Assumed Knowledge (Geometry and graphs)€¦ · C Polygons [4.1] D Symmetry [4.2] E Constructing...

Assumed Knowledge

Contents:

A Angles [4.1, 4.3]

B Lines and line segments [4.1]

C Polygons [4.1]

D Symmetry [4.2]

E Constructing triangles

F Congruence [4.1]

G Interpreting graphs and tables [11.1]

Angles are described by their size or degree measure. The different types of angle are summarised in the

following table:

Straight Angle

One complete turn. 1

2turn. 1

4turn.

One revolution = 360o. 1 straight angle = 180o: 1 right angle = 90o.

Obtuse Angle

Less than a 1

4turn. Between 1

4turn and 1

2turn. Between 1

2turn and 1 turn.

An acute angle has size

between 0o and 90o:

An obtuse angle has size

between 90o and 180o:

A reflex angle has size

between 180o and 360o.

ANGLES [4.1, 4.3]A

(Geometry and graphs)

Revolution

Acute Angle Reflex Angle

Right Angle

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Y:\HAESE\IGCSE01\IG01_AT\001IGCSE01_AT.CDR Thursday, 11 September 2008 2:49:05 PM PETER

2 Assumed Knowledge (Geometry and graphs)

There are several ways to label angles.

We can use a small case letter or a letter of the Greek alphabet.

We can also use three point notation to refer to an angle.

For example, the illustrated angle µ is angle PQR or PbQR.

1 Draw a freehand sketch of:

a an acute angle b an obtuse angle c a right angle

d a reflex angle e a straight angle

2 State whether the following angles are straight, acute, obtuse or reflex:

a b c d

e f g h

3 For the angle sizes given below, state whether the angle is:

i a revolution ii a straight angle iii a right angle

iv an acute angle v an obtuse angle vi a reflex angle.

a 31o b 117o c 360o d 213o e 89o f 90o

g 127o h 180o i 358o j 45o k 270o l 150o

4 Find the measure of angle:

a BAC b DAE

c BAD d CAE

e CAF f BAE

5 Which is the larger angle,

KbLM or PbQR?

P

Q

R

�

L

M

K

Q

R

P

F A B

E

D

C

47°

48°

26°

59°

EXERCISE A

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Y:\HAESE\IGCSE01\IG01_AT\002IGCSE01_AT.CDR Wednesday, 10 September 2008 10:51:49 AM PETER

Assumed Knowledge (Geometry and graphs) 3

When we talk about lines and line segments, it is important to state exactly what we mean.

Line AB is the endless straight line passing through the

points A and B.

Line segment AB is the part of the straight line AB that

connects A with B.

The distance AB is the length of the line segment AB.

There are several other important words we use when talking about lines:

² Concurrent lines are three or more lines

that all pass through a common point.

² Collinear points are points which lie in a

straight line.

² Perpendicular lines intersect at right

angles.

² Parallel lines are lines which never intersect.

Arrow heads indicate parallelism.

² A transversal is a line which crosses

over two other lines.

1 Copy and complete:

a Points A, B and C are ......

b The part of line BD we can see is ......

c The line parallel to line BC is ......

d The lines through B and D, C and E, and D and F, are

...... at D.

2 Draw a diagram which shows:

a line AB perpendicular to line CD b X, Y and Z being collinear

c four concurrent lines d two lines being cut by a transversal.

LINES AND LINE SEGMENTS [4.1]B

EXERCISE B

AB

AB

AB

distance AB

AB

CD

transversal

AB

C

D

E

F

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Y:\HAESE\IGCSE01\IG01_at\003IGCSE01_AT.CDR Thursday, 11 September 2008 3:47:43 PM PETER


A polygon is any closed figure with straight line sides which can be drawn on a flat surface.

is a polygon. are not polygons.

A triangle is a polygon with 3 sides.

A quadrilateral is a polygon with 4 sides.

A pentagon is a polygon with 5 sides.

A hexagon is a polygon with 6 sides.

An octagon is a polygon with 8 sides.

TRIANGLES

There are several types of triangle you should be familiar with:

scalene right angled isosceles equilateral

REGULAR POLYGONS

A regular polygon is one in which all sides are equal and all angles are equal.

For example, these are all regular polygons:

equilateral triangle square regular hexagon

POLYGONS [4.1]C

Three sides of

different lengths.One angle a

right angle.

Two sides

are equal.Three sides

are equal.

60° 60°

60°

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SPECIAL QUADRILATERALS

There are six special quadrilaterals:

The following properties of quadrilaterals are useful:

PARALLELOGRAM

In any parallelogram: ² opposite sides are equal in length

² opposite angles are equal in size

² diagonals bisect each other.

RHOMBUS

In any rhombus: ² opposite sides are parallel

² opposite angles are equal in size

² diagonals bisect each other at right angles

² diagonals bisect the angles at each vertex.

RECTANGLE

In any rectangle: ² opposite sides are equal in length

² diagonals are equal in length

² diagonals bisect each other.

KITE

In any kite: ² two pairs of adjacent sides are equal

² the diagonals are perpendicular

² one diagonal splits the kite into two

isosceles triangles.

SQUARE

In any square: ² opposite sides are parallel

² all sides are equal in length

² all angles are right angles

² diagonals bisect each other at right angles

² diagonals bisect the angles at each vertex.

parallelogram rhombusrectangle square trapezium kite

² A parallelogram is a quadrilateral which has opposite sides parallel.

² A rectangle is a parallelogram with four equal angles of 90o.

² A rhombus is a quadrilateral in which all sides are equal.

² A square is a rhombus with four equal angles of 90o.

² A trapezium is a quadrilateral which has exactly one pair of parallel sides.

² A kite is a quadrilateral which has two pairs of equal adjacent sides.

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Y:\HAESE\IGCSE01\IG01_at\005IGCSE01_AT.CDR Tuesday, 18 November 2008 12:06:59 PM PETER


1 Classify these triangles:

a b c

d e f

2 Draw diagrams to illustrate:

a a quadrilateral b a pentagon c a hexagon

d an octagon e a regular quadrilateral f a regular triangle

g a parallelogram h a rhombus i a trapezium

3 List with illustration:

a the three properties of a parallelogram b the four properties of a rhombus.

LINE SYMMETRY

A figure has line symmetry if it can be reflected in a line so that each

half of the figure is reflected onto the other half of the figure.

A square has 4 lines of symmetry.

Example 1 Self Tutor

For the following figures, draw all lines of symmetry:

a b c

SYMMETRY [4.2]D

A figure has a line

of symmetry if it can

be folded onto itself

along that line.

For example, an isosceles triangle has

one line of symmetry, which is the line

from its apex to the midpoint of its base.

EXERCISE C

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Y:\HAESE\IGCSE01\IG01_at\006IGCSE01_AT.CDR Thursday, 13 November 2008 9:10:18 AM PETER


a

1 line of symmetry

b

no lines of symmetry

c

2 lines of symmetry

ROTATIONAL SYMMETRY

A figure has rotational symmetry if it can be mapped onto itself more than once

as it rotates through 360o about a point called the centre of symmetry.

The flag of the Isle of Man features a symbol called a triskelion

which has rotational symmetry.

Every time you rotate the triskelion through 120o it fits onto itself.

This is done 3 times to get back to the starting position. We say

that its order of rotational symmetry is 3.

The number of times an object will fit onto itself when rotated

through 360o (one complete turn) is called its order of rotational

symmetry.

1 How many lines of symmetry do the following have?

a b c d

e f g h

i j k l

2

3 Draw a triangle which has:

a no lines of symmetry b one line of symmetry c three lines of symmetry.

DEMO

Which of the following

alphabet letters show line

symmetry?

EXERCISE D

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4 Draw a quadrilateral which has:

a no lines of symmetry b c two lines of symmetry

d four lines of symmetry.

5 What is the order of rotational symmetry for each of these figures?

a b c

d e f

6 Find the order of rotational symmetry of the Bauhinia

blakeana flower on the flag of Hong Kong.

7 Which of the following letters show rotational symmetry of order greater than 1?

Here are three such triangles:

The information given is insufficient to draw a triangle of one particular shape.

However, if we are asked to accurately draw triangle ABC in which

AB = 3 cm, BC = 2 cm and AC = 4 cm, one and only one

triangular shape can be drawn.

The easiest way to draw this triangle is to use a ruler and compass

construction.

Everyone using this construction would draw the same figure.

CONSTRUCTING TRIANGLESE

4 cm

2 cm

3 cmA B

C

2 cm

3 cmA B

C

2 cm

3 cmA B

C

2 cm

3 cmA B

C

one line of symmetry

If several people were asked to accurately draw triangle ABC in which AB cm and BC cm, many

different shaped triangles would probably be drawn.

= 3 = 2

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Y:\HAESE\IGCSE01\IG01_at\008IGCSE01_AT.CDR Thursday, 11 September 2008 2:48:28 PM PETER


1 Construct a triangle which has sides of length:

a 2 cm, 3 cm and 3 cm b 2 cm, 3 cm and 4 cm

c 2 cm, 3 cm and 5 cm d 2 cm, 3 cm and 7 cm.

2 Copy and complete:

“The sum of the lengths of any two sides of a triangle must be ............ the length of the third side”.

3 Draw a triangle ABC with all sides greater

than 6 cm in length and with the angles

at A, B and C being 60o, 50o and 70o

respectively.

Place a ruler and set

square as shown in the

figure. Slide the set square

along the ruler to the left,

keeping the ruler firmly in

place.

a Why does the hypotenuse of the set

square produce lines parallel to BC?

b Locate X on AB and Y on AC such that

XY = 4 cm.

You should now have a triangle which has angles of 60o, 50o

and 70o, and where the side opposite the 60o angle is 4 cm long.

In mathematics we use the term congruent to describe things which have the same shape and size. The

closest we get to congruence in humans is identical twins.

1 Which of the following figures are congruent?

A B C D E

2 Which of the following geometric figures are congruent?

A B C D E F G H

I J K L M N O

CONGRUENCE [4.1]F

X Y

B C

A

60°

50° 70°

slide

EXERCISE E

EXERCISE F.1

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Y:\HAESE\IGCSE01\IG01_AT\009IGCSE01_AT.CDR Friday, 12 September 2008 9:40:39 AM PETER


3 Here are some pairs of congruent geometric figures.

a b

c d

e f

For each pair:

i Identify the side in the second figure corresponding to the side AB in the first figure.

ii Identify the angle in the second figure corresponding to AbBC in the first figure.

CONGRUENT TRIANGLES

Two triangles are congruent if they are identical in every respect except for position.

The above triangles are congruent.

We write ¢ABC »= ¢XYZ, where »= reads “is congruent to”.

When writing this congruence statement, we label the vertices that are in

corresponding positions in the same order.

So, we write ¢ABC »= ¢XYZ but not ¢YXZ or ¢ZYX.

If two triangles are equiangular (have all three angles equal), they are not necessarily congruent.

If one triangle was

cut out with scissors

and placed on the

top of the other, they

would match each

other perfectly.

A

B C P

QR

A

B C

DE

F

G

H

A

BC

M

N

O

A

B

C

D

Q

R

S

T

A

B

C

D

P

M

N

O

A

B

C�

X

Y

Z�

A DC F

B E

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For example, these triangles are equiangular but clearly

triangle B is much larger than triangle A.

For example, triangles C and D have two equal sides and the

same non-included angle, but they are not the same triangle.

One and only one triangle can be drawn if we are given:

² two sides and the included angle between them

² one angle is a right angle, the hypotenuse, and one other side

² two angles and a side.

There are, however, four acceptable tests for the congruence of two triangles.

TESTS FOR TRIANGLE CONGRUENCE

Two triangles are congruent if one of the following is true:

² All corresponding sides are equal in length. (SSS)

² Two sides and the included angle are equal. (SAS)

² Two angles and a pair of corresponding sides are

equal. (AAcorS)

² For right angled triangles, the hypotenuses and one

pair of sides are equal. (RHS)

The information we are given will help us decide which test to use to prove two triangles are congruent. The

diagrams in the following exercise are sketches only and are not drawn to scale. However, the information

on them is correct.

80°

60° 40°

B

80°

60° 40°

A

C D

If we are given two sides and a non-included angle, more than one triangle can be drawn.

DEMO

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Are these triangles congruent? If so, state the congruence relationship and

give a brief reason.

a b

c d

a ¢ABC »= ¢QRP fSSSg b ¢ABC »= ¢LKM fRHSg

c ¢ABC »= ¢DFE fAAcorSg

d The two angles ® and ¯ are common, but although AC equals XZ,

these sides are not corresponding. fAC is opposite ® whereas XZ is

opposite ¯.g

So, the triangles are not congruent.

1 In each set of three triangles, two are congruent. The diagrams are not drawn to scale. State which pair

is congruent, together with a reason (SSS, SAS, AAcorS or RHS).

a b

c d

® ¯(alpha) and (beta)

are Greek letters and

are often used to

show equal angles.

EXERCISE F.2

A B

C P

Q

R

Z

� �X YA

B C� �

B

A C� �

F

D E� �

A

B

C

K L

M

AB

C

A

B

C

A B

C

AB

C

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e f

g h

2 Are the following pairs of triangles congruent? If so, state the congruence relationship and give a brief

reason.

a b

c d

e f

A

A

A

A

B

B

B

B

C

C

C

C

A

B

C

D

E

F

P

Q

R

X

Y

Z

A

B

CD

E

F

B

A

C

K

L

M

QP

RD

E

F

A

BC

60°

DE

F

30°

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g h

i j

Magazines and newspapers frequently contain graphs and tables which display information. It is important

that we interpret this information correctly. We often use percentages in our analysis.


A survey was carried out amongst retired people

to see if they were worried about global warming.

The results were collated and a bar chart drawn.

a How many retirees were not concerned about

global warming?

b How many retirees were surveyed?

c What percentage of retirees were:

i a little concerned

ii very worried about global warming?

a 40 retirees were not concerned about global warming.

b 40 + 20 + 35 + 50 + 55 = 200 retirees were surveyed.

c i The percentage ‘a little concerned’

= 20

200£ 100%

= 10%

ii The percentage ‘very worried’

= 55

200£ 100%

= 27:5%

INTERPRETING GRAPHS AND TABLES [11.1]G

X

YZ

DE

F

A

BC��

DE

F

��

C

B

A

X

Y

ZA

B

C

R

QP

0 10 20 30 40 50 60

Not concerned

A little concerned

Concerned

Worried

Very worried

Number of people

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Y:\HAESE\IGCSE01\IG01_AT\014IGCSE01_AT.CDR Thursday, 11 September 2008 2:50:13 PM PETER


1

2

3 The graph shows the change in temperature at the ‘Weather

Centre’ from 6 am to 6 pm on a particular day.

a What was the temperature at:

i 6 am ii 6 pm iii noon?

b Over what period was the temperature:

i decreasing ii increasing?

c What was the maximum temperature and at what time

did it occur?

4 The number of children diagnosed with diabetes in a

large city over time is illustrated in the graph alongside.

a How many children were diagnosed in:

i 1996 ii 2006?

b Find the percentage increase in diagnosis of the

disease from:

i 1996 to 2006 ii 2000 to 2008.

EXERCISE G

0 1 2 3 4 5 6 7 8 9 10

AB–

AB+

B–

B+

A–

A+

O–

O+

Days supply

BLOOD STOCKS

Blo

od

gro

up

The graph shows the world’s largest

rice growers and the quantity of rice

they harvested in .

What tonnage of rice was

harvested in:

Vietnam Burma in ?

What was the total tonnage

harvested from the countries

included?

What percentage of the total rice

harvest was grown in China?

What percentage of rice does India

grow compared with China?

2007

2007

a

i ii

b

c

d

6 am 9 am noon 3 pm 6 pm

15

20

25

30

35

40Temperature (°C)

Time

94 96 98 00 02 04 06 08

30

40

50

60

70

80

Children with diabetes

Number diagnosed

year

The graph alongside shows the number of days’ supply of

blood for various blood groups available in a major hospital.

We can see that for blood type B+ there is enough blood for

transfusions within the next 5 days of normal usage.

a How many blood groups are there?

b What blood type is in greatest supply?

c If supply for 2 or less days is ‘critical’, what blood types

are in critical supply?

d What percentage of the available blood is AB+ or AB¡?

0

��

��

��

��

Country

World rice harvest

Metric tonnes of rice

production in 2007

Chin

aC

hin

a

India

India

Indones

ia

Ban

gla

des

h

Vie

tnam

Thai

land

Burm

a

Phil

lipin

es

Bra

zil

Japan

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Y:\HAESE\IGCSE01\IG01_AT\015IGCSE01_AT.CDR Thursday, 13 November 2008 9:11:11 AM PETER


Type of fruit Frequency

Apple 20

Banana 24

Grapes 3

Orange 11

Mandarin 10

Nectarine 7

Pear 2

Peach 3

5 At a school, the year 11 students were asked to nominate their favourite

fruit. The following data was collected:

a How many year 11 students were there?

b What was the most popular fruit?

c What percentage of year 11 students choose mandarins as their

favourite fruit?

d The school canteen sells apples, bananas and pears. What

percentage of year 11 students will be able to order their favourite

fruit?

6 The table below displays the percentage of people with each blood type, categorised by country.

Blood Type

Austria 30% 7% 33% 8% 12% 3% 6% 1%

Canada 39% 7% 36% 7% 8% 1% 2% 1%

France 36% 6% 37% 7% 9% 1% 3% 1%

Hong Kong 40% 0:3% 26% 0:3% 27% 0:2% 6% 0:2%

South Korea 27% 0:3% 34% 0:3% 27% 0:3% 11% 0:1%

United Kingdom 37% 7% 35% 7% 8% 2% 3% 1%

USA 37% 7% 36% 6% 9% 1% 3% 1%

7

Age Total Crashes Fatalities

16¡ 24 73 645 243

25¡ 34 58 062 173

35¡ 44 49 600 123

45¡ 54 36 658 70

55¡ 64 19 833 58

65¡ 74 15 142 58

75¡ 84 7 275 45

85+ 937 9

The table shows car crashes by age group over a one year

period in an English county.

a Which age group has:

i the most crashes ii the least crashes?

b What percentage of crashes does the 55-64 age group

have compared with:

i the 16-24 age group ii all age groups?

Telling Statistics

Crashes by age

0 10 20 30 40 50

5

10

distance (km)

time (minutes)

O+ O¡ A+ A¡ B+ B¡ AB+ AB¡

a What percentage of Canadians have type B+ blood?

b In which countries is A+ the most common blood type?

c Which of the countries listed has the highest percentage of population with type AB+ blood?

d Which blood type is the fourth most common in the USA?

e In which country is B+ more common than A+?

8 The graph alongside shows the distance June travels

on her morning jog. Use the graph to determine:

a the distance she jogs

b the time taken for her jog

c the distance travelled after 15 minutes

d the time taken to jog the first 6 km.

e the average speed for the whole distance.

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Y:\HAESE\IGCSE01\IG01_AT\016IGCSE01_AT.CDR Monday, 3 November 2008 10:11:26 AM PETER


9

a How far is it from:

i London to Aberdeen ii Cambridge to Oxford?

b How far would you need to travel to complete the circuit from Glasgow to Cardiff to Oxford and

back to Glasgow?

c

London

Aberdeen 865

Birmingham 189 692

Bristol 192 823 137

Cambridge 97 753 163 287

Cardiff 250 857 173 73 343

Edinburgh 646 204 472 601 542 632

Glasgow 644 240 469 599 562 633 72

Liverpool 338 581 158 287 330 322 362 356

Manchester 317 570 142 269 246 303 351 345 55

Newcastle-on-Tyne 444 380 319 469 367 501 176 241 274 227

Oxford 90 802 101 119 129 175 583 571 266 248 407

Lo

nd

on

Ab

erd

ee

n

Birm

ing

ha

m

Bristo

l

Ca

mb

rid

ge

Ca

rdiff

Ed

inb

urg

h

Gla

sg

ow

Liv

erp

oo

l

Ma

nch

este

r

Ne

wca

stle

-on

-Tyn

e

Oxfo

rd

LondonLondon

AberdeenAberdeen

BirminghamBirmingham

BristolBristol

CambridgeCambridge

CardiffCardiff

EdinburghEdinburghGlasgowGlasgow

LiverpoolLiverpoolManchesterManchester

Newcastle-on-TyneNewcastle-on-Tyne

OxfordOxford

Distances are in kilometres.

If you could average km/h on a trip from Cardiff to London, how long would it take, to the

nearest minutes?

85

5

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Y:\HAESE\IGCSE01\IG01_AT\017IGCSE01_AT.CDR Monday, 3 November 2008 10:13:07 AM PETER

18 ANSWERS

EXERCISE A

1 a b c

d e

2 a acute b reflex c straight d obtuse

e reflex f straight g acute h obtuse

3 a acute b obtuse c revolution d reflex

e acute f right angle g obtuse h straight

i reflex j acute k reflex l obtuse

4 a 59o b 48o c 85o d 74o e 121o f 133o

5 KbLM

EXERCISE B

1 a ...... are collinear. b ...... is a line segment.

c ...... is line DF. d ...... are concurrent at D.

2 a b

c d

EXERCISE C

1 a right angled, scalene b obtuse angled, isosceles

c obtuse angled, scalene d equilateral

e acute angled, isosceles f right angled, isosceles

2 a b c

d e f

g h i

3 a ² Opposite sides are equal in length.

² Opposite angles are equal in size.

² Diagonals bisect each other.

b ² Opposite sides are parallel.

² Opposite angles are equal in size.

² Diagonals bisect each other at right angles.

² Diagonals bisect the angles at each vertex.

EXERCISE D

1 a 2 b 4 c 2 d 4 e 2 f 0 as is not a rhombus

g 10 h infinite i 3 j 5 k 1 l 3

2 A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, Y

3 a b c

4 a b

c d

5 a 4 b 2 c 3 d 2 e 8 f 4

6 5 7 H, I, N, O, S, X, Z

EXERCISE E

1 a b

c cannot form a triangle

A B

C

DX

Y

Z

transversal

3 cm

3 cm2 cm

4 cm

3 cm2 cm

2 cm 3 cm

5 cm

IB MYP_3 ANS


0 05 5

25

25

75

75

50

50

95

95

100

100 0 05 5

25

25

75

75

50

50

95

95

100

100

Y:\HAESE\IGCSE01\IG01_AT\018IGCSE01_AT.CDR Wednesday, 26 November 2008 1:53:21 PM TROY

ANSWERS 19

d cannot form a triangle

2 The sum of the lengths of any two sides of a triangle must be

greater than the length of the third side.

EXERCISE F.1

1 A and D; B and E 2 A and O; E, I and M; F and H

3 a i FG ii FbGH b i ON ii ObNM

c i QR ii QbRP d i TS ii TbSR

e i ON ii ObNM f i FE ii FbED

EXERCISE F.2

1 a A and C fSSSg b A and B fRHSg

c B and C fAAcorSg d A and C fSASg

e A and C fSASg f B and C fRHSg

g A and C fSSSg h B and C fAAcorSg

2 a ¢PRQ »= ¢ZXY fSASg b ¢ABC »= ¢LKM fSSSg

c ¢ABC »= ¢FED fAAcorSg

d ¢ABC »= ¢EDF fAAcorSg

e ¢ABC »= ¢FED fAAcorSg

f Only one pair of sides and one angle are the same ) ¢s

may or may not be congruent (not enough information).

g ¢ABC »= ¢PQR fSSSg

h ¢s are similar fall angles equalg but may or may not be

congruent (not enough information).

i ® and ¯ are common to both, however sides EF and CB are

equal but not corresponding ) ¢s are not congruent.

j ¢DEF »= ¢ZYX fRHSg

EXERCISE G

1 a 8 b AB¡ c O+, O¡, A+, A¡ d 44:4%

2 a i 35 000 000 t ii 20 000 000 t

b ¼ 550 000 000 t c ¼ 34:5% d ¼ 76:3%

3 a i 17:5oC ii 25oC iii 32oC

b i 1 pm to 6 pm ii 6 am to 1 pm c 34oC at 1 pm

4 a i 38 ii 69 b i 81:6% ii 40%

5 a 80 students b banana c 12:5% d 57:5%

6 a 8% b France c South Korea

d 0¡ e Hong Kong

7 a i 16 - 24 ii 85+ b i 23:9% ii 7:45%

8 a 9 km b 50 min c 4 km d 30 min e 10:8 km/h

9 a i 865 km ii 129 km b 1379 km c 2 h 55 min

7 cm

2 cm 3 cm

IB MYP_3 ANS


0 05 5

25

25

75

75

50

50

95

95

100

100 0 05 5

25

25

75

75

50

50

95

95

100

100

Y:\HAESE\IGCSE01\IG01_AT\019IGCSE01_AT.CDR Wednesday, 26 November 2008 1:53:24 PM TROY

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