Math Smart - 7

© Ben is confused about rounding decimals. Help Ben by completing the table.



a) 10.34 rounded to
one decimal place
b) 129.1 rounded
is 10.3.
to the nearest 10
is 13.

c) When the digit to
the right of the place value
rounded
am rounding it to is less than 5,
one
I add 1 to the digit in the place
value I am rounding it to.




Ben's statement Correct / Incorrect Reasoning
a) Correct The digit in the hundredths column is less than 5.
Add nothing to the digit in the tenths column and
drop all the digits to right.
b)
c)

d)

© Round the following measurements,

a) 4.32 £ to the nearest litre, b) 16.55 s to one decimal place.
c) 23.96 s to one decimal place, d) 2432 ml to the nearest litre.
e) 542 cm to the nearest metre, f) 6709 m to the nearest kilometre.
g) 4450 g to the nearest kilogram. h)* Challenge! 4.682 kg to the nearest 100 grams.

O Read the measurements. Round each measurement and fill in the blanks.
a) A table has a length of 2.28 m. The length of the table rounded to the nearest
whole metre is .
b) An envelope has a width of 176 mm. The width of the envelope to the nearest
10 mm is .
c) A tea pot is filled with 378 ml of tea. The volume of tea in the pot is filled to the
nearest 100 ml is .
d) A kitchen scale reads a mass of 589 g. The mass shown on the scale to the nearest
10 g is .
e)* Challenge! The time on a clock is 6:39 a.m. The time to the nearest hour is .
f)* Challenge! The temperature reading on a thermometer is 94. 9°C. The temperature on the
thermometer rounded to the nearest 10®C is .

CHAPTER 4.5


In this chapter
Estimation and
Pupils should be able to:
• know when to round up
Approximation of
or down after division
when the context
requires a whole number
Decimals in Word
answer


Problems





When working with practical, real-life situations, you sometimes need to check if
the answer is reasonable and makes sense in the situation.


Think and Share


The manager of a shopping mall wants the parking bays to be 3 metres wide.

How many parking bays could be painted in a car park that is 77 metres wide?
77 -r 3 = 25.67 parking bays.

Using the rules of rounding, you would round the answer to 26 parking bays.
However, there is insufficient space for 26 parking bays. How many parking
bays should be painted?








o Spotlight 4.5.1 Estimation and approximation
in everyday lives

The «symbol is used to
show an approximation. Do you think estimation and approximation could be useful in everyday life

situations? Rounding helps you to quickly estimate a quantity. By rounding
2.99 =
3
actual values to more managable numbers, you can estimate the answers
to many problems. Estimation allows us to arrive at an answer that is "close
enough" to the actual answer. It also helps us to check if our answers are
reasonable.












Decimals

I have 8.3 kg of flour. I used 4.5 kg of the flour. How much flour
I
do have left?
O Which expression would you use if you wanted to calculate exactly how
much flour you have left?
a) 8.3-4.5 b) 8.3 + 4.5 c) 8.3 x 4.5 d) 8.3+ 4.5
O Calculate the exact amount of flour you have left.

O Which expression would you use to estimate how much flour you have
left?
a) 8-4 b)9-4 c)9-5 d)8-5
O Calculate the estimated amount of flour you have left.
O Compare the exact amount of flour to the estimated amount of flour.
Are the two answers the same?



Estimation helps us to work out the answer to a problem mentally or quickly. A
good estimation is close to the actual value and helps us to check the validity of
our answer.
There are times when you work out problems and the answer is a decimal. It does
not make sense to have a decimal for some quantities.




have worked
out that need one and
I
a halP (1.5) tins oQ paint
to paint my bedroom.



You cannot
buy hal(^ a tin
oP paint!






Sometimes the decimal may need to be rounded up or down to give an answer
that is appropriate. For example, the answer could be 1.5 tins of paint, but you
cannot buy half of a tin of paint. You would have to round the answer up to
2 tins of paint.












o

O Mitch needs 60.24 kg of potting soil to fill all his plant pots. He can only
buy potting soil in 5 kg bags.
Mitch uses rounding to estimate the amount of potting soil he needs to
the nearest kilogram.
60^5= 12
Mitch calculates that he needs 12 bags of potting soil.
a) Does Mitch have enough potting soil to fill his plant pots?
Explain your answer.
b) What should Mitch do?
Mitch should round the amount of soil to kg, instead of
rounding kg.
c) Calculate the number of bags of potting soil Mitch should buy.
d) How many kilograms potting soil will Mitch have left over?



In this investigation, we use estimation to make the numbers more manageable
and to get a sensible answer.



Check My
Understanding


Use your calculator to find the answers:
2.85 X 8.23
a) 5.16 X 3.81 b) 19.76^2.8
5.32 + 3.74
Another way of checking
© Show how you use estimation to check your calculations for Question 1,
your answer Is to do the
© Use estimation to decide quickly which is the correct answer for each
calculation a second time
calculation.
on the calculator. The
problem with this is that a) 14.1x2.3 b) 19.7 + 4.8 c) 11.3 + 13.9
you could repeat the
A. 3.243 A. 4.104 A. 2.52
mistake a second time.
B. 32.43 B. 41.04 B. 25.2
This is why it is useful to
use estimation to check C 324.3 C. 410.4 C. 252
your answer. D. 3243 D. 4104 D. 2520
O Harry writes down the following equation:
14.62 X 4 = 584.8
a) Use estimation to explain why Harry is not correct.

b) Calculate the correct answer and then describe the mistake that
Harry made.








UNIT 4 I Decimals

Example 1


There are 220 pupils in Stage 7. They are going on a school trip.

The buses that they are using have a maximum capacity of 50 people each. How
many buses are needed?

220 4- 50 = 4.4
We cannot say that we need 4.4 buses since we count buses in whole numbers.
Since the digit in the tenths place is 4, 4.4 Is usually rounded down to 4. But what
happens when we only have 4 buses?

4 X 50 = 200. 4 buses can only sit 200 pupils. 20 pupils would be left behind.

In this case, we need to round 4.4 up to 5 In order to fit 220 pupils.
So, 5 buses are needed.




llMI SIUI""



!■





















Check My
Understanding


O Mrs Sng needs 6.2 balls of wool to knit a jersey. Each ball of wool costs
$3.15. What Is the total cost of the wool for the jersey?
© You have 12.4 m of red ribbon and 4.8 m of green ribbon. Show how
you would estimate the total length of ribbon you have altogether by
rounding to the nearest metre.
O Kla has 78 sweets. She wants to pack the sweets Into boxes to sell. She
packs 5 sweets into each box. How many boxes will Kla have?

Revision


Write these numbers in a place-value table: b) Check that your answers are reasonable by
estimating and comparing your estimate
a) 49.6 b) 29.69 c) 2.04
to your answers in a).
d) 0.140 e) 340.05 f) 1794.32
c) Mei paid S$30.50 for material to make
g) 600.294
cushions. How much did the material for
each cushion cost?
In the number
a) 4.1432, which digit Is in the hundredths Peggy worked a total of 47.28 hours over 6
place? days.
b) 3.876, what is the value of the digit 8? a) What Is the average number of hours that
c) 5.084, what is the place value of the Peggy worked per day?
digit 5? b) How many hours will Peggy work if she
6 Write two ones and thirty-four hundredths In works 8 days at the same average
numerals. rate?
O Nick's three best practice times for the 100-m Tony has 5.6 kg of coffee. He packs the coffee
sprint are as follows: into 8 jars of equal mass.

13.95 seconds 13.08 seconds 13.69 seconds a) What Is the mass of coffee in each jar?
a) Write Nick's practice times from fastest to b) If each jar has a mass of 120 g when it Is
slowest. empty, what is the total mass of one jar of
b) What is the difference between Nick's coffee?
fastest practice time and his slowest c) What is the total mass of 8 jars of coffee?
practice time?
<E) Use estimation to explain which of the
Subtract 38.45 from 64.12 and round to the
calculations Is incorrect.
nearest tenths. The answer is .
a) 15.2X6.2 = 94.24
o Calculate.
b) 34.1X2.9 = 9.88
a) i) 152-f 100 =
c) 21.7-f 4.1 =52.93
ii) 24.5-r 100 =
ill) 30.21 -r 100 = <D *Challenge! A classroom wall is 7.8 m long. A
classroom desk is 170 cm long. Estimate how
b) I) 2 X 2.4 =
many classroom desks will fit along the wall.
ii) 4 X 2.4 =
ill) 5 X 2.4 =
170 cm
i) 1.2 = 3.6
ii) 3.71 =11.13
\
lii) 2.04 = 6.12
Mei makes curtains and cushions. She uses
0.5 m of material to make a cushion, 1.55 m of
material to make a short curtain and 2.25 m
of material to make a long curtain,
a) Calculate the length of material Mei will
need to make
I) 2 cushions,
ii) 3 long curtains,
iii) 4 short curtains.



UNIT 4 I Decimals

Decimals
Place-value table









1 2 9 4 • 7 6 5


Multiplying by 10, 100 or 1000
'1'" ; • !-• L'Lk
X 10 0 • 2 3 X 100 0 • 2 3 --6 X 1000 0 • 2 ^6
2 • 3 6* 2 3 • 2 3 6m
When you multiply by 10, When you multiply by 100, each When you multiply by 1000, each
each digit moves one place digit moves two places to the left. digit moves three places to the left.
to the left. This makes the This makes the number greater. This makes the number greater.
number greater.
Dividing by 10, 100 or 1000


^10 1 8 2 • 1 8-^3 2 • ■rIOOO 1 8_ ^3 2 •
1 8 ^3 • 2 o o 1 *^8 • 3 2 1 • '8 3 2
When you divide by 10, each digit When you divide by 100, each digit When you divide by 1000, each digit
moves one place to the right. This moves two places to the right. This moves three places to the right. This
makes the number smaller. makes the number smaller. makes the number smaller.

Ordering decimals

Ascending order Descending order
The decimals arranged 6 • 4 5 The decimals arranged in
in ascending order from 5 • 5 2 descending order from greatest
smallest to greatest is: 6 • 4 8 to smallest is:
5.52 6.45 6.48 6.48 6.45 5.52



Estimating and
Rounding decimals approximating


round down round up ak&calcuiations


obtain an answer that makes sense
in the word problem and Is close
enough to the actual answer
19 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 20
check whether your answer is
reasonable

Operations with decimals



Addition " Subtraction


Line up the decimai points

18.^200 Add zeros as placeholders


0.030 Add or subtract digits in 7 580
the columns
+0.004 -0.265

18 ,234 Insert a decimal point in 7,315
1 the answer in line with the
other decimal points

Multiplication X Division



Line up the digits, ignore the decimai Count the number of digits to
points, and multipy the right of the decimai points in
Count the number of digits to the right the problem to place the decimal
of the decimal points in the problem to point in the correct position in the
place the decimal point in the correct answer
position in the answer
2 digits to the right
221 .84
of the decimal point
7.04 2 digits to the right
of the decimal point
Place the decimal point
X3 2 digits from the right
10i9 2
21.12 Place the decimal point
2 digits from the right 2 21.84
l_A_y





Mathematics Connect



A computer processes information in nanoseconds. A nanosecond is one
billionth of a second. One nanosecond is written as 0.000000001 seconds
in decimal form.

The smaller the speed, the faster the computer is able to access its i.i i;'i
memory. A computer processing at a speed of 10 nanoseconds is faster
than a computer operating at a speed of 60 nanoseconds.

An average computer now processes at a speed of 2 Ghz, which is 2 billion
processes/second, equating to 2 processes per nanosecond!



jrr;.'
UNIT 4 I Decimals
U:

UNITS
J




Me^urement




Why do we measure? There are many different reasons why we need to
measure. Measuring helps us make sure that the things we make, come out
as planned. When baking you need to measure the amount of Ingredients In
a recipe. A shop uses scales to weigh the food they sell. They need to know
how heavy or light something Is. A seamstress needs to measure the length of
fabric she needs to make clothing. She will use a measuring tape to make sure
everything Is the right size. A police officer measures the speed of a car to make
sure vehicles are not speeding. And you may be wondering and looking at your
watch to find out how long It will be until your lesson finishes.

Whether we need to measure weight, time, distance, size, temperature or
calories, we will need to use the correct measuring tools and the various
methods associated with them.

What kind of things do you think are measured at this building site?


You will learn about;
this
Metric units
nit
Conversions between different metric units
Choosing suitable units of measurement
> V
Reading scales on analogue and digital
measuring instruments

la
111 ,VV



















•\J

CHAPTER 5.1


In this chapter
etrlc Units
Pupils should be able to:
• know abbreviations for
metric units
Scientists and mathematicians can use measurement to describe the quantity of
• know relationships
something. Quantity is a measure of how much there is, or how many there are.
between metric units,
You want to buy a laptop bag for your laptop. You have seen two laptop bags in a
and convert between
shop. Which bag would you buy? Why?
metric units
What about the other items that you need to carry in your bag? Will they fit?
^ Thlrtt^d Sfii!
^ RECALL

Did you think to start by measuring the |
Convert these
dimensions of your laptop?
measurements.
Think about 260 mm
O 10.5 cm =
• what instrument you would use to I
measure with, t
• which dimensions you would measure
kg
and
• which unit of measure you would use.
kg 378 mm

Once you knew the dimensions of the laptop, did you think to measure the
O 12i59ml=: dimensions of the two laptop bags?


B





35 cm 47 cm
Did you notice that the
unit of measure of the
laptop is millimetres,
and the unit of 24 cm
32 cm
measure of the laptop Solution
bags is centimetres? Convert the unit of measure of the laptop to centimetres:
It is easier to compare
Dimension Laptop Laptop bag A Laptop bag B
dimensions when the
Width (cm)
same unit of measure
is used —think about Height (cm)
what you can do to
You need to buy laptop bag because neither the height nor the width
solve this problem.
of the laptop bag is big enough to fit the laptop.

The metric system


The standard base unit for measuring length is the metre, the base unit for
measuring mass Is the gram, and the base unit for measuring capacity is the litre.
Sometimes the quantity we are measuring is too big or too small to measure
using the base unit.

In the metric system, prefixes are used to indicate the size of a unit in comparison
to the standard base unit.

Length Mass Capacity

Kilometre Kilogram Kilolitre
Hectometre Hectogram Hectolitre larger units

Decametre Decagram Decalitre
base unit metre gram litre

decimetre decigram decilitre

centimetre centigram centilitre smaller unit
millimetre milligram millilitre




Common prefixes in the metric system
Some prefixes give you
Kilo —> Kilo in kilometres means 1000 an idea about the size
of the unit compared to
Hecto—>- Hecto in hectometres means 100
the base unit.
Deca —>■ Deca in decametres means 10

deci —> deci in decimetres means io
There are standard
centi — cent! in centimetres means abbreviations for units in
the metric system.
milli —*- milli in millimetres means
Length
Metric unit Abbreviation
Check My kilometre km
Understanding metre m
centimetre cm
O Arrange the following lengths in ascending order. millimetre mm

1 kilometre 1 centimetre 1 millimetre 1 metre 1 decimetre Mass
Metric unit Abbreviation
O Arrange the following masses in descending order.
tonne t
1 gram 1 hectogram 1 centigram 1 kilogram 1 milligram kilogram kg
O Arrange the following capacities in ascending order. gram g
Capacity
1 decalitre 1 decilitre 1 litre 1 milHlitre 1 kilolitre
Metric unit Abbreviation
litre i
millilitre ml

Converting measurements


In the example where you had to choose a laptop bag that was big enough for
your laptop, some dimensions were given in millimetres and some were given in
centimetres. In order to compare the dimensions, you needed to convert all units
to the same unit of measure.
Converting larger units into smaller units

This is the laptop bag that suited the dimensions of your laptop. The dimensions
of the laptop bag are 32 cm and 47 cm. How can you convert these dimensions to
millimetres?
Look at your ruler. There are 10 mm in 1 cm.

centimetres




47 cm

millimetres
Since 1 cm = 10 mm, we
centimetre
multiply by 10 to convert
X 10
32 cm centimetres to millimetres.
millimetre



How many millimetres are there in 35 cm?
35 cm X 10 = 350 mm
How many millimetres are there in 24 cm?
24 cm X 10 = 240 mm
German mathematician
Carl Friedrich Gauss
Converting smaller units into larger units
(1777 - 1855), placed the
'second' as a base unit.
This is the laptop that you needed to find a suitable laptop bag for. The
He used this to describe
dimensions of the laptop are 378 mm and 260 mm. How can you convert these
the Earth's magnetic field
dimensions to centimetres?
in terms of millimetres,
grams, and seconds.
There are 10 mm in 1 cm.
Using these units in his
calculations, allowed him Since 1 cm = 10 mm, we divide by 10 to
260 mm
to give dimensions based
convert millimetres to centimetres.
on mass, length
centimetre
and time to the
magnetic field millimetre3* 10
I of the Earth.

378 mm




How many centimetres are there in 378 mm?
378 mm -7 10 = 37.8 cm
How many centimetres are there in 260 mm?
260 mm 10 = 26 cm

UNIT5 I Measurement

The table summarises how you can convert smaller units to larger units of
measure, and larger units to smaller units of measure.

Length Mass Capacity

Kilometre Kilogram Kilolitre
converting 10 converting
bigger units Hectometre Hectogram Hectolitre smaller units
to smaller to bigger
Decametre Decagram Decalitre
units in units
metre gram litre

decimetre decigram decilitre

centimetre centigram centilitre
millimetre milligram millilitre





A shop is 4.6 km from Anna's house. What is the distance from Anna's house to
the shop, in metres?

Since 1 km = 1000 m, multiply by 1000 to convert kilometres to metres.
4.6 km X 1000 = 4600 m

Investigate!

o Spotlight
To convert length from:
a) m —cm you x by 100
Have some fun and sing
cm - - m you by 100
along with this metric
b) m — mm you by system conversion song!
■*- m you by
mm
c) m — km you by
km - ► m you by i
d) cm - - km you by
km - cm you by

e) mm - km you by
km - mm you by
© To convert capacity from:
a) i—ml you by
b) ml —> i you by
O To convert mass from:

a) g —*- kg you by
b) kg —*- g you by
O There are 1000 kg in a tonne (t). Convert from
a) t —kg you by
b) kg —t you by

Check My
Understanding


Use the rules you have discovered in the previous investigation.
O Convert these lengths to metres.
a) 3 km b) 4.2 km c) 12.5 km d) 0.5 km e) 300 cm 550 cm
g) 58 cm h) 1500 cm i) 4000 mm j) 6850 mm k) 35 mm 8 mm

© Convert these lengths to centimetres,
a) 6 m b) 8.2 m c) 16.8 m d) 0.56 m e) 0.06 m f) 0.01 m

g) 70 mm h) 18 mm i) 64 mm j) 60 mm k) 498 mm 1) 0.7 mm
O Convert these lengths to millimetres,
a) 4 m b) 1.6 m c) 15.2 m d) 0.35 m e) 17.5 cm f) 0.6 cm

g) 54 cm h) 0.05 cm I) 1.6 km j) 16 km k) 0.5 km 1) 0.05 km

O Convert these lengths to kilometres.
a) 1600 m b) 9530 m c) 14600 m d) 350 m
e) 95 m f) 500 cm g) 5000 cm h) 50 000 cm
O Isaac wants to tie two pieces of shoelaces together because one is not long enough. One shoe lace is
30 cm long and the other is 250 mm long.
a) What is the total length of the shoelaces In centimetres?
b) What is the total length of the shoelaces In millimetres?
O Pepita is 140 cm tall. Dion is 1.62 m tall. How much taller is Dion in cm?

Convert these capacities to litres,
a) 3000 ml b) 8000 ml c) 50 000 ml d) 3765 ml
e) 406 ml f) 500 ml g) 250 ml h) 98 ml

o Convert these capacities to millilitres.
a) Si b) 401 c) 360^ d) U
e) 7.56^ f) 10.U g)0.54£ h) 0.4 i
O The capacity of a large bucket is 8.421. A smaller bucket has a capacity of 1000 ml. How many small
buckets can be poured into the larger bucket?
Jennifer makes 2.17 i of drinks for a party. She mixes ice tea and orange juice to make the drink. She
used 1420 ml of ice tea. How many litres of orange juice did she use?
Convert these masses to kilograms.
a) 1000g b) 8000g c) 70 000 g d) 54 000 g

e) 768 g f) 230 g g) 100 g h) 98 g
Convert these masses to grams,
a) 1 kg b) 6 kg c) 40 kg d) 74 kg
e) 8.81 kg f) 0.65 kg g) 0.5 kg h) 0.054 kg

© Convert these masses to tonnes.
a) 1000 kg b) 6000 kg c) 50 000 kg d) 34 000 kg Recall that
e) 128 kg f) 566 kg g) 399 kg h) 76 kg 1 tonne = 1000 kg


Measurement
I

® A grocery bag contains packets of flour, rice and sugar. The grocery bag E
O
has a mass of 17.42 kg. The mass of the flour is 7.3 kg and the mass of the 1 cm 1 ml 1
O
sugar is 4552 g. What is the mass of the rice?
IE) The mass of a lion at a zoo is 135 kg. A cat has a mass of 500 g. How many
times greater is the mass of the lion than the cat?
IE) Prishan bought a 2-kg bag of grapes. He ate 125 g of grapes on his way
1 cm lOOOg
home. How many grams of grapes did he have left?
® Use <, > or = to compare each pair of quantities.
a) 500 ml 254 ml b) 400g 375 g c) 56 m 73 m
d) 450 ml 0.6 £ e) 0.8 kg 80 g f) 20 mm 200 cm
1 mm 1000 kg
g) 8t 8000 kg h) 201 g 0.2 kg i) 986 mm 0.9 m






Try and Apply i u. 1000 kg

PAIR WORK
Let's play measurement dominoes!
— •— -'S;
O Pfint and cut out the measurement dominoes on the right.
© Place your ten dominoes face-down on your desk. 1 kg 1000 i
O Choose five dominoes from your set of ten dominoes as your playing set.
Do not show your playing set to your partner.
O Toss a coin to decide who starts the game.
Q Player A chooses one of their dominoes and places it face-up on the desk. 1 kg 1000 ml
O It possible, Player B should choose a domino from their playing set to
match either side of the displayed domino. If Player B does not have a
matching domino, they must pick up another domino from their face •
down set.
10 mm
O If this domino matches the displayed domino, it can be left facing up. 1000 ^
L_— .
O It the domino does not match, it must be added to the playing set of
dominoes. It is then the turn of Player A.
O Continue playing until one player has all of their dominoes displayed. This
player is the winner. ig
If neither player can display all their dominoes, the player with the least
number of dominoes is the winner.
0.1 cm 1tonnej



i,



•
1000 tonnes 100 000 cm i

CHAPTER 5.2

In this chapter
Suitable Units of
Pupils should be able to:
• choose suitable units
of measurement to
Measurement
estimate, measure,
calculate and solve
problems In everyday
To find a suitable unit of measurement of something, we should first estimate
contexts
how big it is. Imagine the size of the objects you are estimating or imagine
measuring them in relation to other objects.



Think and Shan

Should an elephant and a mouse be measured in grams or tonnes?
• The average mass of a mouse is 20 g. The mass of a mouse in tonnes
would be 0.00002 t (2 hundred thousandth tonnes).
• The average mass of an African elephant is 5 t.The mass of an African
elephant in grams would be: 5 000 000 g (5 million grams).
It does not make sense to measure the mass of a mouse in tonnes or an
elephant in grams. We need to select the unit of measurement carefully to
work with sensible numbers.




Check My
^ RECALL Understanding


O Write in the suitable units of measure.
O A barrel can hold
a) The length of a bank note is 12
more water than a
b) The thickness of a coin Is 3
bucket Would you
c) The height of a classroom wall Is 3000
use a barrel or a
bucket of water to d) The capacity of a mug is 250 .
wash your car? e) The mass of a car is 900 .
f) The mass of a pen is 30 .
O A football field is
much larger than a g) The capacity of a teaspoon is 0.005
garden. Which unit of
© Match each item to its correct unit of measure,
measurement would
you use to measure item Unit of measure
the football field?
Car length • litres
Mass of a cell phone • metres
Amount of water in a bathtub • milliliters
Length of a pencil tip • grams

Amount of water in a raindrop • millimetres

Measurement

This bus is very
i
heavy, think tonnes
is the most suitable
O Read what each learner
unit oP measurement
says about measuring the
Yes this bus A bus IS
I
mass of a bus. is heavy. think we really heavy. It would
should use grams be better to use
O Work in a group.
as our unit oP kilograms to measure
Who do you agree with
measurement. Its mass.
and why?
O Discuss your answer
with the class.








5.21 Estimating measurements


O Spotlight Estimation is a key skill. You are estimating length, capacity and mass when you
ask questions such as:
Basic estimates in real
• Do have enough material to make this dress?
I
life
• Is there enough milk in the fridge for the rest of the week?
• A long ruler is
• How many people can use an elevator that can carry a maximum mass of
approximately 30 cm
1000 kg?
long.
• A long stride is
approximately 1 m Investigate
long.
PAIR WORK
• A teaspoon holds What you need: a ruler, tape measure, bathroom scale, kitchen scale,
approximately 5 ml of measuring jug, measuring spoons, a bag of potatoes, a coffee mug, a bucket,
liquid.
an empty milk carton and a tablespoon.
• A mug holds
approximately 250 ml O Complete the tables.
of liquid.
Difference Did you use a What
• An orange has a mass Actual between suitable base measuring
of about 100 g. Mass Estimate measure estimated unit? Explain Instrument
ment and actual how you did you
measurement estimated. use?
a) A bag of
potatoes
b) A pencil
case
c) An adult
human
body
d) A maths
book
e) A school
bag

Difference Did you use a What
Actual between suitable base measuring
Capacity Estimate measure estimated unit? Explain instrument
ment and actual how you did you
measurement estimated. use?
a) A water
bottle
b) A coffee
mug
c) A bucket
d) A table
spoon
e) An empty
milk
carton


Difference Did you use a What
Actual between suitable base measuring
Length Estimate measure estimated unit? Explain instrument
ment and actual how you did you
measurement estimated. use?
a) A school
desk
b) A classroom
door
c) Length of a
classroom
d) A shoe
e) A finger
nail




I would use litres to
measure the capacity
Read what each learner oP this teaspoon.
says about measuring the
capacity of the teaspoon.
Write what you think is
the correct answer in the
empty speech bubble.
Discuss your answer with
your partner.



No, we use litres to
measure the capacity oP a much larger k
container. I would choose millimetres as the
unit oQ measurement Por the capacity
oQ the teaspoon.



Measurement

CHAPTER 5.3

In this chapter
Reading Scales
Pupils should be able to:
• read the scales on a
range of analogue
and digital measuring Measuring instruments are used to measure physical quantities such as length,
instruments mass and capacity. They can display the measurement in analogue or digital form.

Length Mass Capacity
^ RECALL C

a
O Use your ruler to find Analogue
the length of the
eraser.

I tape measure mass meter measuring jug

The length of the
eraser is cm. Digital

© Find the mass of the
chicken.
To find the length, mass or capacity using an analogue measuring instrument, you
read from a scale. To measure these physical quantities on a digital measuring
instrument, you read the measurement directly from the digital display.
The scales on analogue instruments are number lines. Scales may be straight or
curved.
% '
I'joo U *" 200
To read a measurement on an analogue scale, work out the interval. Use the
interval to read the measurement indicated.
50(H

I'M 1000 Example 1
JIM
Read the measurement shown on the scale.
The mass of the chicken is

O To work out the exact mass
shown, you need to work
grams
This scale must be read out the interval of the scale.
in a clockwise
Count the number of spaces
direction.
between two long lines.
There are 5 spaces between
The mass shown
is between 600 g two long lines.
and 700 g.
100g-r5 = 20g
Each interval of the scale
represents 20 g.
@ The reading on this scale is 640 g.

Check My
Understanding

PAIR WORK
Write down the readings from each scale. Check your partner's answers.
O How long is the — line? cm

cm


0 1 2 3 4 5 6 7 8

lllllllllllllllllll


@ How many millilitres? ml O How many grams?



350 mf
300 ml
grams
290 ml
200 ml
ISO ml
100 ml





O Read the length shown on each of these digital scales,

a) b)







O Read the capacity shown on
this digital scale.
THIS SALE






O Three girls stood on a bathroom scale. The girls stepped off the bathroom scale one at a
time. The readings on the bathroom scale are given below. What is the mass of each girl?



140 160 140 160
« 100
220-r
40 260 f
20. 0 0 28<),,>,/

a) All three girls on b) Two girls on bathroom c) Only one girl on
bathroom scale. scale. bathroom scale.




UNIT 5 Measurement

O Draw an arrow to show the indicated mass on each analogue scale.
Check your partner's answers.

a) 3.5 kg b) 4700 g c)* Challenge! 650 g









kilograms and grams kilograms and grams kjlograms and grams






O Look at the measuring instrument.
,\\\H '////
\\ 100 120 //
140 V

'60 \ 160
odometer
T 1 7 5 6 9 '
180 speedometer


200.




a) Read the speed indicated on the speedometer in Mark's car. What speed is Mark traveling?
b) The speed limit is 80 km/h. Is Mark exceeding the speed limit?
c) Read the distance indicated in kilometers on the odometer in Mark's car.
Find the distance the car covered.






GROUP WORK
You teacher will give you a variety of measuring instruments such
as a tape measure, meter stick, mass meter, measuring jug and measuring cup
to measure some items around the classroom.

Record your findings in the table.

Physical Interval
Measuring Unit of
quantity on scale (if Measurement
instrument measurement
measured applicable)

Revision


o Convert these lengths to metres: <E» The distance that John rode on his bike is
a) 0.023 km b) 15.5 cm c) 1mm shown on the digital distance meter. His sister
Cara rode 1200 metres more than John. Show
© Convert these lengths to centimetres:
how far Cara rode on the blank digital scale.
a) 3.92 m b) 102 mm c) 0.01mm
© Convert these lengths to millimetres:

a) 2.8 cm b) 5 km c) 1.06 km QJj
o Convert these lengths to kilometres:
a) 254 m b) 3421 cm
® Write down the readings on these scales,
© Convert these capacities to litres:

a) 600 ml b) 60 ml a)
Q Convert these capacities to mlllilitres:
a) A.St b) 0.08£
© Convert these masses to kilograms:
a) 1496 9 b) 10 g
mm
© Convert these masses to grams:
a) 6.876 kg b) 0.003 kg b)

© Convert these masses to tonnes:
grams
a) 5619 kg b) 9 kg
Use <, > or = to compare each pair of
quantities.
a) 45 ml 0.6 £
b) 0.43 kg 340 g
c) 2 mm 20 cm

© Match each Item to its correct unit of
C)
measure.
Item Unit of measure
Mass of this book • • kilograms

Mass of a truck ^ 350 ml
grams
full of sand 300 ml
Mass of a packet ^ 250 ml
tonnes
of crisps
200 ml
© I need 1 litre of water for my science 150 ml
experiment. 1 have a 150-mi and a 25-ml
loom'
container. How can I use these containers to
measure 1 litre of water?

© Tom made 2.7 £ of tomato soup. He served
1.5 £ of the soup. He then froze half of the ml
remaining soup. How many millilitres of soup
did Tom freeze?

■r
I
d) e) 70 f)
[ 50 25

30

10
kilograms and grams
tonne



kg ml











A smart watch provides far more data than just time. Smart watches include data on
the distance you walk, the amount of calories you burn and the number of steps that
you take. The pedometer indicates the number of steps that you have taken in a
specified time period. This data is displayed in digital form on the smart watch.





Help
Sheet
1

Abbreviations and unit conversions

xlOOO xlOO xio X 1000 X 1000

km m cm mm kg 9 ml
kilometre metre centimetre millimetre kilomgram gram millilitre


V 1000 ^ 100 ^ 1000 1000

Measurement scales To read a measurement on an analogue scale,
work out the interval. Use the interval to read
You read from a scale to find length, mass or
capacity. the measurement indicated.
The scales on analogue instruments are
number lines. Scales may be straight or curved
To measure length, mass or capacity on a
digital measuring instrument, you read the
measurement directly from the digital display.







Analogue

UNIT 6













and Their






Prooerties






You Will learn about:
Types of angles
Estimating and measuring the size of an
angle
Drawing angles
Drawing parallel and perpendicular lines
Sum of angles at a point, on a straight
line, in a triangle and in a quadrilateral
Vertically opposite angles
Solving geometrical problems using side
and angle properties
The Great Pyramid of Giza is the-
largest and oldest of the three
pyramids in the Giza pyramid n
complex. Each side of the Great
Pyramid of Giza rises at an angle
of 51.5 degrees C).
t

•?s.



n i.







«to.
• n sr.
-'tjai! ♦'ft—^ - ■ 'i c^'
.. , -k- r.- :
t-*-
Jt

UNIT 6 I Angles and Their Properties

In this chapter
CHAPTER 6.1
Pupils should be able to:
• identify, describe and

Types of Angles estimate the size of
angles and classify them
as acute, right or obtuse

6.1.1 Naming angles • identify, describe and
estimate the size of
angles and classify angles
An angle is formed when two line segments share the same vertex (corner). The as reflex
line segments form the arms of the angle.

Angles are measured in degrees. An angle measures the amount of turn. We use a
protractor to measure the amount of turn of an arm from its original position.
We use the letters of the alphabet to label angles. Capital letters are used at the
ends of the arms of the angle, and at the vertex of the angle. vertex <;angle
In Figure 6.1, the letters A, B, C and 0 are used. The letter O is the vertex of the
angles.

In the diagram
• the lines OA, OB and OC make up the arms of the angles. The lines are:
OA, OB and OC

• there are three angles namely AOB, BOC and AOC.
The symbol (") is used to show which letter is the vertex of the angle.

Lower-case letters can also be used to name the angle.
For example, ^a, /.band Ac. Notice that Ac= Aa+ Ah
s Figure 6.1

1.2 Types of angles ^ RECALL



In Stage 5, we learnt to identify acute, right and obtuse angles, and a straight
Write down the
line. We will now learn about reflex angles and a revolution.
measurements of the
angles below.




An acute angle measures A right angle is equal An obtuse angle
between 0® and 90°. to 90°. measures between
90° and 180°.






A straight line is
equal to 180°
A reflex angle measures A revolution
between 180° and 360° is equal to 360°

Check My
Understanding


O Classify the angles as acute, obtuse or reflex angles,
a) 330" b) 21" c) 65° d) 18r
e) 257" f) 102° g) 275° h) 175'

Q Fill in the blank(s) in each statement. Match each statement to the correct diagram.



angle is bigger than 180°,
but smaller than








Acute angles are less than

• •






A angle is equal to 90'








The angle on a straight line is








A revolution is equal to









An obtuse angle is larger than

but smaller than • •












UNIT 6 Angles and Their Properties

In this chapter
CHAPTER 6.2
Pupils should be able to:
• estimate the size of acute,
stimating. Measuring obtuse and reflex angles
to the nearest 10 degrees
• use a ruler, set square and
and Drawing Angles and draw acute, obtuse
protractor to measure
and reflex angles to the
nearest degree
6.2.1 Estimating angles

^ RECALL


Investigate!
Can you estimate the
size of the marked
O Use your protractor to find out the size of the angle between each pair angles without using a
of iines. Then label the angle between each pair of lines to show the
protractor?
amount of turn from 0°.

45" degrees

Amount
/Lm = An =
of turn
Q Identify if the angles
original position are acute, right or
obtuse angles. Then
measure them to check
your answers


Q Can you identify which pair of lines make an acute or obtuse angle?
Which pairs of lines make a reflex angle?


Now that we have seen how some angles look like at regular turns, can you
estimate the size of an angle?

Estimating the size of an angle means making an educated guess about an angle.
\
First, decide what type of angle it is.
• Does it look smaller than 90° or a right angle?
• Does it look like a right angle which is 90°?
• Does it look bigger than 90° or a right angle?
• Does it look like a straight line which is 180°?
• Does it look bigger than 180°?
• Does it look more or less than 3 right turns which is 270°?
• Does it look close to a full turn which is 360°?
/lr=

o Spotlight


Paper Pocket Protractor GROUP WORK
Use this link to find out
What you need: thick art paper, scissors, a pencil, coloured pencils and a
more about making a
protractor.
paper pocket protractor.
Instructions
Q Get into groups of 3 or 4.
Q Each member of the group has to draw two shapes on the paper. Each
shape must have at least a right angle, an angle of 45°, and an angle of
30° or 60°.
Q Fold a piece of paper to make corners that are 45°, 30°, 60° and 90°.
Q Then use it to measure the angles in your shapes.


You can use the special angles learnt in previous stages to help you in your
estimations.










N B



Example 1


Estimate the size of ABC.
We can use a clock face
^ Use a protractor to
to help us estimate the
find out the actual
angle.
size. It is 33°.
This estimation is
acceptable because
it is within 10° of
the actual size of the
angle.
Solution
This is an acute angle. The size of this angle is between 0° and 90°.

It looks closer to 30° than 45°.

So, we can estimate that the size of angle ABC is about 30°.













Angles and Their Properties

Measuring angles




We use a protractor marked in degrees H to measure angles.






Outer scale
Starts on the left Inner scale
at 0' and goes Starts on the right at 0°
clockwise to 180 and goes anti-clockwise
to 180°.
Baseline shows where 0° and
180 ° are on both scales.
centre

There are two scales on a protractor: an inner scale and an outer scale.
The Inner scale measures in an anti-clockwise direction from 0 on the right-hand
side to 180° on the left-hand side.

The outer scale measures in a clockwise direction from 0 on the left-hand side to
180° on the right-hand side.

Make sure that you read from the correct scale when measuring angles.





Measure this acute angle.





Solution
Place the protractor on the angle. Align the centre of the
protractor with the vertex of the angle as shown. The
baseline of the protractor should line up with one of the
arms of the angle.

Read from the scale that starts at the 0° mark that aligns
with the arm of the angle. Here, we read the inner scale.
The size of this acute angle is 33°.



Example 3

Measure this reflex angle.

Solution
This angle is greater than 180°.
But the protractor can only measure up to 180°.

Extend one of the arms by drawing a straight line.
This line divides the angle into two parts.





















The green part of the angle is 180° (straight line).
Use your protractor to measure the orange part
of the angle. It is 75°.

So, the reflex angle is 180° + 75° = 255°.


Check My
Understanding is:



Q Complete the sentences. Then match each clock to the correct sentence.




• This is a 180-degree angle and is a

angle.





This is a full circle (360°) and is a

turn.





This is a right angle (90°) and is a
angle.






This is a 270-degree angle and is an
angle.






Angles and Their Properties

0 Complete the table.


Actual Was your
measurement estimate within
Angle Type of angle Estimate
(use a 10° of the actual
protractor) measurement?




















c)







d)








e)










" %





g)








h)

Q Use your protractor to measure the following angles.

a) X b)













d)













e) f)









Q Use your protractor to measure the following angles.

a) ^ b)












c) d)












e) f)













Angles and Their Properties

6.2.3 Drawing acute angles



Let us draw ZDEF = 56°.

To draw an acute angle:


1 Use your ruler. Draw a line EF.
This is one arm of the angle.
Mark point E as the vertex.



Place the centre of the protractor on line
EF. Make sure the centre mark of the
protractor falls on point E, the vertex.
Make sure the baseline of the protractor is
sitting on the arm of the angle.





5 Find the 56° mark on the inner scale. Mark
a point at the edge of the protractor. Label
the point, D.



-if!
Use a ruler to draw a straight line from the
vertex, E, to the point, D.
This forms the second arm of the angle.
The angle (zlDEF) is 56°.





Drawing obtuse angles



Similarly, we can draw zlPQR = 168°.

To draw an obtuse angle:



' Use your ruler. Draw a straight line PQ and
mark point Q as the vertex.





Next, place the protractor on line PQ. Make
sure that the centre mark of the protractor
falls on point Q, the vertex.

5 Find the 168° mark on the outer scale of the
protractor. Mark a point and label it as R.





Remove the protractor. Draw a straight line to
join point R to point Q. The angle (^PQR) is
168°.



Drawing reflex angles



Draw an angle aPOR = 295°.
To draw a reflex angle:


1
Use your ruler. Draw a straight line OP.
This forms one arm of the angle.



Mark point O as the vertex. Extend the line
0
from the vertex to form a straight line (180®).


295' 180'
Calculate the extra angle that is needed. For
example, to draw an angle of 295®, you need ,115'
180® + 115°.

295°
Place the protractor on line OP. Align the
centre mark of the protractor with point O
the vertex. Make sure the baseline of the
protractor is sitting on line OP. Find the 115^
mark on the outer scale of the protractor.
Mark the point. The angle (^POR) is 295°.



Check My
Understanding


9 Use your ruler and protractor to draw the following angles,

a) 30® b)47® c) 109® d) 98® e) 67® f) 151' g) 147® h) 126®
Q Use your ruler and protractor to draw the following reflex angles.

a) 190° b)209® c) 236® d) 317® e) 342® f) 359® 9) 333® h) 227®





Angles and Their Properties

In this chapter

Pupils should be able to:
• calculate the sum of
angles at a point, on
a straight line and in
a triangle, and prove
that vertically opposite
angles are equal; derive
and use the property
that the angle sum of
^ RECALL a quadrilateral is 360
degrees


XOKis a straight line. Find Ld.
Spotlight



Adjacent angles are
angles that have the
1 same vertex and share a
^d=180° -
common side.

.^shared
common side
I


Q shared vertex
Aim: Find the sum of angles on a straight line in each diagram.
We know Aa is adjacent
Q /AOC is a straight line. ^ AOD is a straight line.
to Zb since:
B • both angles share a
common vertex 0
• both angles share a
common side 08
• both angles lie on
opposite sides of the
common side 08.
a) Use your protractor to a) Use your protractor to
measure the size of angles measure the size of angles
oand b. a, band c. Angles are measured in
degrees. Remember to
Aa =
Zo =
write the degree symbol
Zb =
zc
=
b) What is the sum of angles
a and b? b) What is the sum of angles
a, b and c?
Conclusion
The sum of angles on a straight line is

iSpotlight
Q In the diagram, AOD is a straight line. Calculate angle a.
B


C


a
Z0 + Zb = 90° A nX43° D
Two angles that add 0
up to 90° are called
complementary angles. 90° + o + 43° = 180° (zls on a str. line) Remember to state the reason
'Z.S on a str. line'.
0= 180°-43°-90°
= 47°

Q In the diagram, PR is a straight line. Calculate the value of b in the diagram.
00
^x+Z.y=180° 1|
Two angles that add
up to 180° are called '
supplementary angles. 2b° + 10°p>;? lb" + 10° + 3b° = 180° (^s on a str. line)
5b° = 180° -10°
= 170°
b =™
Check My 5
= 34
Understanding


O In each diagram, there is one straight line. Calculate the size of the unknown angle.


a) b) d) e)
^9^
Wd
124" \
90°l_ e



f) 9) h) i) j)

>
V
X51° N. 124° _ 78°AA70°
42V'V---/

k) 1) m) n) 0)


boy \34\n
k m jyy>^
r^Y30° "1



Conclusion The angles on a straight line add up to



UNIT 6 Angles and Their Properties

Sum of angles In a triangle

We have learnt that a triangle Is a two-dimensional shape that has three straight
sides and three angles. We call the angles inside a triangle interior angles.


Lines AB, BC and AC make up triangle ABC {^ABQ.
Points A, B and C are the vertices of the triangle.

AABC, LACBand ABAC are the interior angles of AABC.








Let us find out what the sum of the angles in a triangle is.
Q Mark each angle in the triangles and cut them out.








50 mm 50 mm
5.5 cm





6 cm 78 mm

Q Use your protractor to measure the size of each interior angle.
Name the angles and write down the measurements below.


AA AB








^ Place the angles next to one another.

Do the angles fit exactly onto a straight line?



Q What is the sum of all the interior angles of each triangle?





♦Challenge! What triangle is this?





Conclusion
The sum of the interior angles of a triangle is

^ RECALL


Identify the types of triangles.
a) A b)














Example 2


Calculate the value of o.



o° + 8r+ 56°= 180° {Z sum of A)
0° = 180°-81°-56°
= 43°
The sum of the interior 0 = 43
angles of a triangle is 180°.


Check My

Understanding

Q Calculate the unknown angle(s) in each diagram.

a) b) c)














d) e) f)



















Angles and Their Properties

Q Calculate the unknowns.

a) c b) B C)




64° /
6^


4


d) e) ( Remember to State
'A sum of A'.

35\





Xso" r
/ \ 1 1
D



6.3.2 Sum of angles at a point




Investigate!


The arms of these angles are drawn from the same point.
Measure the size of the angles in each diagram to find out what is the sum of angles
at a point.
B














a) How many angles can you see?. a) How many angles can you see?.
b) Use your protractor to measure b) Use your protractor to measure
the size of each angle. the size of each angle.
i) a6c = i) AOD =
ii) A6B = ii) d6c =
,
lii) b6c = iii) a6b = .

c) What is the sum of the angles?. iv) b6c =
c) What is the sum of the angles?
Conclusion
The sum of the angles at a point is

Example 3


Q Calculate the size of BOC


Remember to state the
210'
reason 'zis at a point'.
104° + 110° + BOC = 360° Us at a point)

BOC = 360°-104°-110°
Notice that the angles = 146°
meet at point 0 and
fit together without
leaving any gaps or Q Calculate the value of o.
overlaps.

The sum of angles at a
2a °+102°+180° = 360° Us at a point)
point is 360°. We write
'zls at a point' in the 2a° = 360°-102°-180°
working. = 78°
o° = _78°
2
0 = 39
Check My
Understanding


Q Calculate the value of a. Q Calculate the value of b.















Q Calculate the size of angle c. Q Angles AMB, BMCand CMD are all the same

size. Angle AMD is twice as big as each of the
other angles. Work out the size of each angle.




















UNIT 6 Angles and Their Properties

Speed Challenge 9

PAIR WORK
Race your partner to climb the levels for angles at a point!
0 Start at level 1. Calculate the unknown angles.
0 Check your answers with your partner once you are done for each level.
If your answer is incorrect, stay in the level and rework the calculations,
o Once you have calculated all the level 1 angles correctly, move on to level 2.
0 Continue until you have worked out all the angles in level 4 correctly.

A B C D E

■<k'
A~^°
(
X
( LA—" '
^~~237°
223°

w/
'(p=>- 12oMio^ <T>
<
2-
160° 170° UJ
160°



/30?\ <
\ /A Lrr w 2.
X
Ni
X
^^^100^^ \ 115° ^






—X X




5.3.3 Vertically opposite angles



Vertically opposite angles are the angles opposite
each other when two straight lines cross.

AB and CD are straight lines that intersect at point M.
The point of Intersection is the vertex.

AMD = CMB AMC = DMB
vertex







CD

Investigate


AC and BD are straight lines that
Intersect at point O.
Prove that vertically opposite angles
are equal at a point.

O Calculate angle a.





Q Calculate angle b.






Q Calculate angle c.





Q What do you notice about the size of angles a and c?



0 What do you notice about the size of angles DOC and AOBl




Conclusion:
Vertically opposite angles are .
There are pairs of vertically opposite angles at a point.




Example 4


in the diagram, AB and CD are straight lines.
Verticaily opposite angles
are equal. If -40D = 137°, calculate the size of:

O COS, 0 /IOC and Q BOD.

Solution
0 c6B = AdD (vert. opp. zls)
Remember to state the reason
= 137° 'vert. opp. ^s'.
0 A6D + A6C=^80° s on str. line)
/^6C= 180°-137°

= 43°
0 BOD = AOC (vert. opp. /Ls)
= 43°


UNIT 6 Angles and Their Properties

Check My
Understanding


Q Find angle a. 0 Calculate the size of angle b. 0 Calculate the size of angle c.

100°
53





0 Solve for x. 0 Solve for m.













o Journal Writing


Read each comment carefully. State whether
you agree or disagree with each comment.
Justify your answer.

two straight lines one straight line no straight lines
Z.o= Z.b because the
angles are vertically
/icand Z-c/are not
opposite.
equal because there is Z.e= /./because
only one straight line. the angles are
vertically opposite.

A






Check My
Understanding

0 ^06 and COD are straight lines. 0 Lines AB and CD are straight lines.

B £




63°
r





Calculate the size of each angle. Calculate the size of each angle.
a) AOC b) BOD a) AdD b) AOC
0

Q Lines and CH are straight lines.
Solve for a and b.




4if + w












Sum of angles in a quadrilateral



A quadrilateral is a closed shape that has four sides, four vertices and four interior
angles. The sides of a quadrilateral are straight lines.







We have learnt that the sum of angles in a triangle is 180°. What about in a
quadrilateral?
Let us find out what is the sum of the angles in a quadrilateral.
O Draw and cut out the quadrilaterals shown below. Draw a diagonal that
A diagonal is a line from
cuts the quadrilateral into triangles.
one vertex of a polygon
to the opposite vertex
in the polygon. It cuts a
quadrilateral into two
triangles.


















Q How many diagonals did you draw?

Quadrilateral A Quadrilateral B






Q Name each angle inside the quadrilaterals.



Angles and Their Properties
I

Q Use your protractor to measure the angles inside the triangles that
make up the quadrilateral.

La = ^g =



/Lc = /-i =

Ad = ^/ =
'.
/Le = A/c =

^/ =
Q Sum of angles = Sum of angles =

Conclusion: The sum of the interior angles in a quadrilateral is








Calculate angle a.


The sum of the interior
angles in a quadrilateral
is 360°.


Solution

0+ 85° + 69° + 110° = 360° (sum of int. As in a quad) Remember to state the reason
= 360°-85°-69°-110° 'sum of As in a quad'.
= 360° - 264°
= 96°




Check My
Understanding


Calculate the size of the unknown angles.

CHAPTER 6.4


In this chapter
Parallel Lin^,
Pupils should be able to:
• start to recognise the
angular connections Perpendicular Lines
between parallel lines,
perpendicular lines and
transversals
and Transversals


6.4.1 Perpendicular lines


Perpendicular lines are straight lines that meet at right angles (90°).
We say they are perpendicular to each other.
The symbol used to
state that two lines are
A square is the symbol
perpendicular is 1.
used on a diagram to
show that two lines are
perpendicular, These two lines are
perpendicular.





Are these lines
^ADC is a right angle.
perpendicular? So, AB is perpendicular to CD.
We write AB 1 CD.

Parallel lines


Parallel lines are two or more lines that are the same distance apart. They will
never meet or intersect.


Arrow heads are the
These two lines
symbols used on a _
are parallel.
diagram to show that
two lines are parallel.


•• Think and Share

Can you see the parallel lines?



»»}}}






UNIT 6 I Angles and Their Properties

Transversal lines



A transversal line is a line that cuts two or more lines. We will learn about the
angles that are formed when two parallel lines are cut by a transversal line.

transversal

[1
7
parallel lines parallel lines
LL

This transversal line is
^perpendicular to the
parallel lines.



Investigate!


The symbol used to state I
Let us find out the size of the angles
that two lines are parallel \
formed when a transversal line
is //. I
crosses a pair of parallel lines.





Q Use your protractor to measure:
a) Angles 0 and b b) Angles c and d


Zo = /.b = ^c =
c) Angles e and f d) Angles 5 and h

= ^f=. ^g = Ah-
Try and Apply!
Conclusion:
Colour one pair of
Z. and z. are equal. L, and A are equal.
angles that are equal
^ and ^ are equal. L and Z are equal. in the diagram.




Investigate

Use your protractor to measure the pairs
of angles.
a) Angles 0 and b.

AO - Ab-

b) Angles c and d.
Ac- Ad^

CHAPTER 6.5
In this chapter

Pupils should be able to: Solving Geometrical
• solve simple geometrical
problems by using side
and angle properties to
identify equal lengths Problems
or calculate unknown
angles
5.5.1 Finding unknown angles

in triangles




Try and Apply!



Truss Challenge!

Put your engineering skills to the test. See if
you can calculate all the unknown angles in
this roof truss.

















We have learnt some properties of the sides and angles of triangles earlier.
To solve geometrical problems, we need to make use of the properties of
triangles.











Equilateral Isosceles Right
Size of each angle = 60*^ The angles opposite the The sum of the two acute
equal sides are equal. angles = 90°
These angles are called the
base angles of the triangle.

s Figure 6.2 Angle properties of triangles



Angles and Their Properties

Example 1



In
0 triangle STU, aSTU = 65° and ^SUT=SO°.
fmd^TSU.

Solution
Z7"5t;=180°-65°-80°
= 35°





In
0 triangle XYZ, ZXYZ= 125° and ZYXZ=2S°.
Find ZXZK

Solution
v^XZy=180°-28°-125'
= 27°





0 triangle -460, ZBAC=S2° and ZACB is a right angle.
In
Find AABC

Solution A right triangle has one
right angle. The other two
^-406=90°
acute angles add up to be
Z7\fiC=90°-52^ 90°.
= 48°





0 DEFls an equilateral triangle. Find the sum of ZfDFand ZDFE.


Solution ^
Each angle of an equilateral triangle = ^ degrees = 60 An equilateral triangle
Sum of /.FDFand ZDFE= 60° + 60° has three equal angles
= 120° and three equal sides.
Each angle is 60°.
The sum of ZfDFand ADFE Is 120°.

^ ABC\s an isosceles triangle and ^BAC= 82°.
/^ACB = /.ABC because
triangle ABC has a line of Find ^ABC
symmetry, AD.
Solution
A
^ACB=^ABC
= (180°-82°)-r2
= 49°





0 ABC is an equilateral triangle and ^AED = 85'
An isosceles triangle has
Find ^ADE.
two equal angles and two
equal sides. The angles Solution
opposite the equal sides ^EAD = eO°
are equal. They are called AADE= 180°-60°-85°
the base angles of the
= 35°
triangles.




O IVXris an isosceles triangle. Angle X is a right angle and = 86°.

Find ZZXK
Solution
ZlVXT=90°
zxwr=zxw
= 90° ^ 2
= 45°

In triangle H/XZ, ZlVXZ= 180°-45°-86'=
= 49°
ZZXy= 90°-49°
= 41°



Check My
Understanding


O In triangle KLM, /.KLM = 15° and AKML = 27° Q In triangle PQR, ZPfiQ = 65° and ZQP/? is a right
Find Z.LKM. angle. Find /.PQR.
P
















E UNIT 6 Angles and Their Properties