Geometry is everywhere. Angles, parallel lines, triangles...

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�

NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8

9780170189538

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

MAT08MGWK10027

Skillsheet

Types of angles

MAT08MGSS10015

118 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

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

MAT08MGSS10019

Skillsheet

Starting Geometer’sSketchpad

MAT08MGSS10020

Puzzle sheet

Angles: A dog day

MAT08MGPS00020

Worksheet

Straight angles, rightangles and revolutions

MAT08MGWK00038

1199780170189538


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.

120 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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

1219780170189538


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°


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

MAT08MGWS10028

122 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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

MAT08MGWK10028

Skillsheet

Angles and parallellines

MAT08MGSS10016

Homework sheet

Angle geometry

MAT08MGHS10030

Worksheet

Angles in parallel lines

MAT08MGWK00039

Puzzle sheet

Angles in parallel lines

MAT08MGPS00021

1239780170189538


Example 2


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’

124 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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


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

MAT08MGWS10029

1259780170189538


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

MAT08MGWK10029

Skillsheet

Line and rotationalsymmetry

MAT08MGSS10017

126 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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

1279780170189538


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

MAT08MGWK10030

Worksheet

Triangle geometry

MAT08MGWK10031

Skillsheet

Naming shapes

MAT08MGSS10018

Homework sheet

Symmetry and triangles

MAT08MGHS10031

Puzzle sheet

Classifying triangles

MAT08MGPS00003

128 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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°

1299780170189538


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%

130 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

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

1319780170189538


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

MAT08MGWK10032

Worksheet

Classifyingquadrilaterals

MAT08MGWK10033

Worksheet

Always, sometimes,never true

MAT08MGWK10034

Puzzle sheet

What shape am I?

MAT08MGPS10013

Video tutorial

Classifyingquadrilaterals

MAT08MGVT00002

Puzzle sheet

Singing in the car

MAT08MGPS00004

Worksheet

Classifying trianglesand quadrilaterals

MAT08MGWK00010

‘Adjacent’ means ‘next to eachother’.

132 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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’?


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.

1339780170189538


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

MAT08MGTC00005

Technology

GeoGebra:Quadrilateral sides and

angles

MAT08MGTC00002

Technology

GeoGebra:Quadrilateral diagonals

MAT08MGTC00003

134 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

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

MAT08MGWK10032

1359780170189538


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)

136 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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

MAT08MGWS10030

1379780170189538


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.

138 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

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


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

1399780170189538


Example 5


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.

140 9780170189538

Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Geometry

Example 6


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

1419780170189538



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°


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

142 9780170189538

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

1439780170189538


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

144 9780170189538

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


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


cba

a°

120° 140°

130°

110°

100°145°

160°

b°

b°130°

90°

d°

d° d°

d°d°

See Example 7

1459780170189538


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

146 9780170189538

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


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


cba

ed

130° x°

130° b°

a°

85°

y°

72° n°

70° 160° w°

3 Name each type of angle pair.

cba


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


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

150 9780170189538

Chapter 4 revision


cba

gfed

37°

26°

m°

10°

125°x°

33°

y°

60°

130°d°

50°

b° a°

56°b°

a°114°

b°

a°


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

Geometry is everywhere. Angles, parallel lines, triangles...

Documents

Transcript of Geometry is everywhere. Angles, parallel lines, triangles...