallel. Which of thefollowing is not a special type of parallelogram? Select the correct answer A, B, C or D.
A square B kite C rectangle D rhombus
8 Sketch each of these quadrilaterals, showing its main features.
a rectangle b trapezium c rhombus d kite
9 Which of the following statements are always true? (Explain your answers)a A rhombus is a square.
b A square is a rhombus.
c A rectangle is a parallelogram.
d A parallelogram is a quadrilateral with its opposite sides parallel and equal.
e The diagonals of a parallelogram meet at right angles.
f A square is a rectangle.
g A rectangle is a square.
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Technology Properties of quadrilateralsThis activity will use GeoGebra to construct quadrilaterals. For each quadrilateral, set up yourdrawing page by clicking View. Make sure that Grid is ticked and Axes is not ticked.
Square1 To draw a square, click on Regular Polygon. Use the grid to draw an interval of length
7.5 cm. Select 4 points. Make sure that the labels are showing.
2 Use interval between two points to construct the two diagonals for the square. Selectdistance or length to measure the length of each diagonal in the square. What do younotice?
3 Now measure the Angle of each vertex of the square. What do you notice?
4 Draw another square with side length 9 cm. Repeat steps 2 and 3. List two properties of thesquare.
Rectangle1 To draw a rectangle, click on Polygon. Use the grid to draw a rectangle with sides 6 cm by
3 cm.
2 Use distance or length to measure each side length of the rectangle.
3 Now measure the Angle of each vertex to check your accuracy. Correct any inaccurate sidesusing Move. What symbol is shown on each vertex when the angle is shown as exactly 90�?
4 Draw another rectangle with length 5 cm and width 8.4 cm. Use the instructions given instep 2 above to measure the side lengths and angles and also to correct any inaccurate sidesand/or angles.
5 Use interval between two points to construct the two diagonals for each of your rectangles.Select distance or length to measure the length of each diagonal in every rectangle. What doyou notice?
6 Complete this property: The ____________ in a rectangle are ________.
7 In one rectangle, select Intersect two objects and click on the two diagonals.
Use distance or length to measure the distance from the intersection point to the vertex foreach diagonal. Repeat for the second rectangle. What do you notice?
8 Complete this property: The ____________ of a rectangle _____________ each other.
Technology
GeoGebra: Makingquadrilaterals
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Technology
GeoGebra:Quadrilateral sides and
angles
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Technology
GeoGebra:Quadrilateral diagonals
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Geometry
Parallelogram1 Select Interval with given length from point. Make the interval 6 cm long. Click Show label.2 Now select Interval with given length from point and click point A. Make the interval
4 cm. Use Move and drag the new interval as shown below. Label the new point, C.
3 Click Parallel line and select line AB and point C. Now select AC and point B.
4 Use Intersect two objects to create the missing vertex of the parallelogram. Label the vertex.5 Use distance or length to show the length of each side of the parallelogram. What do you
notice?
6 Complete the following: The ___________ sides of a parallelogram are _________.
7 Now use Angle to find the size of \CAB and \CDB. Repeat for \ACD and \ABD. Whatdo you notice?
8 Complete the following: The ___________ angles of a parallelogram are _________.
9 Now draw the diagonals of the parallelogram. Use distance or length to show the length ofeach diagonal. What do you notice?
10 Complete the following: The diagonals of a parallelogram are _________.
11 Do the diagonals of a parallelogram bisect each other? Repeat step 7 from ‘Rectangle’ tohelp you.
12 Complete the following: The diagonals of a parallelogram _________ bisect each other.
13 Drag any vertices of the parallelogram that you can. Is it possible to draw otherparallelograms with the same dimensions, 6 cm by 4 cm? What do you notice?
14 Use your GeoGebra skills to accurately construct other quadrilaterals such as a rhombus,kite or trapezium.
4-06 Properties of quadrilaterals
Exercise 4-06 Properties of quadrilaterals1 Copy the table below or use the link to print one out. Accurately draw each of the six special
quadrilaterals below on a sheet of paper, then cut them out. Use your shapes to help youcomplete the table of quadrilateral properties.
a Trapezium • One pair of opposite sides are _______________
Worksheet
Properties ofquadrilaterals
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b Kite • Two pairs of adjacent sides are _______________• One pair of opposite _______________ are equal.• Diagonals intersect at _______________
c Parallelogram • _______________ sides are equal.• Opposite _______________ are parallel.• Opposite angles are _______________• Diagonals _______________ each other.
d Rhombus • All _______________ are equal.• _______________ sides are _______________• _______________ angles are _______________• Diagonals bisect each other at _______________ angles.• Diagonals _______________ the angles of the rhombus.
e Rectangle • Opposite sides are _______________• Opposite sides are also _______________• All angles are _______________• Diagonals are _______________• Diagonals _______________ each other.
f Square • All sides are _______________• All angles are _______________• Opposite _______________ are parallel.• Diagonals are _______________• Diagonals bisect each other at _______________
2 Which quadrilaterals have each property?
a Opposite sides are equal b Diagonals cross at right anglesc Opposite angles are equal d One pair of opposite sides are parallele Diagonals bisect each other f Opposite sides are parallelg Adjacent sides are of different lengths h Two equal diagonalsi All angles are equal j All sides are equal
3 Copy and fill in the blanks tofind the values of a and b.
a° b°
70°
a þ 70 ¼ 180 (_____ angles on _______ lines)a ¼ _______b ¼ ________ (opposite ________ of a parallelogram)
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4 Find the value of each pronumeral.
123°
26°
24°
87°
41°
105°
110°
88°
33°
5 cm
7 m4 m
7 cm
m°a°
b°
x°
x°
a°
b°l°
m° n° a°
a°a
c
b°
a
b
b°
y°
y°b ca
d e f
g h i
j k
5 I am a quadrilateral with opposite sides equal and parallel.My diagonals are equal and I have two axes of symmetry.Which quadrilateral am I? Select the correct answer A, B, C or D.
A parallelogram B rectangle C square D rhombus
6 I am a quadrilateral with opposite sides equal.My diagonals bisect each other and meet at right angles.Which quadrilateral am I? Select A, B, C or D.
A parallelogram B trapezium C rectangle D rhombus
7 A rectangle is a quadrilateral with four right angles.Which one of the following is a special type of rectangle? Select A, B, C or D.
A square B kite C parallelogram D rhombus
8 A rhombus is a quadrilateral with all sides equal.Which one of the following is a special type of rhombus? Select A, B, C or D.
A square B kite C parallelogram D rectangle
Worked solutions
Exercise 4-06
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Technology Exterior angle of a triangleIn this activity we will use GeoGebra to discover an important property about the interior andexterior angles of any triangle. To set up your drawing page, click View and make sure that Gridis selected and Axes is not selected.
1 Draw a triangle using the polygon tool.
Right-click on each vertex and select Show label if no labels are showing.
2 Select Ray through two points from the third icon menu and draw a ray from A
through C.
3 Select New Point from the second icon menu. Insert the new point on ray AC outsideinterval AC. Show label D.
4 Use Angle from the eighth icon menu to find the size of \ABC, \BAC and \BCD.
5 Calculate \ABC þ \BAC.
6 Compare your answer from question 5 with the size of \BCD. What do you notice?
7 Copy and complete: The __________ angle of a triangle is _______ to the sum of the _____opposite _________ angles.
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4-07 Angle sums of triangles andquadrilaterals
Angle sum of a triangle
Summary
The angle sum of a triangle is 180�
a þ b þ c ¼ 180
a°
b°c°
Example 4
Find the value of each pronumeral, giving reasons.
ba
y°x°
42°
39°
67°
Solutiona xþ 42þ 39 ¼ 180 (Angle sum of a triangle)
x ¼ 180� 42� 39
¼ 99
b yþ 67þ 67 ¼ 180 (Angle sum of an isosceles triangle)
y ¼ 180� 134
¼ 46
The exterior angle of a triangle
Summary
The exterior angle of a triangle is equal to the sumof the two interior opposite angles.
z ¼ x þ y x°
y°
z°
Worksheet
Find the unknownangle 2
MAT08MGWK10035
Video tutorial
Angle sums of trianglesand quadrilaterals
MAT08MGVT10007
Video tutorial
Angle in polygons
MAT08MGVT00006
Homework sheet
Quadrilaterals andangle sums
MAT08MGHS10032
Quiz
Shapes and angles
MAT08MGQZ00006
Puzzle sheet
Mixed angles
MAT08MGPS00023
Puzzle sheet
Angles in triangles
MAT08MGPS00022
Technology
GeoGebra: Exteriorangle of a triangle
MAT08MGTC00007
Worksheet
Angles in triangles
MAT08MGWK00040
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8
Example 5
Find the value of each pronumeral, giving reasons.
ba31°
83°112°
51°m°
k°
Solutiona m ¼ 83þ 31 (Exterior angle of a triangle)
¼ 114
b k þ 51 ¼ 112 (Exterior angle of a triangle)
k ¼ 112� 51
¼ 61
Angle sum of a quadrilateralAny quadrilateral can be divided into two triangles along one of its diagonals. Because the anglesin each triangle add to 180�, the angles in both triangles add to 2 3 180� ¼ 360�.u� þ v� þ w� ¼ 180� and x� þ y� þ z� ¼ 180�) Angle sum of quadrilateral ¼ 180� þ 180�
¼ 360�
u°v° x°
y°z°w°
Summary
The angle sum of a quadrilateral is 360�.
a þ b þ c þ d ¼ 360a° b°
c°
d°
This property is true for both convex and non-convex quadrilaterals.
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Geometry
Example 6
Find the value of each pronumeral, giving reasons.
91°
79°25°
37°
230°
68°
q°k°
ba
Solutiona qþ 91þ 79þ 68 ¼ 360 (Angle sum of a quadrilateral)
qþ 238 ¼ 360
q ¼ 360� 238
¼ 122
b k þ 25þ 37þ 230 ¼ 360 (Angle sum of a quadrilateral)
k þ 292 ¼ 360
k ¼ 360� 292
¼ 68
Exercise 4-07 Angle sums of triangles and quadrilaterals1 Find the value of each pronumeral.
a b c50° 92°
28°
65° 65°
85°a° b°
c°
d e f
g h i
57°
32°
60° 140°
30°
130°
f °
e°
h°i°
d °
g°
See Example 4
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8
2 Find the value of each pronumeral.
a b c
d e f
g h i
85°
42°
105°
70°
130°
80°118°
110°120° 150°
35°a°
d°
g° i°
b°
e°
c°
f °
68°
h°
3 Find the value of each pronumeral.
a b c
d e f
g h i
62°62°
118°
118°
71° 82°
98°
46°59°
72°115°
83°
48°220°
54°
15°
40° 40°140°
110°
99°
a°
d°
g°
b°
e°
c°
f °
h°i°
4 Which of the following is the value of u? Select the correct answerA, B, C or D.
A 50 B 60 C 70 D 80u°
50°
60°
See Example 5
Worked solution
Exercise 4-07
MAT08MGWS10031
See Example 6
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12
Geometry
5 Which of the following is the value of r?Select A, B, C or D.
A 25 B 35C 50 D 100
r°
130°
6 Which equation is correct for the triangle on the right?Select A, B or C.
A n ¼ l þ m B n ¼ k þ m C n ¼ l þ kn°
k°
m° l °
7 Which of the following is the value of x?Select A, B, C or D.
A 115 B 65C 127 D 122
x°
110°
115°
°58
Investigation: Angle sum of a polygon
We know that the angle sum of a triangle is180� and that the angle sum of a quadrilateralis 360�, but how do we find the angle sum ofother convex polygons?A hexagon can be divided into four trianglesby drawing the diagonals from one vertex.
3
14
2
The sum of the angles in a hexagon ¼ ðangle sum of a triangleÞ3 4
¼ 180�3 4
¼ 720�
An octagon can be divided into six triangles bydrawing the diagonals from one vertex.
3
2
1
45
6The total of the angles in an octagon ¼ 180� 3 6 ¼ 1080�.1 How is the number of sides related to the number
of triangles formed in a shape?2 Copy and complete the following sentences.
The angle sum of a polygon with n sides isA ¼ 180 3 (number of sides � _______)�or: A ¼ 180 3 (n � _______)�
3 Find the angle sum of each polygon.
a 11-sided polygon b 20-sided polygon c 14-sided polygon
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8
4-08 Extension: Angle sum of a polygonA convex polygon has all vertices pointing outwards.A regular polygon has all sides the same length and all angles the same size.
Example 7
These shapes are all hexagons (six-sided).
a Which hexagons are convex?b Which hexagon is regular?
i ii iii
Solutiona Hexagons ii and iii are convex because all of their vertices point outwards.b Hexagon iii is regular, because all its sides are equal and all its angles are equal.
Summary
The angle sum of a polygon with n sides is given by the formula A ¼ 180(n � 2)�.This property applies to both convex and non-convex polygons.
Example 8
Find the size of one angle in a regular hexagon.
SolutionA hexagon has six sides (n ¼ 6).
Angle sum of a hexagon ¼ 180ð6� 2Þ�
¼ 180 3 4�
¼ 720�For a regular hexagon:
one angle ¼ 720�46
¼ 120�
[ Each angle in a regular hexagon is 120�.
Worksheet
Angle sum of a polygon
MAT08MGWK10036
Technology worksheet
Angle sum of a polygon
MAT08MGCT10002
Worksheet
Angles in regularpolygons
MAT08MGWK00041
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12
Geometry
Exercise 4-08 Angle sum of a polygon1 Copy these polygons and, beneath each drawing, write their correct name (chosen from the
list). Say if each polygon is regular or irregular, convex or non-convex.
hexagon nonagon heptagondecagon octagon trianglequadrilateral pentagon
fed
hg
cba
2 Copy and complete this table.
Polygon Number of sides Sum of angles inside polygonhexagonheptagonoctagonnonagondecagon
3 Find the sum of the interior angles in:
a a 15-agon b a 20-agon c a 25-agon d a 100-agon
4 Find the value of each pronumeral.
cba
a°
120° 140°
130°
110°
100°145°
160°
b°
b°130°
90°
d°
d° d°
d°d°
See Example 7
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8
fed
g
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°
5 Find the size of one interior angle in each regular polygon.
a square b equilateral triangle c regular hexagond regular octagon e regular decagon f regular pentagong regular dodecagon
6 Find the number of sides of the polygon that has an angle sum of:
a 2160� b 5760� c 4320� d 9180� e 22 140�
Mental skills 4B Maths without calculators
Converting decimals and percentages to fractions
1 Consider each of the following examples.
a 0:35 ¼ 357
20100¼ 7
20(two decimal places, two zeros in the denominator)
b 0:8 ¼ 84
510¼ 4
5(one decimal place, one zero in the denominator)
c 0:64 ¼ 6416
25100¼ 16
25
d 0:22 ¼ 2211
50100¼ 11
502 Now convert each decimal to a fraction.
a 0.75 b 0.28 c 0.3 d 0.14 e 0.06f 0.85 g 0.32 h 0.49 i 0.56 j 0.9k 0.72 l 0.65 m 0.2 n 0.24 o 0.53
3 Consider each of the following examples.
a 26% ¼ 2613
50100¼ 13
50b 40% ¼ 402
5100¼ 2
5
c 8% ¼ 82
25100¼ 2
25d 95% ¼ 9519
20100¼ 19
20
See Example 8
Worked solution
Exercise 4-08
MAT08MGWS10032
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12
Geometry
Power plus
As questions become more complex, it may not be possible to find the answer in one step.It may be necessary to find another angle first.1 Find the value of x in each diagram. (Give reasons for all steps.)
ba
dc
fe
hg
ji
lk
x°105°
120°76°
x°52°
40°
x°
84°
x°
40°
36°
72°
6 cm x cm
x°60°
86°
x° x°46° x°
25°
x°
80° 85°
68°
x°
72°
x°
30°50° 110° x°40°
4 Now convert each percentage to a fraction.
a 76% b 10% c 80% d 45% e 88%f 56% g 75% h 31% i 68% j 5%k 60% l 54% m 6% n 49% o 82%
1479780170189538
NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8
Chapter 4 review
n Language of maths
acute
alternate
angle sum
axis/axes
co-interior
complementary
convex
corresponding
diagonal
equilateral
exterior angle
interior angle
isosceles
kite
obtuse
parallelogram
quadrilateral
reflex
rhombus
rotational symmetry
scalene
supplementary
trapezium
vertically opposite
1 What are supplementary angles?
2 Name the types of angles associated with parallel lines cut by a transversal.
3 What type of triangle has one angle that is greater than 90�?
4 What is the angle sum of a quadrilateral?
5 Illustrate the difference between an interior angle and an exterior angle of a triangle.
6 What does equilateral mean? What is the common name for an ‘equilateral parallelogram’?
n Topic overview• Do you think this chapter is useful? Why? What did you learn in this chapter?• How confident do you feel with geometry?• List anything in this chapter that you did not understand. Show your teacher.
Copy and complete this mind map of the topic, adding detail to its branches and using pictures,symbols and colour where needed. Ask your teacher to check your work.
GEOMETRY
Type
s of t
rian
gles
Prop
ertie
s Angle sum
Exterior angle
Types of quadrilaterals
Types of angles
Angles andparallel lines
Properties
Angle sum
Triangles
Quadrilaterals
Line and rotationalsymmetry
Angles
A
ngle
geo
metr
y
Puzzle sheet
Geometry crossword
MAT08MGPS10014
Worksheet
Mind map: Geometry
MAT08MGWK10037
148 9780170189538
1 Name each type of angle.
cba
2 Find the value of each pronumeral, giving reasons.
cba
ed
130° x°
130° b°
a°
85°
y°
72° n°
70° 160° w°
3 Name each type of angle pair.
cba
4 Find the value of each pronumeral, giving reasons.
ba
dc
100°
m° 85°
x°
100° y° x° 88°
p°
m°
See Exercise 4-01
See Exercise 4-01
See Exercise 4-02
See Exercise 4-02
1499780170189538
Chapter 4 revision
5 Copy each shape and mark in its axes of symmetry.
a b c
6 For each shape in question 5, decide whether or not it has rotational symmetry. If it does,state the order of rotational symmetry.
7 Classify each of these triangles by its sides and angles.
a b c d
e f g h i
8 Draw a neat diagram of each of the following quadrilaterals.
a a parallelogram b a kite c a trapezium
9 Name all the quadrilaterals that have the following properties.a all sides equal in lengthb no parallel sidesc all angles right anglesd two pairs of opposite angles equale diagonals bisect each other
10 Find the value of each pronumeral.
cba
ed f
110°
20°
a°48°
a°
75°
75°
y°
69° 58°
y°
25°121°
x°37°
m°
See Exercise 4-03
See Exercise 4-03
See Exercise 4-04
See Exercise 4-05
See Exercise 4-06
See Exercise 4-07
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Chapter 4 revision
11 Find the value of each pronumeral.
cba
gfed
37°
26°
m°
10°
125°x°
33°
y°
60°
130°d°
50°
b° a°
56°b°
a°114°
b°
a°
12 Find the value of each pronumeral.
a b c
131°
67°120°
60° 50°
215°
12°83° 104°
x°
m°
y°
See Exercise 4-07
See Exercise 4-07
1519780170189538
Chapter 4 revision