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Geometrical fi gures are everywhere. Triangles, quadrilaterals and other polygons can be found all around us, in our homes, on transport, in architecture, in art and in nature. In this chapter, we will study the properties of angles and shapes and use geometrical instruments to construct them accurately.
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1 Is each marked pair of angles alternate, corresponding or co-interior?
a b c
d e f
2 Find the value of the variable in each of these diagrams. (Give reasons for your answers.)
a b c
d e f
3 Find the value of the pronumeral(s) each time, giving reasons:
a b c
d e f
Exercise 3-02
Ex 2
a°
70°
d°
135°a° 61°
m°
120° p°
91° m°68°
61°a°
b°
110°
c°d°
115°
b°a°
50° 60°
h°g°m° 80°
p°q°
89°
y°
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g h i
4 Which of the following is the value of y in this diagram?Select A, B, C or D.A 85 B 40C 35 D 65
5 For each diagram, decide whether AB is parallel to CD. If it is, then prove it.
a b c d
x°
w°y°
72°
a°55° 45°
n°60° 50°
40°105°
y°
Ex 3
48°48°
E
F
B
D
A
C 100°100°
EF
B D
A C
120°
58°
E
F
B
D
A
C
115°65°E
F
B D
A C
Can you identify the ways the idea of parallel lines has been used in the design of this terminal building and its surrounds, at Munich airport, Germany?
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Line and rotational symmetrySymmetrical objects appear balanced and usually are pleasing to look at. In general, an object is symmetrical if it looks the same after it has undergone a change of position or a movement. The two main types of symmetry are line symmetry and rotational symmetry.• Line symmetry: When a shape is folded along a line, called the
axis of symmetry, the two halves of the shape fit exactly on top of each other.
• Rotational symmetry: When we rotate (spin) a shape it fits exactly on itself at least once before it completes a whole revolution. The number of times the shape fits on itself in one revolution is called its order of rotational symmetry.
1 Copy these shapes and mark their axes of symmetry.
a b c d
2 Decide whether each of the shapes below has rotational symmetry. State the order of rotational symmetry for the shapes that do.
a b c d
3 Use computer software to create a shape, pattern or letter that has one axis of symmetry.
This shape has one axis of symmetry.
This shape has rotational symmetry of order 4.
Applying strategies and communicating Worksheet3-03
Symmetry
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3-03 Classifying trianglesTriangles can be classified in two ways: by their sides or by their angles.
1 Classify each triangle according to its sides and angles.
2 Sketch a triangle that is:a right-angled and isoscelesb equilateralc scalene and obtuse-angledd acute-angled and scalenee right-angled and scalenef isosceles.
3 Which of the following describes this triangle when we classify it by sides and angles?Select A, B, C or D.A isosceles and obtuse-angledB isosceles and acute-angledC scalene and obtuse-angledD scalene and acute-angled
4 Is it possible to draw an obtuse-angled equilateral triangle? Discuss 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.
a b
c d
e f
42°
42°
Skillsheet3-05
Line and rotational symmetry
x°
42°
38° 38°
7 cm
y cm
x°
4.8 m
ml mm
60°
60°
60°
17.2 m
a m
b m
r°
15°10 cm
10 c
m
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Just for the record
It’s all Greek or Latin to me!Many of our geometrical words come from Greek or Latin. Latin was the language of the ancient Roman Empire.
Explain what this sentence means, and illustrate with a diagram: ‘A rhombus is equilateral but not equiangular’.
Word Origin Meaning
Equilateral Latin: aequus latus Equal sides
Equiangular Latin: aequus angulus Equal corners
Isosceles Greek: isos skelos Equal legs
Scalene Greek: skalenos Uneven leg
Acute Latin: acutus Sharp
Obtuse Latin: obtusus Dull or blunt
Reflex Latin: reflexus Bent back
Triangle Latin: tri angulus Three corners
Rectangle Latin: rectus angulus Right corners
Quadrilateral Latin: quadri latus Four sides
Polygon Greek: poly gonon Many angles
Diagonal Greek: dia gonios From angle to angle
Trapezium/Trapezoid Latin/Greek: trapeza Small table
Working mathematically
Angle sum of a triangleWhat value do the sizes of the three angles in a triangle always add to? In groups of two to four, complete the following activity.
1 Draw a large triangle on paper, cut it out and label its three angles a, b and c.
2 Tear off the three angles and arrange them next to each other, so their points meet.
3 What type of angle do they form? How many degrees in this type of angle?4 To see if this works for all triangles, repeat the above steps for a variety of
differently-shaped triangles.
a°
b°c°
a°
b°
c°
Applying strategies and reasoning
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a° + b° + c° = 180°The sum of the three angles is called the angle sum of the triangle.
Formal proof5 We can use parallel lines to prove that the angle sum of a triangle is 180°.
Step 1: Draw any triangle ΔABC, with angles of size a°, b° and c°.
Step 2: Draw a line DE through A and parallel to CB.
a Which angle in ΔABC is equal to ∠DAC? Why?b Which angle in ΔABC is equal to ∠EAB? Why?c What does ∠DAC + a° + ∠EAB equal? Why?d What does this show about a° + b° + c°?e What does this show about the angle sum of a triangle?f Check by measuring with a protractor.
a° b°
c°
A
C B
a°
c° b°
AD E
C B
a°
c° b°
The angle sum of a triangle is 180°. !
Example 4
Find the value of each a bpronumeral, giving reasons:
Solutiona t + 70 + 63 = 180 (Angle sum of a triangle.)
t = 47b x + 58 + 58 = 180 (Angle sum of an isosceles triangle.)
x = 64
t°
63°
70°
x°
58°
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1 Find the value of each pronumeral.
a b c d
e f g
h i j
k l m
n o p
q r s
Exercise 3-04
Ex 4
a°
70° 60°
b°
75° 75°
c°
110° 40°
d°
80° 40°
e°
28° 28°
f °
40°g°37°
76°
h°
32°126°i°
47°81°
j°
53°
37°
k°
60° 60°
l °60° m° 36°
79°
n°23°
18°
q°
11°
m°
y°
30°x°
42°
x°80°
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2 Using what you know about angles, find the value of each pronumeral below:
a b c
3 Which of the following is the value of m?Select A, B, C or D.A 115 B 100C 50 D 130
x°
70°
50°
a°
20° c°b°
a°
40°
50°
m°
130°
Geometry3-05
Exterior angle ofa triangle
Working mathematically
Exterior angle of a triangleAn exterior angle of a triangle is created by extending one side of the triangle. ‘Exterior’ means ‘outside’, while ‘interior’ means ‘inside’.
What is the relationship between the exterior angle of a triangle and the interior angles?
An example
1 Which is the exterior angle in this triangle: a or b?
2 Find the values of a and b.
3 What is the relationship between 34°, 30° and b°?
4 Copy and complete: An exterior angle of a triangle is equal to the ________ of the interior opposite ________.
exterior angle
exterior angle
a° b°
30°
34°
Applying strategies and reasoning
Geometry3-05
Exterior angle ofa triangle
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3-05 The exterior angle of a triangle
Formal proofWe can use parallel lines to prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.Step 1: Draw any triangle ΔABC, with angles of size a°, b° and c°.
Step 2: Extend side CB to D to create exterior angle d°.
Step 3: Draw a line, EF, through A and parallel to CD.
1 a Which angle in ΔABC is equal to ∠EAC? Why?b What is the size of ∠EAB?
2 a What type of angles are ∠EAB and ∠ABD?b What is the size of d°?
3 a What do Questions 1 and 2 show about the exterior angle of a triangle?b Check by measuring with a protractor.c Tear off the angles marked a° and c° and check whether, when placed together,
they match the angle marked d° in size.
A
C B
a°
c° b°
A
C DB
a°
c° b° d°
AE F
C DB
a°
c° b° d°
Worksheet3-04
Triangle geometry
An exterior angle of a triangle is equal tothe sum of the two interior opposite angles.
z = x + y!
x°
y°
z°
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1 For each triangle below, name:i the exterior angle
ii the two interior opposite angles for the exterior angle.
a b c
2 Find the value of each pronumeral and state whether it represents an interior or exterior angle.
a b c
d e f
Exercise 3-05
Example 5
Find the value of each pronumeral, giving reasons:
a b
Solutiona x = 45 + 41 (Exterior angle of a triangle.)
= 86b m + 54 = 116 (Exterior angle of a triangle.)
m = 62
x°
41°
45°m°
116° 54°
b°
c°a°
d °
q°
t°
p°
r°
w°
z°
x° y°
Ex 5
18°
86°
h°
100°84°
y°70° 25°
e°d °
130°
14°b°
a°
45°
m°
83° 31°
y°
x°
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3 Find the value of each variable:
a b c
d e f
4 Find the value of each variable:
a b c
5 Which of the following is the value of p?Select A, B, C or D.A 110 B 125C 55 D 140
6 Another exterior angle proofComplete the missing reasons in this proof.Consider any triangle XYZ in which the angles area°, b° and c°. Produce the line YZ to the point W.a° + b° + c° = 180° (________________________)
82 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952
2 What am I? (Note: There may be more than one answer each time.)a Opposite sides are equal.b My diagonals meet at right angles.c Opposite angles are equal.d Opposite sides are not equal.e Diagonals bisect each other.f Opposite sides are parallel.g Adjacent sides are of different lengths.
3 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 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.
c • ______________ sides are equal.• Opposite __________ are parallel.• Opposite angles are ___________.• Diagonals __________ each other.
d • All _________ are equal.• __________ sides are ________.• __________ angles are _________.• Diagonals bisect each other at ________ angles.• Diagonals ________ the angles of the rhombus.
e • Opposite sides are _________.• Opposite sides are _________.• All angles are _________.• Diagonals are _________.• Diagonals _________ each other.
f • All sides are _________.• All angles are _________.• Opposite _________ are parallel.• Diagonals are _________.• Diagonals bisect each other at _________.
Parallelogram
Rhombus
Rectangle
Square
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4 a Which types of quadrilateral have a pair of opposite sides parallel?b Which types of quadrilateral have two equal diagonals?
5 Find the value of each variable:
a b c
d e f
g h i
j k
6 My opposite sides are equal.My diagonals bisect each other.My diagonals meet at right angles.Which one of the following quadrilaterals am I?Select A, B, C or D.A parallelogram B trapezium C rectangle D rhombus
m°
123° b°
a°
26°
y°
x°
24°
l°
a°
b°
41°
y°x°
87°
m° n°
105°
33°
b°
a°
5 cm
7 cm
a
b
7 m4 m
ca
110°
88°a°
b°
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3-07 The angle sum of a quadrilateralAny quadrilateral can be divided into two triangles along one of its diagonals. Because the angles in each triangle add to 180°, the angles in both triangles add to 2 × 180° = 360°.
u° + v° + w° = 180°, and x° + y° + z° = 180°∴ Angle sum of quadrilateral = 180° + 180°
= 360°
Working mathematically
Angle sum of a quadrilateralWhat value do the four angles in a quadrilateral always add to? In groups of two to four, complete the following activity.
1 Draw any large quadrilateral on paper, cut it out and label its four angles a°, b°, c° and d°.
2 Tear off the four angles and arrange them so their points meet.
3 What type of angle do they form? How many degrees in this type of angle?
4 To see if this works for all quadrilaterals, repeat the above steps for a variety of differently-shaped quadrilaterals.
b°a°
d ° c°
d °c°
b° a°
Applying strategies and reasoning
Worksheet3-08
Find the missingangle 2
Worksheet3-09
Deductive geometry
x°
w°
u°
z°
v°
y°
The angle sum of a quadrilateral is 360°.!
Example 6
Find the value of m in this quadrilateral:
Solutionm + 75 + 60 + 100 = 360 (Angle sum of a quadrilateral.)
m + 235 = 360m = 125
75°
100°
60°
m°
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2 Which of the following is the value of x?Select A, B, C or D.A 95 B 105C 85 D 115
70°
95°x°
Working mathematically
Angle sum of a convex polygonWe know that the angle sum of a triangle is 180° and that the angle sum of a quadrilateral is 360°, but how do we find the angle sum of other convex polygons?A hexagon can be divided into four triangles by drawing the diagonals from one vertex.
The sum of the angles in a hexagon = (angle sum of a triangle) × 4= 180° × 4= 720°
An octagon can be divided into six triangles by drawing the diagonals from one vertex.
The total of the angles in an octagon = 180° × 6= 1080°
1 Copy and complete the following sentence.The angle sum of a convex polygon with n sides is:
A = 180 × (number of triangles − ___)°or: A = 180 × (n − ___)°
2 Find the angle sum of:a an eleven-sided polygonb a 20-sided polygonc a 14-sided polygon.
32
14
3
2
1
45
6
Applying strategies and reasoning
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3-08 Angle sum of a convex polygonA convex polygon has all vertices pointing outwards. A regular polygon has all sides the same length and all angles the same size.
Worksheet3-10
Angle sum ofa polygon
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.
The angle sum of a convex polygon with n sides is given by the formula A = 180(n − 2)°. !
Example 8
Find the angle sum of a pentagon.
SolutionIn a pentagon, the number of sides, n, is 5.A = 180(5 − 2)°
= 180 × (3)°= 540°
∴ The angle sum of a pentagon is 540°.
Stage 5
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1 Copy these polygons and, beneath each drawing, write the correct name (chosen from the list). Say if each polygon is regular or irregular, convex or non-convex.hexagon nonagon heptagondecagon octagon trianglequadrilateral 13-agon pentagon
a b c
d e f
g h
Exercise 3-08
Example 9
Find the size of one angle in a regular hexagon.
SolutionA hexagon has six sides (n = 6).Angle sum of a hexagon = 180(6 − 2)°
= 180° × (4)° = 720°
For a regular hexagon:x° = 720° ÷ 6
= 120°∴ Each angle in a regular hexagon is 120°.
x°
Ex 7
Stage 5
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Construct an isosceles triangle with two equal 3 cm sides and one 4 cm side.SolutionStep 1: Step 2: Step 3:
Draw a rough sketch. Use a ruler. Use a ruler and compasses.
Step 4: Step 5:
Use a ruler and compasses. Use a ruler to complete the triangle.
4 cm
3 cm 3 cm
4 cm 4 cm
3 cm
4 cm
3 cm
4 cm
3 cm 3 cm
Example 12
Construct a trapezium, DEFG, where DE || GF, DE = 4 cm, EF = GF = 2 cm, and ∠F = 135°.Solution
Step 1: Step 2: Step 3:
Draw a rough sketch. Use a ruler. Use a protractor and ruler.
Step 4: Step 5:
Use a protractor and ruler: use a 45° angle because co-interior angles in parallel lines add to 180°.
Complete the trapezium.
4 cm
2 cm
D E
FG2 c
m135°
2 cm FG135°
2 cm
E
FG2 c
m
135°
45°4 cm
2 cm
D E
FG2 c
m135°
4 cm
2 cm
D E
FG2 c
m
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1 Construct each of these triangles accurately.
a b c
2 Copy and complete this sentence:In any triangle, the longest side is always ____________ the largest angle, while the shortest side is always _____________ the ___________ angle.
3 Construct each triangle accurately. Draw a rough sketch first.a ΔABC with BC = 5 cm, AB = 4 cm and ∠B = 90°b ΔWXY with XY = 3.5 cm, ∠X = 70° and ∠Y = 55°c ΔFGH with GH = 6.2 cm, FH = 4.9 cm and FG = 5.5 cm
4 The triangle inequality rule says that: ‘If you add any two sides of a triangle, the combined length is always greater than the length of the third side.’ (This inequality can be written as a + b � c.)a Test that this inequality is true for all the triangles you constructed in Question 3.b Why is it impossible to construct a triangle with sides of length 10 cm, 6 cm and
18 cm?
5 Construct each quadrilateral accurately.
a b
c d
6 Construct a rhombus with side lengths 5 cm and an included angle of 40°.
7 Construct a quadrilateral LMPQ, with LM = 5 cm, LQ = 3 cm, MP = 4 cm, ∠L = 120° and ∠M = 110°.
8 Construct trapezium DEFG, so DE || FG, DE = 4 cm, EF = GF = 7 cm and ∠F = 75°.
9 Computer software, such as The Geometer’s Sketchpad or Cabri Geometry, can be used to accurately draw triangles and quadrilaterals. Use the link provided to see how.
Exercise 3-09
Ex 10
Ex 11
7 cm
6 cm
V U
T
4 cm
5 cmZY
X
70° 30° 3.5
cm
F
ED105°
4.5 cm
Ex 12
6 cm
4 cm
M L
J K
60°
S Q
P
R
95°
4 cm
7 cm
7 cm
5.5
cm
D C
BA
55°
5 cm
5 cm
6.5 cm
IH
F G
63°50°
Geometry3-09
Constructions involving triangles and quadrilaterals
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1 Draw a line and mark a point, L, on it. Construct a perpendicular line that passes through L:a using a set squareb using compassesc using a protractor.
2 Draw a line and mark a point, X, above it. Construct a perpendicular line that passes through X:a using compasses b using a set square.
3 Copy each of these diagrams and use compasses to construct a perpendicular line through P in each diagram.
a b
Exercise 3-10
Example 16
Use a set square to construct a line parallel to AB through point P.
Solution
Step 1: Step 2: Step 3:
Position the ruler through P and the set square along the line AB.
Slide the set square along the ruler to point P.
Rule the line parallel to AB through P.
P
A B
P
A B
P
A B
P
A B
Ex 13
Ex 14
P
P
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c d
4 Copy these diagrams and construct a line parallel to AB through X each time.i using compasses ii using a set square.
a b c
5 a Draw a point, P, on a line, XY, and use compasses to construct a perpendicular through P.
b When we construct the perpendicular through P, we are really constructing an ‘invisible’ isosceles triangle. Draw the ‘invisible’ triangle on your diagram from part a.
c Which property of the isosceles triangle proves that the line through P is a perpendicular?
6 a Draw a point, P, above a line AB and use compasses to construct a perpendicular through P.
b Can you ‘see’ the ‘invisible’ rhombus? Draw it.c Which property of the rhombus proves that the line through P is perpendicular
to AB?7 a Draw two intervals that are parallel and of different lengths.
b Join their ends to make a quadrilateral.c What type of quadrilateral have you constructed?
8 a Draw an interval and mark its midpoint.b Draw a different-sized interval through the midpoint
of the first interval, perpendicular to it and with the same midpoint (as shown on the right).
c Join the ends of the interval to make a quadrilateral.d What type of quadrilateral have you constructed?
9 a Draw a large triangle, ABC.b From each vertex, construct a line that is
perpendicular to the opposite side of the triangle. (These lines are called the altitudes of the triangle.)
c What do you notice about the altitudes of a triangle?
P
P
Ex 15
Ex 16
X
A
BX
A
B
X
A
B
A
C
B
Geometry3-11
Medians in a triangle
Geometry3-12
Altitudes in a triangle
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10 Computer software, such as The Geometer’s Sketchpad or Cabri Geometry, can be used for accurate constructions with lines. Use the link provided to see how.
3-11 Angle and interval constructionsA ruler and compasses can be used to bisect (cut in half) an angle or an interval. They can also be used to construct special angles.
Geometry3-13
Parallel and perpendicular lines
Worksheet3-13
Bisecting intervalsand angles
Geometry3-14
Constructions involving parallel lines
Example 17
Bisect the interval AB.
Solution
Step 1: Step 2: Step 3:
Open your compasses more than half the length of AB and draw a large arc from A.
Draw another arc the same distance from B.
Use a ruler and mark the midpoint M. Check with a ruler that M is halfway.
A B
A B A B A BM
Example 18
Bisect ∠CAB.
Solution
Step 1: Step 2: Step 3:
Use compasses to draw a large arc from A.
Draw another arc the same distance from D.
Draw another arc the same distance from E.
A B
C
A B
C
D
E A B
C
D
E A B
C
D
E
F
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Step 4:
Use a protractor to check that the angle has been bisected.
Use a ruler to join F to A to cut the angle in half.
A B
C
D
E
F
Example 19
Construct a 60° angle.
SolutionConstruct an ‘invisible’ equilateral triangle, because each of its angles will be 60°.
Step 1: Step 2: Step 3:
Use compasses from A. Use compasses from C. Use a ruler.
AC
AC
AC
D
60°
Example 20
Construct a 120° angle.
SolutionConstruct two ‘invisible’ equilateral triangles that share a side.
Step 1: Step 2: Step 3:
Use compasses from B. Use compasses from C and D. Use a ruler.
B C B
ED
C B
ED
C
120°
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Language of mathsalternate angle sum bisect co-interiorcompasses complementary construct correspondingdiagonal equilateral exterior angle interior angleinterval isosceles kite parallelparallelogram perpendicular protractor quadrilateralrectangle rhombus scalene set squaresquare supplementary trapezium vertically opposite
1 What word means ‘to cut in half’?
2 What type of triangle has one angle that is greater than 90°?
3 Illustrate the difference between an interior angle and an exterior angle of a triangle.
4 What is ‘equilateral’? What is the common name for an ‘equilateral parallelogram’?
5 Which geometrical instrument is used to draw circles and mark equal lengths?
6 Explain how you would prove the angle sum of a quadrilateral?
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 this diagram and use it to form your chapter summary.
GEOMETRICALFIGURES
Type
s of
tria
ngle
s
Pro
pert
ies Angle sum
Exterior angle
Types of quadrilaterals
Types of angles
Angles andparallel lines
Bisecting intervals
and angles
Triangles, quadrilaterals, angles
Perpe
ndicu
lar a
nd
para
llel li
nes
Properties
Angle sum
Triangles
Quadrilaterals
Constructions
AnglesAng
le g
eom
etry
Chapter 3 reviewWorksheet
3-14
Geometrical figures crossword
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Chapter revision1 Name each type of angle shown:
a b c
2 Find the value of each pronumeral, giving reasons.
a b c
d e
3 Name each type of angle pair indicated.
a b c
4 Find the value of each pronumeral, giving reasons.
a b
c d
Exercise 3-01
Exercise 3-01
130° x°
130° b°
a°
85°
y°
72° n°
70° 160° w°
Exercise 3-02
Exercise 3-02
100°
m° 85°
x°
100° y° x° 88°
p°
m°
Topic testChapter 3
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5 Classify each of these triangles by its sides and angles.
a b c d
e f g h i
6 Find the value of each pronumeral.
a b c
d e f
7 Find the value of each pronumeral.
a b c
d e f g
Exercise 3-03
Exercise 3-04
110°
20°
a° 48°
a°
75°
75°
y°
69° 58°
y°
25° 121°
x° 37°
m°
Exercise 3-05
37°
26°
m°
10°
125° x°
33°
y°
60°
130° d°
50°
b° a°
56° b°
a° 114°
b°
a°
03 NCM8_4 SB TXT.fm Page 103 Friday, September 19, 2008 11:22 AM
104 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952
8 What type of quadrilateral am I? (There may be more than one answer each time.)a My opposite angles are equal.b My diagonals bisect each other at right angles.c My opposite sides are equal.d All of my sides are equal.e My opposite sides are parallel.f My diagonals bisect my interior angles.
9 Find the value of each pronumeral.
a b c d
10 Name each of the following polygons. State if they are regular or irregular and findtheir angle sum.
a b
c d
11 Find the size of the interior angles in each of these regular polygons.
a b c
Exercise 3-06
Exercise 3-07
109° 115°
66° y°
131° 88°
60° x° 141° 95°
98°
m°
121°
88°
y°
Exercise 3-08
Exercise 3-08
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