space and geometry 3web2.hunterspt-h.schools.nsw.edu.au/studentshared/mathematics/year 8... ·...

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3 SPACE AND GEOMETRY

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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In this chapter you will:

• use angle relationships to fi nd unknown angles in diagrams

• use angle properties to identify parallel lines• recognise and classify types of triangles by sides

and angles• justify that the angle sum of a triangle is 180° and

that any exterior angle equals the sum of the two interior opposite angles

• investigate the properties of special quadrilaterals, including line and rotational symmetry

• classify special quadrilaterals on the basis of their properties

• establish that the angle sum of a quadrilateral is 360°

• (Stage 5) fi nd the angle sum of a convex polygon• apply geometrical facts, properties and

relationships to fi nd unknown sides and angles in diagrams

• construct various types of triangles and quadrilaterals using geometrical instruments

• draw a perpendicular to a line from a point on or off the line using geometrical instruments

• draw a line parallel to a given line, through a point, using geometrical instruments

• use a ruler and compasses to bisect an interval and an angle

• use a ruler and compasses to construct angles of 60° and 120°

Wordbank

• angle sum the total of the angle sizes in a shape

• bisect to cut in half• compasses geometrical instruments for marking

equal lengths or constructing circles• exterior angle an angle outside a polygon

formed by extending a side of the polygon• midpoint the point at the centre of an interval,

at equal distance from its ends• perpendicular an interval or line constructed at

right angles (90°) to another interval or line• supplementary two angles that add to 180°

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64 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

Start up

1 Draw two different examples of:a an acute angle b a right anglec an obtuse angle d a 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 What is the complement of:a 16°? b 38°? c 88°? d 2°?

4 Write the supplement of:a 29° b 169° c 100° d 33°

5 Name the types of angles marked below:

a b c d

3-01 Angle geometryClassifying angles

Right angle Straight angle Revolution

90° (quarter-turn)Note that a right angle is marked

with a box symbol.

180° (half-turn) 360° (complete turn)

Acute angle Obtuse angle Reflex angle

Less than 90° Greater than 90° but less than 180°

Greater than 180° but less than 360°

Skillsheet3-01

Types of angles

Geometry3-01

Angle vocabulary

Skillsheet3-02

Starting The Geometer’s Sketchpad

Skillsheet3-03

Starting CabriGeometry

Worksheet3-01

Brainstarters 3

03 NCM8_4 SB TXT.fm Page 64 Friday, September 19, 2008 11:22 AM

65ISBN: 9780170136952 CHAPTER 3 GEOMETRICAL FIGURES

Angle facts

Adjacent angles Complementary angles Angles in a right angle

Angles next to each other, sharing a common arm and a

common vertex.∠ABD and ∠DBC are adjacent.

Angles that have a sum of 90°, for example 35° and 55°.

Complementary angles (add to 90°).

a + b = 90

Supplementary angles

Angles that have a sum of 180°, for example 140° and 40°.

Angles on a straight line Angles at a point Vertically opposite angles

Supplementary angles (add to 180°).

x + y = 180

(In a revolution) add to 360°.a + b + c + d = 360

Equal angles.w = y and x = z

B

A

C

D

a° b°

x° y°

a° b°

c° d°

w° x°

y° z°

Example 1

Find the value of each pronumeral, giving reasons:

a b

Solutiona a° + 135° = 180° (Angles on a straight line.)

a° = 45°

b a° = 55° (Vertically opposite angles.)b° + 55° = 180° (Angles on a straight line.)

b° = 180° − 55°= 125°

a° 135° 55°

b° a°



1 Classify each type of angle shown:

a b c

d e f

g h i

2 Find the value of each pronumeral, giving reasons:

a b c

d e f

g h i

j k l

Exercise 3-01

Ex 1

x° 20°

a° 68° a°

b° 110°

y° 75° a°

161° x°

148° 95°

a°

b° c° 128° x°

160°

35° x°

a° b°

c° 45°

k°

32° n° 110°



m n o

p q r

3 Write the values of the variables in these diagrams. (Give reasons for your answers.)

a b c

d e f

g h

4 Points A, B and C lie on a straight line. BE bisects ∠DBF. Which of the following is the size of ∠EBC?Select A, B, C or D.A 95° B 110°C 85° D 140°

w° 112°

92°

q° 118° y°

320°

p°p°

28°p°

65°42°a°

Geometry3-02

Revolutions andstraight angles

m°

60°65°

m°

a°a°

80°

x° 100° a°

b° c°

35°

65°a°

a° a°45°

p°p°

m° m°m°

30°40°

AB

C

D

E

F

03 NCM8_4 SB TXT.fm 7 Friday, September 19, 2008 11:22 AM


3-02 Angles and parallel linesWhen parallel lines are crossed by another line (called a transversal), special pairs of angles are formed.

Alternate angles Corresponding angles Co-interior angles

Alternate angles on parallel lines are equal.

Corresponding angles on parallel lines are equal.

Co-interior angles on parallel lines are supplementary

(add to 180°).

Worksheet3-02

Find themissing angle 1

Geometry3-03

Angles andparallel lines

Skillsheet3-04


Example 2


a b

Solutiona y = 62 (Alternate angles on parallel lines.)

b a + 110 = 180 (Co-interior angles on parallel lines.)

a = 70

62°

y°

110°a°

Example 3

Prove that the lines AB and CD are parallel.

Solution∠AEF and ∠EFD are alternate angles.∠AEF = ∠EFD = 76°∴ AB || CD (Alternate angles are equal.)

76°

76°

E

F

B

DC

A



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°



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?



Skillsheet3-05

Line and rotational symmetry

Working mathematically

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



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.

a b c d

e f g h

i j k l

Classifying by sides

Equilateral triangle Isosceles triangle Scalene triangle

Three equal sides(Also three equal angles,

each 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 triangle

Three acute angles(less than 90°)

One obtuse angle(between 90° and 180°)

One right angle (90°)

Exercise 3-03

Skillsheet3-06

Two-dimensionalshapes

Geometry3-04

Triangles

60°60°

60°



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


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



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


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



3-04 The angle sum of a triangle

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°



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°



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


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°


Geometry3-05

Exterior angle ofa triangle



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°



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


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°



3 Find the value of each variable:

a b c

d e f


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° (________________________)

∴ a° + b° = 180° − c°but: ∠XZW = 180° − c° (____________________)

∴ ∠XZW = a° + b°

e°59°

126°

h°

11°

162° m°

130° 140°

x° 140°p°

130°w°

46°

m°

44°

52°

146°

b°a°

38°68°c°

b°

a°

p°

70°

a°

b°c°

X

Y

Z W



3-06 Classifying quadrilateralsThere are many special types of quadrilaterals. We will now review what we already know about these shapes.

1 Accurately draw an example of each of the following shapes. Use your shapes to help you complete the sentences.

Exercise 3-06

a • One pair of opposite sides are __________.

b • Two pairs of adjacent sides are __________.• One pair of opposite __________ are equal.• Diagonals intersect at _____________.

Mental skills 3

Multiplying by 9, 11, 99 or 101We can use expanding when multiplying by a number near 10 or near 100.

1 Examine these examples:a 25 × 11 = 25 × (10 + 1) b 14 × 9 = 14 × (10 − 1)

= 25 × 10 + 25 × 1 = 14 × 10 + 14 × (-1)= 250 + 25 = 140 − 14= 275 = 126

c 27 × 101 = 27 × (100 + 1) d 18 × 8 = 18 × (10 − 2)= 27 × 100 + 27 × 1 = 18 × 10 + 18 × (-2)= 2700 + 27 = 180 − 36= 2727 = 144

2 Now simplify these:a 16 × 11 b 33 × 11 c 29 × 9 d 45 × 9e 62 × 11 f 7 × 101 g 18 × 101 h 36 × 99i 19 × 8 j 45 × 12 k 21 × 102 l 6 × 98

Maths without calculators

Worksheet3-05

Properties of quadrilaterals

Worksheet3-07

Always, sometimes, never true?

Geometry3-07

Studyingquadrilaterals

Worksheet3-06

Classifyingquadrilaterals

Geometry3-06

Quadrilaterals

Skillsheet3-05


Skillsheet3-06

Two-dimensional shapes

Trapezium

Kite

03 NCM8_4 SB TXT.fm Page 81 Wednesday, October 1, 2008 1:24 PM


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



4 a Which types of quadrilateral have a pair of opposite sides parallel?b Which types of quadrilateral have two equal diagonals?


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°

03 NCM8_4 SB TXT.fm Page 83 Monday, September 29, 2008 1:07 AM


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°


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°


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°




a b c

d e f

g h i

j k l

m n o

p q r

Exercise 3-07

Ex 6

110°

85°

75°

a°

120°

120°60°

b° 115°

50°

55°c°

r°

62°

118°

118°

m°

15°

160°160°

l °

65°

55°

109°

n° 70°

20°

220°

p°42°

21°

y°

x°17°

k°

s°

20° 15°

25°

t°

38°

38°

142°

b°

33° n°

l° m°

112° 67° a°b°

131°

130°

28°

x°

y°

40°

45°

m°

72° 130°a°

c° b°



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°


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




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



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



2 Copy and complete this table:

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 angles marked by the pronumerals in these polygons. Angles marked with the same pronumerals are equal in size.

a b c

d e f

g

5 Find the size of one interior angle in these regular polygons. a square b equilateral triangle c hexagond octagon e decagon f pentagon g dodecagon

6 Find the number of sides of the polygon if the total number of degrees of the interior angles is:a 2160° b 5760° c 4320° d 9180° e 22 140°

Polygon Number of sides Sum of angles inside polygon

hexagon

heptagon

octagon

nonagon

decagon

Ex 8

a°

120° 140°

130°

110°

145°

160°

b°

b°130°

90° 100°

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°

Geometry3-08

A five-pointed star

Ex 9

Stage 5



3-09 Constructing triangles and quadrilaterals

Compasses

Protractor

Set square

90 100 110 120 130

140150

160170

180

8070

6050

4030

2010

0

90 80 70 6050

4030

2010

0

100110

120

130

140

150

160

170

180

Example 10

Construct a triangle with two sides of length 3 cm and 4 cm and an included angle of 100°.

Solution

Step 1: Step 2: Step 3:

Draw a rough sketch. Use a ruler to draw the 4 cm side.

Use a protractor to measure 100°.

Step 4: Step 5:

Use a ruler to draw the 3 cm side. Complete the triangle.

The ‘included’ angle sits between the two given sides.

100°

3 cm

4 cm 4 cm100°

4 cm

100°

3 cm

4 cm100°

3 cm

4 cm

Geometry3-09

Constructing triangles

Worksheet3-11

Constructions in diagrams

Worksheet3-12

Constructions in words



Example 11

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


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



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



3-10 Constructing perpendicular and parallel linesWe can apply properties of geometrical figures to construct perpendicular and parallel lines.


The perpendicular bisector in an isosceles triangleΔABC is an isosceles triangle with AB = AC.It has one axis of symmetry, AD.

1 CD = DB. Why?

2 ∠ADC = ∠ADB. Why?

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.’ Explain this property in your own words.

A

BCD

Reflecting and reasoning

Geometry3-13

Parallel and perpendicular lines

Example 13

Use compasses to draw a perpendicular through the point B on the line.

Solution


Use compasses to draw two arcs from B.

Open compasses wider to draw an arc from C.

Use compasses to draw an arc with the same distance from D.

Step 4:

Join B to where thetwo arcs cross.

Use a protractor or set square tocheck that the line is

perpendicular (at 90°) to CD.

B

C DB C DB C DB

C DB



Example 14

Use compasses to draw a perpendicular through the point P above the line.

Solution


Use compasses from P to mark two arcs with the same radius on the line.

Use compasses from Q and R to mark two intersecting arcs with the same radius below the line.

Join P to where the two arcs cross.

Use a protractor or set square to check that the line is perpendicular to QR.

P

P

Q R

P

Q R

P

Q R

Example 15

Use compasses to draw a line through X which is parallel to the given line.

Solution

Step 1: Step 2:

Use compasses from X to mark two large arcs at Y and Z.

Use compasses from Y to mark an arc with the same radius at A on the line.

Step 3: Step 4:

Use compasses from A to mark an arc with the same radius to cross the arc at Z.

Join XZ to construct a line parallel to AY.

X

X

Y

Z X

AY

Z

X

AY

Z X

AY

Z



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


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



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



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


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


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



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°.


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.


Use compasses from B. Use compasses from C and D. Use a ruler.

B C B

ED

C B

ED

C

120°



1 Draw any three intervals and bisect them.

2 Draw any three angles and bisect them.

3 Construct an angle of 90° by bisecting a straight angle.

4 Construct an angle of 90° and then bisect it to form a 45° angle.

5 Draw an interval of length 10 cm and use compasses to divide it into four equal sub-intervals. Check by measuring with a ruler.

6 a Use compasses to construct (without measuring) an isosceles triangle (any size). Label it ABC.

b Bisect angle A. Extend the line to cut BC at D.c What do you notice about ∠ADB?d What do you notice about CD and DB?

7 a Draw ∠PQR (any size). Use a pair of compasses to bisect the angle with the interval TQ.

b Check with a protractor that ∠PQR has been bisected by TQ.c When we are bisecting an angle, we are constructing an ‘invisible’ rhombus, with PQ

and QR being two of its sides. Draw the ‘invisible’ rhombus.d Which property of the rhombus proves that TQ bisects ∠PQR?

8 Computer software, such as The Geometer’s Sketchpad or Cabri Geometry, can be used to accurately bisect angles. Use the link provided to see how.

9 a Construct an angle of 60°.b Bisect the angle you constructed to form a 30° angle.

10 a Construct an angle of 120°.b Bisect the angle you constructed.

11 Construct an angle of 15°.

12 a Draw an interval, BC, of length 4 cm.b Construct an angle of 120° at B.

13 a Mark any point, P, on an interval, DE.b Construct a 120° angle, ∠APD, by first constructing a 60° angle ∠APE. Explain what

you did.

Exercise 3-11

Ex 17

Ex 18

B

A

C D

Geometry3-15

Bisecting an angle

Ex 19

Ex 20



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 of the following diagrams. (Give reasons for all steps.)

a b

c d

e f

g h

i j

k l

x°105°

76° 120°

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°



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


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



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



5 Classify each of these triangles by its sides and angles.

a b c d

e f g h i


a b c

d e f


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°



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.


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



12 Use geometrical instruments to construct each shape.

a b c

13 Construct a quadrilateral, WXYZ, where WX ⊥ XY, WX = 8 cm, XY = 4 cm, ZY = 5 cm and WZ = 4.5 cm

14 Copy each diagram and construct the perpendicular to AB going through P.

a c

b

15 Copy each diagram and construct the line parallel to BC going through P.

a b

16 Construct the following angles:a 60° b 120° c 90°

17 a Draw any three angles and then bisect them.b Draw any three intervals and then bisect them.

Exercise 3-09

60°

5 cm30°

60 mm 45 mm

45 mm

45 m

m

Exercise 3-09

Exercise 3-10

A

B

P

A

B

P

A

B

P

Exercise 3-10

CB

P

C

BP

Exercise 3-11

Exercise 3-11


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