Chapter 7
Chapter 8
Chapter 9
Rate and Ratio
7A
p.2
7B
p.9
7C
p.24
Angles in Triangles and Polygons
8A
p.36
8B
p.46
8C
p.57
8D
p.66
8E
p.74
Introduction to Deductive Geometry
9
p.83
For any updates of this book, please refer to the subject homepage:
http://teacher.lkl.edu.hk/subject%20homepage/MAT/index.html
For mathematics problems consultation, please email to the following address:
lkl.mathematics@gmail.com
For Maths Corner Exercise, please obtain from the cabinet outside Room 309
1
F2B: Chapter 7A
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 1
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 2
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 3
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
7A Level 1
Maths Corner Exercise
7A Level 2
Maths Corner Exercise
7A Level 3
Maths Corner Exercise
7A Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
2
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
7.1
Lesson Worksheet 7A
(Refer to §7.1)
Rate
A rate is a comparison of two quantities of different kinds by division.
It usually has a unit.
e.g. (a) Patrick runs 8 m per second.
Running speed = 8 m/s
∴
(b) 3 cups of ice cream are sold at $60.
$60
Price rate =
= $20/cup
3 cups
∴
1. Express the following rates with the units given in brackets.
Situation
Rate
(a)
There are 10 beds in 2 rooms. (beds/room)
(
(
(b)
6 cans of coke are sold at $42. ($/can)
(
(
(c)
A printer can produce 90 pages in 5 minutes.
(pages/min)
(d)
) beds
=(
) rooms
) beds/room
)
=
)
After 4 months, a plant grows 60 cm taller.
(cm/month)
○→ Ex 7A 1–4
Example 1
3 dozen eggs are sold at $54. Find the price
rate of the eggs in
Instant Drill 1
A company receives 182 phone calls in
2 weeks. Find the frequency of phone calls in
(a) $/dozen eggs,
(b) $/egg.
(a) calls/week,
(b) calls/day.
$54
3 dozen eggs
= $18/dozen eggs
$54
(b) Price rate =
3 dozen eggs
$54
=
3 × 12 eggs
= $1.5/egg
Sol (a) Price rate =
Sol (a) Frequency =
(
(
) calls
) weeks
=
(b) Frequency =
3
1 week
=
days
2. The selling price of 3 boxes of cereal is
$120. Each box is of 2 kg. Find the price
rate of the cereal in
(a) $/box,
3. A train travels 9 000 m in 15 minutes. Find
the speed of the train in
(a) m/min,
(b) km/h.
(b) $/kg.
○→ Ex 7A 10–14
Example 2
The typing speed of Kelly is 60 words/min.
How many words can she type in 3 minutes?
Instant Drill 2
Chicken wings are sold at $35/kg. Find the
selling price of a pack of 4 kg chicken wings.
Sol Number of words
Sol Selling price
= 60 × 3
= 180
=(
=
4. Samuel works 8 hours a day. If his wage
rate is $46/h, find his wage in a day.
)×(
)
5. Mandy swims at a speed of 2.5 m/s. How
far does she swim in 1 minute?
○→ Ex 7A 6, 7
6. A machine produces chopsticks at a rate of
200 pairs/h. How many hours does it take
to produce 1 200 pairs of chopsticks?
7. Suppose the exchange rate between H.K.
dollars and U.S. dollars is 7.8 HKD/USD.
How many U.S. dollars can 117 H.K.
dollars be changed to?
It means that 7.8
HKD
can be changed to
1 USD.
○→ Ex 7A 8, 9
4
‘Explain Your Answer’ Questions
8. 4 red pens are sold for $35 and 10 blue pens are sold for $52. On average, which type of pens
is more expensive? Explain your answer.
Price rate of red pens =
Price rate of blue pens =
∵ $
(>/=/<)$
∴ On average, (red / blue) pens are more expensive.
9. The download count of software A in a week is 4 130 while the download count of software B
in 15 days is 8 550. Is the daily download rate of software A higher than that of software B?
Explain your answer.
Level Up Question
10. The water temperature in a cooking pot rises at a rate of 2.6°C/min.
(a) Find the time taken for the water temperature to rise by 65°C.
(b) By how many °C does the water temperature rise in half an hour?
5
New Century Mathematics (2nd Edition) 2B
7
Rate and Ratio
Consolidation Exercise 7A
Level 1
Express the following rates with the units given in brackets. [Nos. 1−
−4]
1.
There are 36 balls in 3 bags. (balls/bag)
2.
There are 200 needles in 5 boxes. (needles/box)
3.
The total weight of 20 pencils is 180 g. (g/pencil)
4.
Tom finishes 30 questions in 15 min. (questions/min)
5.
Complete the table below.
Distance
Time
(a)
200 m
10 s
(b)
280 km
(c)
Speed
140 km/s
4 min
20 m/min
6.
Some candies are sold at $36/kg. How much is 0.5 kg candies?
7.
The salary of a worker is $42/hour. What is his salary after working for 5 hours?
8.
Regina is reading a comic book and her reading rate is 2 pages/min. If there are 70 pages in the book,
how long will she take to finish reading the book?
9.
The petrol consumption rate of a car is 20 km/L. How much petrol is consumed by the car in
travelling 50 km?
10. Two dozen eggs are sold at $48. Find the price rate in $/egg.
11. Henry can draw 30 circles in 20 seconds. Find his rate of drawing circles in circles/min.
12. The selling price of 3 bottles of 2L-lemon tea is $60. Find the price rate of the lemon tea in the
following units.
(a) $/bottle
(b) $/L
13. A shop is open 6 days a week and 50 weeks a year. If the weekly profit of the shop is $6 000, find the
(a) daily profit,
(b) annual profit.
14. A peak tram finishes a journey of 1.4 km in 8 minutes. Find its speed in the following units.
(a) m/min
(b) km/h
6
Level 2
15. A computer program can run 42 times in 12 minutes.
(a) How long does it take to run the program 7 times?
(b) How many times can the program run in 2 hours?
16. The price of 15 apples is $90.
(a) Find the price rate in $/apple.
(b) Sally has $150. Does she have enough money to buy two dozen apples? Explain your answer.
17. Robot T walks 1 980 m in 11 minutes.
(a) Find the walking speed of Robot T in
(i) m/s,
(ii) km/h.
(b) Robot U walks 3 km in
1
hour. Which robot, T or U, has a higher walking speed? Explain your
4
answer.
18. In shop A, the price of 6 oranges is $21. In shop B, the price of 8 oranges is $32.
(a) Find the price rate of the oranges in each shop in $/orange.
(b) The oranges in shop B are now sold at a discount of 15%. Steven wants to buy oranges with the
lower price rate. Which shop, A or B, should he visit? Explain your answer.
19. There are 1 840 books in a library. A librarian starts to sort all the books. It is known that he has sorted
120 books in the first 30 minutes.
(a) Find his rate of sorting books in books/h.
(b) If he continues to sort books at this rate, can he finish the rest of the books in the next
7 hours? Explain your answer.
20. Suppose 202.5 Thailand Bahts are equivalent to 45 H.K. dollars.
(a) Find the exchange rate between Thailand Bahts and H.K. dollars in Thailand Bahts/HKD.
(b) How many Thailand Bahts can 140 H.K. dollars be changed to?
(c) How many H.K. dollars can 172.8 Thailand Bahts be changed to?
21. A ship sails 125 km in 150 minutes.
(a) Find the speed of the ship in km/h.
(b) How many hours does it take to travel 90 km?
(c)
If the speed of the ship is decreased by 3 m/s, how many kilometres can it travel in
120 minutes?
7
Consolidation Exercise 7A (Answer)
1.
2.
3.
4.
12 balls/bag
40 needles/box
9 g/pencil
2 questions/min
5.
Distance
Time
Speed
(a)
200 m
10 s
20 m/s
(b)
280 km
2s
140 km/s
(c)
80 m
4 min
20 m/min
14. (a)
(b)
15. (a)
(b)
16. (a)
(b)
17. (a)
18.
6.
7.
8.
9.
10.
11.
12.
$18
$210
35 min
2.5 L
$2/egg
90 circles/min
(a) $20/bottle
(b) $10/L
13. (a) $1 000
(b) $300 000
19.
20.
21.
8
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(c)
(a)
(b)
(c)
175 m/min
10.5 km/h
2 min
420
$6/apple
yes
(i) 3 m/s
(ii) 10.8 km/h
U
shop A: $3.5/orange, shop B: $4/orange
B
240 books/h
no
4.5 Thailand Bahts/HKD
630 Thailand Bahts
38.4 HKD
50 km/h
1.8 h
78.4 km
F2B: Chapter 7B
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 4
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 5
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 6
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 7
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 8
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 9
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 10
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
9
Book Example 11
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 12
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
7B Level 1
Maths Corner Exercise
7B Level 2
Maths Corner Exercise
7B Level 3
Maths Corner Exercise
7B Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
10
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
7.2A
Lesson Worksheet 7B
(Refer to §7.2)
Meaning of Ratio
(a) A ratio is a comparison of two quantities of the same kind by division.
It has no unit.
p
(b) A ratio of a quantity p to a quantity q can be written as p : q or ,
q
where p, q ≠ 0 and they are called the terms of the ratio.
Note that
p:q≠q:p
in general.
e.g. Ratio of the lengths of A to B
= 3 cm : 5 cm
 3
= 3 : 5  or 
 5
5 cm
3 cm
Cancel out the units.
A B
1. Determine whether each of the following is a ratio.
If yes, put a ‘’ in the box and express the ratio in the form of p : q in the space provided.
If not, put a ‘’ in the box.
Are the two
$8
9 L quantities of the
(a)
(b)
same kind?
$11
4s
2 kg
17 cm
(c)
(d)
2 kg =
50
1g
g
7.2B
Properties of Ratio
(a) If k ≠ 0, then
(i) a : b = a × k : b × k
(ii) a : b =
a b
:
k k
(b) A ratio p : q is in its simplest form if both p and q are integers and
they have no common factor other than 1.
e.g. 3 : 2 is in its simplest form but 6 : 4 is not.
Example 1
Simplify the following ratios.
1 1
(a) 21 : 12
(b)
:
3 2
Sol (a) 21 : 12
Divide each term by the
H.C.F. of 21 and 12, i.e.
21 12
=
:
3.
3
3
We call 21 : 12 and 7 :
=7:4
4 two equal ratios.
1 1
(b)
:
3 2
Multiply each term
1
1
= × 6:
× 6 by the L.C.M. of 3
3
2
and 2, i.e. 6.
=2:3
Instant Drill 1
Simplify the following ratios.
1 2
(a) 30 : 36
(b)
:
5 3
Sol (a) 30 : 36
30
36
=
:
(
) (
)
=
1 2
(b)
:
5 3
1
2
= ×(
):
×(
)
5
3
=
11
2. Simplify the following ratios.
8
40
(a) 8 : 40 =
:
(
) (
)
(e) 3 :
2
=
5
3=
3
1
=
Convert both
terms
into integers first.
(b) 20 : 15 =
(c)
(d)
1 1
1
: =
×(
2 7
2
=
):
1
×(
7
(f) 3.3 : 2.7 = 3.3 × 10 : 2.7 × 10
= 33 : (
)
33
(
)
=
:
(
) (
)
=
)
(g) 0.6 : 0.12 = 0.6 × (
=
1 5
: =
4 8
) : 0.12 × (
)
○→ Ex 7B 1
Example 2
Simplify 1 m : 50 cm.
Sol
1 m : 50 cm
= 100 cm : 50 cm
= 100 : 50
100 50
=
:
50 50
=2:1
3. Simplify 2 kg : 800 g.
Instant Drill 2
Simplify 20 s : 1 min.
Express both terms
in the same unit first.
Sol
20 s : 1 min
= 20 s : (
)s
=
Cancel out the
unit.
4. Simplify 1.5 cm : 65 mm.
○→ Ex 7B 2
12
Example 3
If x : 12 = 3 : 4, find x.
Sol x : 12 = 3 : 4
x
3
=
12 4
3
x=
× 12
4
=9
a:b=
a
b
Instant Drill 3
If 5 : 6 = 25 : y, find y.
Sol
5 : 6 = 25 : y
5
(
)
=
(
)
(
)
=
Find the unknowns in the following ratios. [Nos. 5–6]
5. (y – 2) : 9 = 2 : 3
6. 5 : a = 20 : (a + 3)
○→ Ex 7B 4, 5
In each of the following, x and y are non-zero numbers. Find x : y. [Nos. 7–8]
x
Rewrite it as
7. 5x = 8y
8.
– 4y = 0
x ?
3
‘ = ’.
y
?
○→ Ex 7B 6, 7
9. There are 80 pets in a pet shop, of which 32 are dogs and the rest are cats.
(a) Find the ratio of the number of dogs to the total number of pets.
(b) Find the ratio of the number of dogs to the number of cats.
○→ Ex 7B 11–13
13
7.2C
Three-term Ratio
(a) A ratio of three quantities a, b, c (a three-term ratio) can be expressed as
a : b : c.
e.g. Consider a : b : c = 5 : 2 : 4. We have:
(i) a : b = 5 : 2
(ii) b : c = 2 : 4, i.e. 1 : 2
a : b : c = 5 : 2 : 4
(iii) a : c = 5 : 4
a : b : c = 5 : 2 : 4
a : b : c = 5 : 2 : 4
(b) If k ≠ 0, then
(i) a : b : c = a × k : b × k : c × k
Example 4
Simplify the following ratios.
1 1 1
(a) 2 : 10 : 4
(b)
: :
4 3 2
Sol (a) 2 : 10 : 4
H.C.F. of 2, 10 and
2 10 4 4
=
:
:
2
2 2 2
=1:5:2
1 1 1
L.C.M. of 4, 3 and 2
(b)
: :
= 12
4 3 2
1
1
1
=
× 12 : × 12 : × 12
4
3
2
=3:4:6
10. Simplify the following ratios.
(a) 20 : 25 : 40
20
25
40
=
:
:
(
) (
) (
)
=
(b)
0.6 : 1.5 : 2.7
(ii) a : b : c =
a b c
:
:
k k k
Instant Drill 4
Simplify the following ratios.
2 1 2
(a) 16 : 8 : 4
(b)
:
:
3 6 9
Sol (a) 16 : 8 : 4
16
8
4
=
:
:
(
) (
) (
)
=
2 1 2
(b)
:
:
3 6 9
2
1
2
×( ):
× ( ):
×(
=
3
6
9
=
(c)
)
42 L : 21 L : 63 L
=
(d)
300 m : 3 km : 900 m
=
=
Express all the
terms in the
same unit first.
○→ Ex 7B 3
14
We can combine two ratios to form a three-term ratio.
p: q: r
1: 3
3: 4
1: 3: 4
e.g. Suppose p : q = 1 : 3 and q : r = 3 : 4.
We have p : q : r = 1 : 3 : 4.
Example 5
If a : b = 4 : 3 and b : c = 2 : 1, find a : b : c.
Sol Method 1
a:b
common term
a:b: c
Make them the same by
4:3
2 : 1 considering their
L.C.M.
Method 2
a:b =
4
L.C.M. of 3 and 2 = 6
3×(
) = 6, 2 × (
)=
6
= 4×2 : 3×2
b:c=
2×3 : 1×3 =
∴ a:b:c=8:6:3
a:b
=2 × (
b:c=
∴ a:b:c=
4×(
)
×
2
:
1
Method 2
=(
):3 × (
3
a:b: c
2:5
4:3
L.C.M. of 5 and 4 =
) :5 × (
×
a:b:c= 2×4 : 2×3 : 1×3
=
8
: 6
:
3
6:3
Instant Drill 5
If a : b = 2 : 5 and b : c = 4 : 3, find a : b : c.
Sol Method 1
b:c=
=8:6
:
×
): (
)=
(
a:b
)
): (
)
=
2
b:c=
a:b:c=(
:
×
×
5
×
4
:
):(
3
):(
)
=
11. If a : b = 5 : 6 and b : c = 9 : 2, find a : b : c.
L.C.M. of 6 and 9 =
Method 1
=5 × (
) :6 × (
b:c=
∴ a:b:c=
9×(
a:b
)
):2 × (
Method 2
a:b =
5
b:c=
=
)=
:
6
9
:
2
a:b:c=
Simplify the
ratio if possible.
12. If a : b = 6 : 1 and a : c = 4 : 5, find a : b : c.
a:b: c
6:1
4:
5
○→ Ex 7B 8
15
7.2D
Dividing a Quantity in a Given Ratio
Example 6
Instant Drill 6
(a) Divide 24 in the ratio of 1 : 2.
(a) Divide 100 kg in the ratio of 3 : 2.
(b) Divide 40 m in the ratio of 4 : 3 : 1.
Sol (a)
24
(b) Divide $63 in the ratio of 2 : 6 : 1.
Sol (a)
100 kg
1
1+ 2
1
= 24 ×
3
=8
First portion = 24 ×
First portion = 100 ×
(
(
= 100 ×
(
2
1+ 2
2
= 24 ×
3
= 16
kg
Second portion =
(b)
40 m
4
m
4 + 3 +1
4
= 40 ×
m
8
= 20 m
First portion = 40 ×
$63
First portion = $63 ×
(
(
= $63 ×
(
(
)+(
)
)
)
)+(
=
3
m
4 + 3 +1
3
= 40 × m
8
= 15 m
Second portion = 40 ×
Second portion =
1
m
4 + 3 +1
1
= 40 × m
8
=5m
Third portion = 40 ×
13. Andrew and Belle share a 400 g steak in
)
=
Second portion = 24 ×
(b)
( )
)+(
)
kg
)
Third portion =
14. Carl and Donna jointly donate a sum of
16
)
the ratio of 3 : 1. How much does Andrew
$3 500 in the ratio of 5 : 2. How much
get?
does Carl donate?
Andrew’s share = 400 ×
(
( )
)+(
)
g
=
15. The numbers of wins, losses and draws of
a football team are in the ratio of 7 : 2 : 3.
If the team played 36 matches altogether,
how many matches did the team draw?
16. A ribbon is divided into two parts in the
ratio of 7 : 3. If the larger part is 56 cm
long, what is the total length of the ribbon?
x cm
larger part (56 cm)
Let x cm be the total length of the ribbon.
( )
x×
= 56
( )+( )
=
○→ Ex 7B 17, 18, 22
17. A drink is made by mixing honey and
green tea in the ratio of 2 : 13. If the drink
contains 40 mL of honey, find the total
volume of the drink.
18. Flora, Gina and Helen share a box of
cookies in the ratio 5 : 3 : 1. If Gina gets
18 cookies, find the total number of
cookies in the box.
○→ Ex 7B 19–21
‘Explain Your Answer’ Question
17
19. Vivian is 25 years old and Wallace is 15 years old now.
(a) Find the ratio of Vivian’s age to Wallace’s age.
(b) Will the ratio of their ages change after 5 years? Explain your answer.
(a) The required ratio =
(b) 5 years later, Vivian will be _____ years old and Wallace will be _____ years old.
∴ The ratio (will / will not) change after 5 years.
Level Up Questions
20. If 2x + 4y = 7y – 3x, find x : y.
21. It is given that the ratio of Jason’s weight to Roy’s weight is 1 : 3, and the ratio of Sam’s
weight to Roys’s weight is 7 : 6.
(a) Find the ratio Sam’s weight : Jason’s weight : Roy’s weight.
(b) If Jason weighs 20 kg, find the total weight of Sam, Jason and Roy.
New Century Mathematics (2nd Edition) 2B
18
7
Rate and Ratio
Consolidation Exercise 7B
Level 1
Complete the following table and give the answers in the simplest form. [Nos. 1−
−3]
1.
x
y
(a)
2
10
(b)
21
6
(c)
1
3
1
4
x
y
(a)
250 mL
1L
(b)
0.8 kg
30 g
(c)
1
km
4
500 m
x
y
z
(a)
5
45
30
(b)
40
12
28
(c)
14 cm
56 mm
35 mm
2.
3.
4.
(b) 5 : b = 10 : 20
(b) (y + 1) : 14 = (y + 6) : 21
In each of the following, a and b are non-zero numbers. Find a : b.
(a) 7a = b
7.
x:y:z
Find the unknowns in the following ratios.
(a) 4 : (x − 1) = 32 : 8
6.
x:y
Find the unknowns in the following ratios.
(a) a : 3 = 27 : 9
5.
x:y
(b) 3a − 4b = 0
In each of the following, x and y are non-zero numbers. Find x : y.
(a)
x y
=
5 3
(b)
19
2x y
− =0
9 4
8.
In each of the following, find x : y : z.
(a) x : y = 1 : 2 and y : z = 2 : 5
(b) x : y = 1 : 9 and y : z = 3 : 1
(c) x : y = 4 : 3 and x : z = 8 : 1
(d) x : z = 3 : 2 and y : z = 7 : 6
In each of the following, divide the given quantity in the ratio as indicated. [Nos. 9−
−10]
9.
(a) Divide 35 in the ratio 1 : 6.
(b) Divide $20 in the ratio 3 : 2.
(c) Divide 180 g in the ratio 4 : 5.
10. (a) Divide 10 in the ratio 1 : 2 : 2.
(b) Divide 80 cm in the ratio 1 : 5 : 4.
(c) Divide 270 mL in the ratio 2 : 3 : 4.
11. The weight of a dog is 15 kg while the weight of a cat is 13.5 kg. Find the ratio of the weight of the
dog to that of the cat.
12. Among 150 towels in a box, 100 of them are black and the rest are white. Find the ratio of the number
of black towels to that of white towels.
13. In a test, the ratio of Joe’s score to Tom’s score is 5 : 2 and the ratio of Tom’s score to Ken’s score is 4 :
3. Find the ratio of Joe’s score to Tom’s score to Ken’s score.
14. In the figure, PQR is a straight wire.
Find the following ratios of lengths.
(a) PQ : QR
(b) PR : QR
(c) PR : QR : PQ
15. The height of building A is 52 m. Building B is 12 m lower than building A, while building C is 20 m
higher than building A. Find the ratio of the height of building A to that of building B to that of
building C.
16. A special drink is made by mixing orange juice and soda water in the ratio 2 : 1 by volume. Find the
volume of orange juice required to produce 750 mL of the special drink.
17. In a school, there are 1 044 students. If the ratio of the number of boys to that of girls is 5 : 7, how
many boys and girls are there in the school?
20
18. There are 168 red balls and some green balls in a bag. The ratio of the number of red balls to that of
green balls is 3 : 4. Find the total number of balls in the bag.
19. Eason, Fred and George share a pack of game cards in the ratio 5 : 9 : 2. If Eason gets 35 game cards,
find the total number of game cards in the pack.
20. In a triangle, the lengths of the three sides are in the ratio 2 : 3 : 4. If the length of the shortest side is
18 cm, find the perimeter of the triangle.
Level 2
21. A bottle of cleaning agent is formed by dissolving 2 g of bleach powder in 0.2 kg of water.
(a) What is the ratio of the weight of bleach to that of water in the bottle of cleaning agent?
(b) What is the ratio of the weight of bleach to that of the cleaning agent?
22. Find x : y in each of the following.
(a) 5x + 8y = 6x + 7y
(b) (x + 4y) : (5x + 2y) = 1 : 2
23. Find x : y in each of the following.
(a)
1 1
: =2:3
x y
(b)
1 3
: =1:2
2x y
24. If x : y = 4 : 3, find
(a)
1 1
: ,
x y
(b) (x + 2y) : (2x − y).
25. If a : b = 1 : 2 and b : c = 1 : 3, find
(a) c : a,
(b) (c − 2a) : (a + b).
26. In each of the following, find a : b : c.
(a) a = 2b and b = 4c
(b) 10a = 5b = c
27. It is given that a : b : c = 1 : 6 : 2. Find the values of a and c in each of the following cases.
(a) b = 90
(b) a + c = 36
28. In a swimming race, the speeds of Leo and Nick are 150 m/min and 2 m/s respectively. Find the ratio
of the speed of Leo to that of Nick.
29. Winnie and Shirley share a pack of candies in the ratio 3 : 7. If Shirley gets 16 candies more than
Winnie does, find the total number of candies in the pack.
21
30. In the figure, ABCD is a rectangle and AB : AD = 7 : 6. If AB is 5 cm
longer than AD, find the area of ABCD.
31. There are 27 Chinese books and some English books on a bookshelf. The numbers of Chinese books
and English books are in the ratio 3 : 7.
(a) How many English books are there?
(b) Ben takes 18 Chinese books and 18 English books away from the bookshelf. He then claims that
the ratio of Chinese books to English books on the bookshelf is still 3 : 7.
Do you agree? Explain your answer.
32. Some cows are shared among 3 farmers A, B and C such that A’s share : B’s share = 3 : 2 and B’s
share : C’s share = 3 : 1.
(a) Find the ratio A’s share : B’s share : C’s share.
(b) It is given that B gets 48 cows. Find the total number of cows being shared.
33. The price of a green apple is $6 and the price of a red apple is $10. James buys some green and red
apples in the ratio 5 : 3. How many apples does he buy if he spends $360 in total?
34. In the figure, the height of △ ABC is h cm with AB as the base.
D is a point on AB such that AD : DB = 2 : 5. Let AB = x cm.
(a) Express the area of △ ABC in terms of x and h.
(b) Find the ratio of the area of △ ADC to that of △ DBC to
that of △ ABC.
22
Consolidation Exercise 7B (Answer)
1.
x
y
x:y
(a)
2
10
1:5
(b)
21
6
7:2
(c)
1
3
1
4
4:3
2.
x
y
x:y
(a)
250 mL
1L
1:4
(b)
0.8 kg
30 g
80 : 3
(c)
1
km
4
500 m
1:2
3.
x
y
z
x:y:z
(a)
5
45
30
1:9:6
(b)
40
12
28
10 : 3 : 7
(c)
14 cm
56 mm
35 mm
20 : 8 : 5
12. 2 : 1
13. 10 : 4 : 3
14. (a) 3 : 2
(b) 5 : 2
(c) 5 : 2 : 3
15. 13 : 10 : 18
16. 500 mL
17. boy: 435, girl: 609
18. 392
19. 112
20. 81 cm
21. (a) 1 : 100
(b) 1 : 101
22. (a) 1 : 1
(b) 2 : 1
23. (a) 3 : 2
(b) 1 : 3
24. (a) 3 : 4
(b) 2 : 1
25. (a) 6 : 1
(b) 4 : 3
26. (a) 8 : 4 : 1
(b) 1 : 2 : 10
27. (a) a = 15, c = 30
(b) a = 12, c = 24
28. 5 : 4
29. 40
30. 1 050 cm2
31. (a) 63
(b) no
32. (a) 9 : 6 : 2
(b) 136
33. 48
1
xh cm2
34. (a)
2
(b) 2 : 5 : 7
4.
(a) 9
(b) 10
5. (a) 2
(b) 9
6. (a) 1 : 7
(b) 4 : 3
7. (a) 5 : 3
(b) 9 : 8
8. (a) 1 : 2 : 5
(b) 1 : 9 : 3
(c) 8 : 6 : 1
(d) 9 : 7 : 6
9. (a) 5, 30
(b) $12, $8
(c) 80 g, 100 g
10. (a) 2, 4, 4
(b) 8 cm, 40 cm, 32 cm
(c) 60 mL, 90 mL, 120 mL
11. 10 : 9
23
F2B: Chapter 7C
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 13
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 14
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 15
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 16
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 17
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 18
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
7C Level 1
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
24
Teacher’s
Signature
___________
(
)
Maths Corner Exercise
7C Level 2
Maths Corner Exercise
7C Level 3
Maths Corner Exercise
7C Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
25
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
7.3A
Lesson Worksheet 7C
(Refer to §7.3)
Similar Figures
In two similar figures, the ratios of the lengths of all the corresponding sides
are equal, i.e. they are proportional.
Example 1
The diagram below shows two similar
rectangles. Find the unknown x.
2
4
x
4
Sol
=
3
2
3
4
x=
×3
x
2
=6
Instant Drill 1
The diagram below shows two similar
parallelograms. Find the unknown y.
8
y
(
)
Sol
=
(
)
(
)
5
y=
4
y
In each of the following, the two given figures are similar. Find the unknowns. [Nos. 1–4]
1.
2.
21
x
a
5
10
9
6
15
3.
16
12
27
4.
x
4
20
y
25
16
y
○→ Ex 7C 1–4
26
7.3B
Scale Drawings
(a) A scale drawing is a reduced or an enlarged drawing of a real object, where
the scale = length in the scale drawing : actual length.
(b) The scale is usually expressed in the form 1 : n:
(i) n > 1 stands for a reduction,
(ii) n < 1 stands for an enlargement.
e.g. Refer to the scale drawing of a tree in the figure.
Suppose the actual height of the tree is 3 m.
Scale = 2 cm : 3 m
2 cm
=
3m
2 cm
=
3 × 100 cm
1
=
150
= 1 : 150
2 cm
i.e. A length of 1 cm in
the drawing represents
150 cm (or 1.5 m) on
the real tree.
5. In each of the following, find the scale of the drawing in the form 1 : n.
(a) Length in a scale drawing = 25 cm,
(b) Length in a scale drawing = 10 cm,
actual length = 800 m
actual length = 1 mm
Scale = (
):(
)
=
○→ Ex 7C 5, 6
6. The distance between two airports on a map is 4 cm. If the actual distance between them
is 200 km, find the scale of the map in the form 1 : n.
1 km = 1 × 1 000 m
= 1 × 1 000 × 100 cm
○→ Ex 7C 10
27
Example 2
The figure is a scale drawing
Instant Drill 2
The length of the ferry in the
of a sofa. If the length of the
photograph is 4.2 cm. Find
sofa in the drawing is 3 cm,
3 cm
find its actual length in m.
Scale
1 : 90
Sol Let x cm be the actual length of the sofa.
3 cm : x cm = 1 : 90
1
3
∴
=
x
90
x = 3 × 90
= 270
∴ The required actual length is 270 cm,
4.2 cm
the actual length of the ferry
Scale
in m.
1 : 800
Sol Let x cm be the actual length of the ferry.
(
):(
) = 1 : 800
∴
=
i.e. 2.7 m.
7.
9 cm
Scale
8. Tracy makes a scale drawing of a building
with a scale of 1 cm : 150 m. If the actual
height of the building is 420 m, find its
1 : 0.018
height in the drawing.
The length of a computer chip in the scale
drawing as shown is 9 cm. Find the actual
length of the computer chip in mm.
○→ Ex 7C 7–9
28
9. The figure shows the floor plan of a meeting room. If the
actual length of the room is 18 m, find the length of the
meeting room on the floor plan in cm.
?
Scale 1 : 250
10. The figure shows a map with a scale of 1 : 6 000.
(a) Use a ruler to measure the distance between two bus stops
A and B on the map.
A
(b) Hence, estimate the actual distance (in m) between A and B.
B
Scale 1 : 6 000
○→ Ex 7C 11, 12
‘Explain Your Answer’ Question
11. On a map with scale 1 : 400 000, the length of a cycling track is 4.5 cm.
(a) Find the actual length of the cycling track in km.
(b) Jason cycles at an average speed of 12 km/h along the cycling track. Can he finish the
whole journey in 2 hours? Explain your answer.
29
Level Up Questions
12. In the diagram, trapeziums ABCD and EFGH are A
similar figures. Find the unknowns x and y.
20
B
x+5
F
12
D
x
13. The figure shows the floor plan of a rectangular hospital ward.
Its scale is 1 : 140. The dimensions of the floor plan are
4.5 cm × 3.5 cm. Find the actual area of the hospital ward in m2.
Find the actual length
and the actual width first.
30
y
E
C
H
8
G
New Century Mathematics (2nd Edition) 2B
7
Rate and Ratio
Consolidation Exercise 7C
Level 1
In each of the following, the two given figures are similar. Find the unknowns. [Nos. 1−
−4]
1.
2.
3.
5.
4.
The figure shows a photo of a dove. The height of the dove in
the photo is 3.2 cm. If the actual height of the dove is 28.8 cm,
find the scale of the photo in the form 1 : n.
31
6.
The figure shows a photo of a butterfly. The wingspan of the
butterfly in the photo is 4.8 cm. If the actual wingspan of the
butterfly is 0.32 cm, find the scale of the photo in the form n : 1.
7.
The figure is a scale drawing of a fish. Its scale is 1 : 4. If the
length of the fish in the drawing is 5 cm, find the actual length
(in cm) of the fish.
32
8.
The length of a cell in a scale drawing is 36 mm. If the scale of the drawing is 1 200 : 1, find the
actual length (in mm) of the cell.
9.
Victor wants to make a scale drawing of a building with a scale of 1 cm : 9 m. If the actual height of
the building is 126 m, what should its height be in the drawing?
10. The distance between cities P and Q on a map is 5 cm. If the actual distance between them is 13 km,
find the scale of the map in the form 1 : n.
11. The distance between two places is 16 km. Find their distance apart (in cm) on a map with scale 1 :
250 000.
12. The scale of a map is 1 : 2 000 000. If the length of a path on the map is 3 cm, find its actual length (in
km).
13. The figure shows the floor plan of a rectangular room with a scale of
1 : 400.
(a) Use a ruler to measure the length and the width of the room on
the plan.
(b) Hence, estimate the actual area of the room.
Level 2
14. In the figure, quadrilateral WXYZ is formed by enlarging
quadrilateral PQRS. Find the perimeter of WXYZ.
15. In the figure, ABCD and FCDE are two similar rectangles. Find
the length of BF.
33
16. Patrick uses a photocopier to reduce Fig. A to
Fig. B. The height of Fig. A is 10 cm while the
height of Fig. B is 6 cm. It is given that the
perimeter of Fig. A is 63 cm.
(a) Find the perimeter of Fig. B.
(b) Patrick further reduces Fig. B so that the
perimeter of the new figure is 18.9 cm. Find the height of that new figure.
17. The figure shows a photo of a temple. It is given that the
height of the temple in the photo is 2.8 cm and the actual
height of the temple is 35 m.
(a) Find the scale of the photo in the form 1 : n.
(b) If the actual height of the entrance is 6 m, what is its
height (in cm) in the photo?
18. The distance between cities A and B on a map is 5 cm. It is given that the actual distance between
them is 300 km.
(a) Find the scale of the map in the form 1 : n.
(b) The distance between another two cities C and D on the same map is 11 cm. Find the actual
distance (in km) between them.
19. On a map, 1 cm represents an actual distance of 500 m.
(a) Find the scale of the map in the form 1 : n.
(b) If the actual length of a tunnel is 2 km, find the length of the tunnel on the map.
20. The figure shows the floor plan of a restaurant with scale
1 : 800. On the plan, VU = UT = RQ = QP = 1 cm and
TS = SR = 2 cm.
(a) Find the actual area (in m2) of the restaurant.
(b) If the cost of tiling each m2 of the floor in the restaurant is $250,
find the total cost of tiling the whole floor.
21. The figure shows a scale drawing of a rectangular board.
(a) Use a ruler to measure the length and the width of the board in the
drawing.
(b) Estimate the actual length and the actual width of the board.
(c) Carol claims that the ratio of the area of the board in the drawing to the
actual area of the board is 1 : 200. Do you
agree? Explain your answer.
34
Consolidation Exercise 7C (Answer)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
15. 22.5 cm
16. (a) 37.8 cm
(b) 3 cm
17. (a) 1 : 1250
(b) 0.48 cm
18. (a) 1 : 6 000 000
(b) 660 km
19. (a) 1 : 50 000
(b) 4 cm
20. (a) 704 m2
(b) $176 000
21. (a) length: 3.6 cm, width: 3 cm
(b) actual length: 7.2 m, actual width: 6 m
(c) no
14
6
9
a = 12, b = 32
1:9
15 : 1
20 cm
0.03 mm
14 cm
1 : 260 000
6.4 cm
60 km
(a) length: 4.5 cm, width: 3.5 cm
(b) 252 m2
14. 72 cm
35
F2B: Chapter 8A
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 1
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 2
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 3
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 4
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
8A Level 1
Maths Corner Exercise
8A Level 2
Maths Corner Exercise
8A Level 3
Maths Corner Exercise
8A Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
36
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
8.1A
Lesson Worksheet 8A
(Refer to §8.1A)
Exterior Angles of a Triangle
Consider
△ABC as shown. BC is produced to D.
BD and AC will form an angle c1 outside the triangle.
c1 is the exterior angle of
△ABC.
a and b are the interior opposite angles of c1.
1.
In each of the following, determine whether x is an exterior angle of
S
x R
(a)
(b) P
x
S
R
P
(c)
△PQR.
Q
Q
(d)
P
S
T
x
R
Q
R
x
S
QRS is a straight line.
2.
Q
P
SRP and TRQ are straight lines.
Refer to the figure. Write down the two interior opposite
q
angles of each of the following exterior angles.
(a) p
b
(b) q
a
An exterior angle of a triangle is equal to the sum of its
two interior opposite angles.
i.e.
In the figure,
c1 = a + b
[Reference: ext. ∠ of
△]
37
c
p
Example 1
Instant Drill 1
In the figure, ACD is a straight line. Find x.
In the figure, BCD is a straight line. Find y.
B
∠ABC and ∠BAC
are the two interior
opposite angles of x.
B
55°
42°
78°
x
D
C
A
C
Sol x = 42° + 78°
ext. ∠ of
△
Sol y = (
A
)+(
)
ext. ∠ of
△
=
= 120°
3.
38°
y
D
In the figure, BCD is a straight line. Find x.
A
x
28°
122°
4.
In the figure, ACD is a straight line. Find p.
D
C
A
C
41°
B
D
100°
p
100° = (
)+( B
)
ext. ∠ of
△
=
5.
In the figure, ACD is a straight line. Find q.
In the figure, CBE and ACD are straight lines.
Find m and n.
5q
D
6.
∠ABC (i.e. m) and
∠______ are the
adjacent angles on
104°
B
the straight line
m
CBE.
B
E
C
2q
A
n
D
30°
C
m+(
8.
In the figure, ABE and DACF are straight
lines. Find x.
In the figure, ACD and FBCE are straight
lines. Find y.
B
E
The two interior
opposite angles of
x are ∠______ and
∠______.
) adj. ∠s on st. line
=
→
○ Ex 8A 1−6
7.
A
)=(
77°
D
E 60°
C
150° D
A
x
F
80°
C
A
B
y
F
→
○ Ex 8A 7−12
38
9.
In the figure, BCD and ECA are straight lines.
10. In the figure, ADC and EBD are straight lines.
Find m and n.
Find p and q.
E
n
B
m
E
B
86°
C
68°
47°
35°
A
D
C
D
26°
A
C
△ABC,
m =(
p
64°
B
m
In
q
A
)+(
)
ext. ∠ of
△
=
In
△CDE,
E
n
m
C
D
→
○ Ex 8A 13−15
Level Up Question
11. In the figure, ADCG, BEC and DEF are straight lines.
F
B
Find p and q.
p
58°
E
140°
q
A
39
D
C
G
New Century Mathematics (2nd Edition) 2B
8
Angles in Triangles and Polygons
Consolidation Exercise 8A
Level 1
In each of the following figures, QRS is a straight line. Find the unknowns in the figures.
[Nos. 1−
−6]
1.
2.
3.
4.
5.
6.
Find the unknown(s) in each of the following figures. [Nos. 7−
−14]
40
7.
8.
9.
10.
11.
12.
41
13.
14.
15. In the figure, ADB, AEC and DEF are straight lines.
Find x.
16. In the figure, QTR is a straight line. Find y.
Level 2
Find the unknown(s) in each of the following figures. [Nos. 17−
−20]
17.
18.
42
19.
20.
21. In the figure, UYZ and UXW are straight lines.
Is UZ parallel to VW ? Explain your answer.
22. Find x in the figure.
43
23. Refer to the figure. Express a in terms of x and y.
24. In the figure, MOL and NOK are straight lines.
Express d in terms of a, b and c.
25. In the figure, HAOED and BCOGF are straight lines. Find
the sum of all the marked angles.
44
Consolidation Exercise 8A (Answer)
1.
3.
5.
7.
8.
10.
12.
13.
14.
90°
18°
25°
a = 36°, b = 78°
137°
40°
28°
p = 80°, q = 40°
x = 116°, y = 21°
2.
4.
6.
15.
17.
18.
19.
20.
21.
23.
24.
25.
142°
62°
37°
9. 79°
11. 47°
45
31°
123°
x = 26°, y = 78°
x = 37°, y = 69°
x = 39°, y = 61°
yes
a=x+y
d=a+b−c
360°
16. 68°
22. 49°
F2B: Chapter 8B
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 5
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 6
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 7
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 8
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 9
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 10
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
8B Level 1
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
46
Teacher’s
Signature
___________
(
)
Maths Corner Exercise
8B Level 2
Maths Corner Exercise
8B Level 3
Maths Corner Exercise
8B Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
47
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
8.1B
Lesson Worksheet 8B
(Refer to §8.1B−C)
Isosceles Triangles
Names of different parts of an isosceles triangle:
Equal sides
I.
Vertical angle
Base
Base angles
Properties of Isosceles Triangles
A
The two base angles of an isosceles triangle are equal.
i.e. In
△ABC, if AB = AC,
A
then ∠B = ∠C.
[Reference: base ∠s, isos.
△]
B
Example 1
Find x in the figure.
C
B
Instant Drill 1
Q
P
C
Find a in the figure.
x
D
40°
Sol
1.
∵
∴
R
PQ = QR
x = 40°
base ∠s, isos.
Find y in the figure.
△
Sol
2.
B
67°
∵
∴
F
DF =
a=
Find x in the figure.
R
x
3
2
y
C
2
A
3
65°
P
3.
In the figure, ACD is a straight line. Find p and
4.
Q
b
Find p
first.
q
B
Q
In the figure, TP // QR. Find a and b.
A
q.
E
a
30°
p
a
C
140°
R
T
56°
P
D
→
○ Ex 8B 1−4
→
○ Ex 8B 16−18, 20
48
In
△ABC with AB = AC and D is a point on BC, if
any one of the conditions below is true, then the
other two conditions must also be true.
(i) ∠BAD = ∠CAD
(ii) BD = DC
(iii) AD ⊥ BC
[Reference: property of isos.
△]
Example 2
Instant Drill 2
In the figure, D is a point on BC such that
In the figure, M is a point on PR such that
BD = DC. Find p and q.
QM ⊥ PR. Find x and y.
Q
A
20°
p
q
Sol In
∵
∴
C
Sol In
D
△ABC,
17°
x
P
y
M
∵
∴
B
AB = AC and BD = DC.
△
property of isos. △
p = 20°
4
△PQR,
R
=
and
.
property of isos.
AD ⊥ BC
i.e. q = 90°
5.
In the figure, EGF is a straight line. DG is the
6.
and q.
angle bisector of ∠EDF. Find x and y.
D
In the figure, M is the mid-point of YZ. Find p
Y
i.e. ∠EDG =
∠
10
i.e. YM =
MZ
q
p
10
X
30°
M
y
E
x G
3
F
Z
→
○ Ex 8B 10−12
49
II.
Conditions for a Triangle to be Isosceles
In
△ABC,
if ∠B = ∠C,
then AB = AC.
[Reference: sides opp. eq. ∠s]
Example 3
Determine whether
△ABC is an isosceles triangle.
Instant Drill 3
Determine whether
B
C
63°
△ABC is an isosceles triangle.
B
Find ∠A first.
126°
27°
63°
Sol
∵
∴
i.e.
7.
∠A = ∠B = 63°
C
A
A
∠A + (
Sol
CA = CB
)+(
) =(
sides opp. eq. ∠s
△ABC is an isosceles triangle.
Find x and y in the figure.
△
58°
5.3
8.
B
5
In the figure, DBC is a straight line. Find p and
C
q.
4
y
64°
x
∠A =
Are there any
equal angles?
A
Is ABC
isosceles?
If yes, which two
sides are equal?
)
A
p
67°
q
C
113°
B
D
→
○ Ex 8B 5, 6
9.
Refer to the figure. Find n, and hence determine whether
△ABC is an
B
isosceles triangle.
70°
C
n
40°
A
D
→
○ Ex 8B 19, 22
50
8.1C
Equilateral Triangles
Each of the three interior angles of an equilateral triangle is 60°.
i.e. In the figure,
if
△ABC is an equilateral triangle,
then ∠A = ∠B = ∠C = 60°.
[Reference: property of equil.
△]
△
Example 4
Instant Drill 4
Find x in the figure.
In the figure,
C
7 cm
A
△ABC is an equilateral triangle. Find
y.
A
x
7 cm
25°
7 cm
∵ AB = BC = AC
∴ △ABC is an equilateral triangle.
∴ x = 60°
property of equil. △
10. In the figure,
△ABC is an equilateral triangle.
y
B
B
Sol
△
Conversely, for ABC,
if ∠A = ∠B = ∠C = 60°,
then ABC is an
equilateral triangle.
Sol
∵ △ABC is an equilateral triangle.
∴ ∠ABC = _______ property of equil. △
11. In the figure,
Find m and n.
C
△ABC is an equilateral triangle
and M is a point on AB. Find p and q.
C
B
30°
A
n+1
m
p
q
A
M
8
2n − 3
B
C
→
○ Ex 8B 7−9
51
△
Does ABC have
the properties of
isosceles triangles?
12. In the figure, BDC and DAE are straight lines.
△ABC is an equilateral triangle. Find p and q.
C
D
p
13. In the figure, AED and BEC are straight lines.
△ABC is an equilateral triangle. Find x and y.
B
Find ∠CBA
first.
D
E
y
x
B
A
40°
25°
C
q
A
E
→
○ Ex 8B 13−15
→
○ Ex 8B 21
Level Up Question
14. Refer to the figure.
A
(a) Find x.
(b) Is
△ABC an equilateral triangle? Explain your answer.
Consider the sizes of its interior
angles.
60°
D
x
75°
C
52
B
New Century Mathematics (2nd Edition) 2B
8
Angles in Triangles and Polygons
Consolidation Exercise 8B
Level 1
Find the unknown(s) in each of the following figures. [Nos. 1−
−9]
1.
4.
7.
2.
3.
6.
5.
8.
9.
Find the unknowns in each of the following figures. [Nos. 10−
−13]
10.
11.
53
12.
13.
Find the unknown(s) in each of the following figures. [Nos. 14−
−17]
14.
15.
16.
17.
18. In the figure, JKL is a straight line.
(a) Is △ KLM an isosceles triangle? Explain your answer.
(b) If a + b = 180°, is △ KLM an equilateral triangle?
Explain your answer.
Level 2
Find the unknown(s) in each of the following figures. [Nos. 19−
−22]
19.
20.
54
21.
22.
23. In the figure, QRS is a straight line, ∠QUR = 30°, UQ = US,
QP // ST and UR ⊥ QS. Find a.
24. In the figure, QUP, QRS, PTR and UTS are straight lines.
Is △ PQR an equilateral triangle? Explain your answer.
25. In the figure, RTP is a straight line, SR = SP and
ST // PQ.
(a) Find a and b.
(b) Is △ PRS an equilateral triangle? Explain your answer.
26. In the figure, ADB is a straight line and AD = BD = CD.
(a) Express ∠CDB in terms of x.
(b) Is AC perpendicular to BC? Explain your answer.
27. In the figure, △ ABC is an equilateral triangle.
(a) Find x.
(b) Is △ ADB an isosceles triangle? Explain your answer.
28. In the figure, ABFE is a square. △ BCF and △ EFD are
equilateral triangles. Find ∠FCD.
55
Consolidation Exercise 8B (Answer)
1.
2.
3.
4.
5.
6.
7.
9.
10.
11.
12.
13.
14.
16.
17.
18.
19.
20.
21.
22.
23.
25.
h = 42°, k = 96°
m = 71°, n = 38°
66°
y = 50°, z = 3
94°
a = 78°, b = 51°
15
8. 2
120°
h = 13, k = 32°
a = 90°, b = 38°
x = 8, y = 90°
y = 30°, z = 105°
79°
15. 35°
a = 58°, b = 116°
x = 30°, y = 60°
(a) yes
(b)
66°
y = 104°, z = 294°
48°
x = 70°, y = 95°
15°
24.
(a) a = 45°, b = 30°
(b) yes
26. (a) ∠CDB = 2x (b)
27. (a) 15°
(b)
28. 15°
56
yes
yes
yes
yes
F2B: Chapter 8C
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 11
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 12
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 13
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 14
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 15
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
8C Level 1
Maths Corner Exercise
8C Level 2
Maths Corner Exercise
8C Level 3
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
57
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Maths Corner Exercise
8C Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
58
Teacher’s
Signature
Mark:
_________
___________
(
)
Book 2B
8.2A
Lesson Worksheet 8C
(Refer to §8.2A)
Sum of Interior Angles of a Polygon
The sum of interior angles of an n-sided polygon is (n − 2) × 180°.
This property holds
for both convex and
concave polygons.
[Reference: ∠ sum of polygon]
1.
Fill in the blanks.
(a) Sum of interior angles of a 14-sided polygon
=(
−
) × 180°
∠ sum of
polygon
=
(b) sum of interior angles of a 22-sided polygon
= _____________________
______________
=
→
○ Ex 8C 1−3
Instant Drill 1
Example 1
Find y in the figure.
Find x in the figure.
A
How many sides
does this
polygon have?
B
y
B
x
A
C
110°
130°
Sol
∵
∴
60°
C
110°
D
ABCD is a quadrilateral.
D
Sol
∵
∴
3.
Find n in the figure.
y + 90° +
= (4 − 2) × 180° ∠ sum of
130° + 60°
polygon
100°
E
ABCDE is a
.
y + 280° = 360°
y = 80°
2.
Find m in the figure.
A
F
m
A
140°
B
n
145°
110° E
120°
F
G
B 136°
150°
D
130°
113°
C
C
59
140°
115°
D
E
→
○ Ex 8C 4−9
4.
Find p in the figure.
5.
A
B
In the figure, ABF is a straight line. Find θ.
D
F
E
150° 110°
p
θ
160°
120°
C
126°
97°
F
B
E
45°
p
C
150°
A
D
→
○ Ex 8C 10−12
Example 2
Instant Drill 2
The figure shows a regular decagon. Find x.
The figure shows a regular 16-sided polygon. Find
x
A decagon has
_____ sides.
Sol Sum of interior angles of a regular decagon
= (10 − 2) × 180°
y.
y
Sol Sum of interior angles of a regular 16-sided
polygon
∠ sum of polygon
=
= 1 440°
∵
All the interior angles of a regular
polygon are equal.
∴
x=
1 440°
10
= 144°
6.
Find the size of each interior angle of a regular 20-sided polygon.
→
○ Ex 8C 13−15
7.
If the sum of interior angles of an n-sided
8.
polygon is 1 620°, find the value of n.
Set up an equation
60
according to the
given conditions.
If the sum of interior angles of an n-sided
polygon is 1 980°, find the value of n.
→
○ Ex 8C 16−18
‘Explain Your Answer’ Question
9.
The figure shows a part of a regular polygon. Steven claims
170° 170° 170°
that the polygon has 36 sides. Do you agree ? Explain your answer.
Sum of interior angles of a regular 36-sided polygon
=
Level Up Question
10. Find the unknown(s) in each of the following figures.
(a)
E
A
113°
275°
B
D
D
(b)
3x 2x
150°
C
104°
x
E
142°
123°
C
B
x
A
New Century Mathematics (2nd Edition) 2B
61
y F
115°
G
8
Angles in Triangles and Polygons
Consolidation Exercise 8C
Level 1
Find the sum of interior angles of each of the following polygons. [Nos. 1−
−3]
1.
Nonagon
2.
14-sided polygon
Find the unknown in each of the following figures. [Nos. 4−
−9]
4.
6.
8.
5.
7.
9.
62
3.
20-sided polygon
Find the unknown in each of the following figures. [Nos. 10−
−12]
10.
11.
12.
Find the size of each interior angle of the following regular polygons. [Nos. 13−15]
13. Regular 10-sided polygon
14. Regular 24-sided polygon
15. Regular 36-sided polygon
In each of the following, the sum of interior angles of a polygon is given. Find the number of sides of the
polygon. [Nos. 16−
−18]
16. 1 080°
17. 3 060°
18. 4 680°
Level 2
Find the unknown(s) in each of the following figures. [Nos. 19−
−24]
19.
20.
63
21.
22.
23.
24.
25. It is given that the sum of interior angles of an n-sided polygon is 4 times that of a heptagon. Find the
value of n.
26. Find the number of sides of a regular polygon if the size of each of its interior angles is
(a) 157.5°,
(b) 165.6°,
(c) 172°.
27. (a) Find the size of each interior angle of a regular pentagon.
(b) If the size of each interior angle of a regular n-sided polygon is 1.5 times that of a regular
pentagon, find the value of n.
28. Is it possible that the size of each interior angle of a regular polygon is 175°? Explain your answer.
29. In the figure, ABCDEF is a regular hexagon and AHGF is a square. Find
x and y.
64
Consolidation Exercise 8C (Answer)
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
1 260°
3 240°
170°
125°
159°
69°
144°
170°
19
24°
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
2 160°
53°
73°
81°
66°
74°
165°
8
28
58°
65
58°
x = 36°, y = 25°
x = 62°, y = 123°
a = 63°, b = 98°
22
(a) 16
(b) 25
(c) 45
(a) 108°
(b) 20
yes
x = 75°, y = 45°
F2B: Chapter 8D
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 16
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 17
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 18
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
8D Level 1
Maths Corner Exercise
8D Level 2
Maths Corner Exercise
8D Level 3
Maths Corner Exercise
8D Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
66
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
8.2B
Lesson Worksheet 8D
(Refer to §8.2B)
Sum of Exterior Angles of a Polygon
Each of the following shows a set of exterior angles of the given polygon.
A polygon has more
than one set of
exterior angles.
e.g.
Triangle
1.
Quadrilateral
Quadrilateral
Pentagon
Draw a set of exterior angles of each of the following polygons.
Step 1: Extend a side of the
polygon at each vertex.
Step 2: Mark the exterior
angles.
(Part (a) has been done for you as an example.)
(a)
(b)
(c)
(d)
The sum of exterior angles of a convex polygon is 360°.
[Reference: sum of ext. ∠s of polygon]
Example 1
Instant Drill 1
Find x in the figure.
Find y in the figure.
D
105°
y
x
E
C
80°
Sol
A
A
B
x + 80° + 60° + 90° = 360° sum of ext. ∠s
x + 250° = 360°
C
50°
60°
Sol y +
of polygon
D
45°
B
65°
=
=
x = 110°
67
2.
3.
Find m in the figure.
m
Find n in the figure.
B
C
122°
45°
83°
C
A
142°
4.
B
5.
Find p in the figure.
A 40°
A
Find x and y in the figure.
C
81°
F
B
116°
y
77°
p
x
p
84°
82°
140°
D
E
C
D
n
A
B
D
→
○ Ex 8D 1−9
6.
Find the size of each exterior angle of a
7.
regular octagon.
Find the size of each exterior angle of a
regular 18-sided polygon.
An regular octagon has ____ sides
and
has (equal / unequal) exterior
Sum of exterior angles =
∵
All the exterior angles of a regular
polygon are
.
∴ Each exterior angle
=
→
○ Ex 8D 10−12
8.
If the size of each exterior angle of a regular n-
9.
68
It is given that the size of an exterior angle of a
sided polygon is 72°, find the value of n.
regular polygon is 10°. Find the number of
sides of the polygon.
Sum of exterior angles =
n×(
)=
=
→
○ Ex 8D 13−15
Level Up Questions
10. Find x in the figure.
79°
x
2x + 9°
x
46°
62°
11. Find x and y in the figure.
29°
110°
y 48°
x
15°
69
New Century Mathematics (2nd Edition) 2B
8
Angles in Triangles and Polygons
Consolidation Exercise 8D
Level 1
Find the unknown in each of the following figures. [Nos. 1−
−6]
1.
2.
3.
5.
4.
6.
Find the unknown(s) in each of the following figures. [Nos. 7−
−9]
7.
8.
9.
70
Find the size of each exterior angle of the following regular polygons. [Nos. 10−
−12]
10. Regular nonagon
11. Regular 18-sided polygon
12. Regular 36-sided polygon
In each of the following, the size of each exterior angle of a regular polygon is given. Find the number of
sides of the polygon. [Nos. 13−
−15]
13. 30°
14. 24°
15. 8°
16. It is given that the size of each interior angle of a regular polygon is 135°.
(a) Find the size of each exterior angle of the regular polygon.
(b) Find the number of sides of the regular polygon.
Level 2
Find the unknown(s) in each of the following figures. [Nos. 17−
−19]
17.
18.
19.
20. In a regular polygon, if the size of each interior angle is 2.5 times the size of each exterior angle, find
the number of sides of the polygon.
21. In the figure, ABCD is a part of a regular polygon. AB and DC are
produced to meet at P.
(a) Find x.
(b) Find the number of sides of the regular polygon.
71
22. In the figure, PQRS is a part of a regular polygon. PQTU and
SRT are straight lines. If RT ⊥ QU, find the number of sides
of the regular polygon.
23. In the figure, ABCDEFGHIJKL is a regular 12-sided polygon.
BAOP and OLK are straight lines. Find ∠POL.
24. In the figure, ABCDE is a part of a regular n-sided polygon.
AB and ED are produced to meet at F such that BC = DC = FC.
If ∠AFE = 72°, find the value of n.
72
Consolidation Exercise 8D (Answer)
1.
3.
5.
7.
8.
10.
12.
92°
63°
80°
a = 71°, b = 63°
111°
40°
10°
2.
4.
6.
14.
16.
17.
19.
21.
22.
24.
142°
41°
46°
9. 152°
11. 20°
13. 12
73
15
(a) 45°
18°
a = 54°, b = 48°
(a) 72°
8
10
15.
(b)
18.
20.
(b)
23.
45
8
18°
7
5
60°
F2B: Chapter 8E
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 19
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
8E Level 1
Maths Corner Exercise
8E Level 2
Maths Corner Exercise
8E Level 3
Maths Corner Exercise
8E Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
74
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
Lesson Worksheet 8E
(Refer to §8.3A−C)
[In this worksheet, only compasses and a straight edge can be used for constructions.]
8.3A
Construction of Angle Bisector
B
Consider a given angle ∠AOB.
S
If OS bisects ∠AOB, i.e. ∠AOS = ∠BOS,
then OS is the angle bisector of ∠AOB.
A
O
1. Follow the steps below to construct the angle bisector of ∠AOB in the figure.
Step 1: Using O as centre and a radius
of suitable length, draw an arc
to cut OA at a point P, and cut
OB at another point Q.
B
Step 2: Using P and Q as centres and a
1
radius longer than PQ, draw
2
two arcs to meet at a point S.
Step 3: Join OS. The line OS obtained is
the angle bisector of ∠AOB.
A
O
2. Bisect each of the following marked angles.
(a)
(b)
F
J
E
H
D
G
○→ Ex 8E 1, 2
75
8.3B Construction of Perpendicular Bisector
P
Consider a given line segment AB.
If PQ is perpendicular to AB and bisects AB,
A
i.e. PQ ⊥ AB and AN = NB,
then PQ is the perpendicular bisector of AB.
N
B
Q
3. Follow the steps below to construct the perpendicular bisector of the line segment AB in the
figure.
Step 1: Using A as centre and a radius
longer than
1
AB, draw an arc on
2
each side of the line segment AB.
Step 2: Using B as centre and the same
radius as in 1, draw an arc on
each side of AB such that they
cut the two arcs drawn in 1 at
two points P and Q.
B
A
Step 3: Join PQ. The line PQ obtained
is the perpendicular bisector of
AB.
4. Construct the perpendicular bisector of each of the following marked line segments.
(a)
(b)
C
G
D
H
○→ Ex 8E 3, 4
76
8.3C
I.
Constructions of Some Special Angles
Construction of Angles of 90° and 45°
5. Follow the steps below to construct angles of 90° and 45°.
Step 1: Draw a straight angle AOB.
(This step has been done for you.)
straight angle
= 180°
Step 2: Bisect ∠AOB into ∠AOC and
∠BOC.
Then ∠AOC = ∠BOC = 90°.
Step 3: Bisect ∠BOC into ∠BOD and
∠COD.
Then ∠BOD = ∠COD = 45°.
A
O
B
II. Construction of Angles of 60° and 30°
6. Follow the steps below to construct angles of 60° and 30°.
Step 1: Draw a line segment AB of
suitable length.
Step 2: Using A and B as centres and AB
as radius, draw two arcs to meet
at a point C.
Step 3: Join AC.
Then we have ∠CAB = 60°.
Step 4: Bisect ∠CAB into ∠CAD and
∠BAD.
Then ∠CAD = ∠BAD = 30°.
Which type of
triangle can we
get if we join BC?
77
Example 1
Instant Drill 1
Construct an angle of 15° by using compasses
and a straight edge only.
Construct an angle of 22.5° by using compasses
and a straight edge only.
Sol
Sol
30° ÷ 2 = 15°
Step 1: Construct an angle of 60°.
Step 2: Obtain an angle of 30° by bisecting
the angle of 60°.
Step 3: Obtain an angle of 15° by bisecting
the angle of 30°.
C
(
)° ÷ 2 = 22.5°
Step 1: Construct an angle of (
)°.
Step 2: Obtain an angle of (
)° by
bisecting the angle in 1.
Step 3: Obtain an angle of 22.5° by bisecting
the angle in 2.
∠BAC = 60°,
∠BAD = ∠CAD = 30°,
∠BAE = ∠DAE = 15°
D
E
15°
A
B
In the figure, ∠BAE = 15°.
In the figure,
Level Up Question
7. The figure below shows a line segment AB.
(a) Construct an angle ∠ABC of size 60° such that C is a point lying above AB.
(b) Construct an angle ∠ABD of size 45° such that D is a point lying below AB.
(c) Find the size of ∠CBD.
(a), (b)
B
A
(c) ∠CBD =
78
.
○→ Ex 8E 5−7
New Century Mathematics (2nd Edition) 2B
8
Angles in Triangles and Polygons
Consolidation Exercise 8E
[In this exercise, use compasses and a straight edge only for constructions.]
Level 1
1.
Bisect each of the following marked angles.
(a)
(b)
(c)
2.
Construct the perpendicular bisector of each of the following marked line segments.
(a)
(b)
(c)
79
Construct each of the following angles. [Nos. 3−
−5]
3.
270°
4.
5.
6.
In the figure, O is a point on AB. Construct a line passing through O and
120°
330°
perpendicular to AB.
7.
(a) Construct the perpendicular bisectors of the line segments
PQ and QR in the figure respectively. Mark the point of
intersection of the two perpendicular bisectors as O.
(b) Use O as centre and OP as radius to draw a circle. Does the
circle pass through the points Q and R?
Level 2
8.
Draw any line segment and divide it into 4 equal parts.
9.
Draw any angle and divide it into 4 equal angles.
10. The figure shows a circle with centre O. Divide the circle into
8 equal parts.
11. (a) Construct a regular pentagon.
(b) Hence, construct a regular decagon.
12. (a) Construct any triangle ABC. Then, construct three line segments AA′, BB′ and CC′, where A′, B′
and C′ are the mid-points of BC, AC and AB respectively.
(b) Do the three line segments AA′, BB′ and CC′ constructed in (a) intersect at one point?
13. Refer to the figure.
(a) Construct a line passing through O and perpendicular to AB.
(b) Hence, construct a line passing through O and parallel to AB.
80
Construct each of the following figures. [Nos. 14−
−15]
(The sizes of the figures constructed can be different from the given figures.)
14.
15.
81
Consolidation Exercise 8E (Answer)
7.
(b) yes
12. (b) yes
82
F2B: Chapter 9
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 1
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 2
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 3
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 4
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 5
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 6
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 7
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
83
Book Example 8
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
9 Level 1
Maths Corner Exercise
9 Level 2
Maths Corner Exercise
9 Level 3
Maths Corner Exercise
9 Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
84
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 2B
9.2A
(I)
Lesson Worksheet 9
(Refer to §9.2)
Proving Geometric Properties Relating to Intersecting and Parallel
Lines
Intersecting lines
Q
(a)
(b)
a
R
O
S
c
b
a
P
P
R
O
d
(c)
a
b
b
c
Q
If POR is a straight line,
If POQ and ROS are
a + b + c = 360°
then a + b = 180°.
[Reference:
straight lines,
[Reference: ∠s at a pt.]
then a = b and c = d.
adj. ∠s on st. line]
[Reference: vert. opp. ∠s]
Example 1
In the figure, PQR is a straight line. Prove that
Instant Drill 1
x + y = 130°.
AO ⊥ OC.
Refer to the figure. If x + y = 270°, prove that
S
P
T
50°
y
x
A
x O
R
Q
Statement
Reason
Sol x + 50° + y = 180°
x + y = 130°
C
y
B
Facts given in
the question
Sol
x + y = 270°
∠AOC + (
)+(
adj. ∠s on st. line
∠AOC + (
given
)=(
)
)=(
=
)
__________
State the
reason(s)
here.
∴
1. In the figure, AOB and COD are straight
lines. Prove x = y + 10°.
A
P
C
y
30°
O
2. In the figure, POR is a straight line and
a = b. Prove that x = y.
Q
20° Q
x
B
Consider the
straight
angles
formed at O.
D
x
P
O
y
a
b
R
S
○→ Ex 9 1−4
85
If the sum of two adjacent angles ∠POQ and ∠ROQ is 180°,
then POR is a straight line.
This theorem can be
used to determine
straight lines.
Q
i.e. In the figure,
if a + b = 180°,
then POR is a straight line.
b
a
P
R
O
[Reference: adj. ∠s supp.]
Example 2
Instant Drill 2
Refer to the figure. Prove that AOC is a straight Refer to the figure. Prove that WOZ is a
line.
straight line.
B
X
Y
142°
38°
A
Sol ∠AOC = ∠AOB + ∠BOC
= 180°
∴ AOC is a straight line.
T
A
D
xx
O
y
Z
4. In the figure, AOB is a straight line.
Q
U
52°
O
adj. ∠s supp.
3. Refer to the figure. Prove that POS is a
straight line.
y
W
55°
Sol ∠WOZ
=
= 142° + 38°
P
73°
C
O
x
O
5x
30°
C
B
(a) Find x.
(b) Prove that COD is a straight line.
R
S
○→ Ex 9 5, 6
86
(II) Parallel lines
(a)
a
A
C
b
B
(b) A
D
C
a
b
B
(c) A
D
C
a
b
B
D
If AB // CD,
If AB // CD,
If AB // CD,
then a = b.
then a = b.
[Reference:
[Reference:
then a + b = 180°.
[Reference:
corr. ∠s, AB // CD]
alt. ∠s, AB // CD]
int. ∠s, AB // CD]
Example 3
In the figure, AFB and EFD are straight lines.
Instant Drill 3
In the figure, AB // CD and ADE is a straight
AB // CD. Prove that x + y = 360°.
line. Prove that x + y = 180°.
E
x
B
B
F
A
D
C
Sol ∠EDC = x
x
D
y
corr. ∠s, AB // CD
∠EDC + y = 360°
∴ x + y = 360°
A
C
y
Sol ∠ADC =
∠s at a pt.
5. In the figure, BA // DC. Prove that
x + y = 180°.
6. In the figure, FDBG is a straight line.
AB // CD and EC // FG. Prove that x = y.
C
A
G
x
A
x
B
E
D
2x
y
C
B
D
y
E
E
F
○→ Ex 9 7−10
87
(a)
a
A
b
C
B
(b) A
D
C
a
b
B
(c) A
D
C
B
a
b
D
If a = b,
If a = b,
then AB // CD.
then AB // CD.
If a + b = 180°,
then AB // CD.
[Reference:
[Reference:
[Reference:
corr. ∠s equal]
alt. ∠s equal]
int. ∠s supp.]
Example 4
In the figure, CDE and FAD are straight lines.
Instant Drill 4
In the figure, AF and CD intersect at E. If
If x + y = 180°, prove that AB // CE.
x + y = 180°, prove that AB // CD.
A
F
x
D
B
x
B
A
y
C
E
E
D
Sol ∠FDE + y = 180°
adj. ∠s on st. line
y F
C
Sol
∠FDE = 180° − y
x + y = 180°
given
x = 180° − y
∴ ∠FDE = x
∴ AB // CE
corr. ∠s equal
7. In the figure, BC // DE. If x + y = 360°,
prove that AB // CD.
y
D
B
x
C
8. In the figure, CD // EF. AQB and QRS are
straight lines. If x = y, prove that AB // CD.
A
x
C
E
Q
B
R
D
y
E
S
F
A
○→ Ex 9 11−13
88
9.2B
Proving Geometric Properties Relating to Triangles
(a)
(b)
a + b + c = 180°
[Reference: ∠ sum of
c=a+b
[Reference: ext. ∠ of
△]
Example 5
Refer to the figure. Prove that x + y = 80°.
△]
Instant Drill 5
In the figure, ACD is a straight line. Prove that
B
y − x = 90°.
y
A
y
A
100°
x
D
C
x
C
B
Sol x + y + 100° = 180°
x + y = 80°
∠ sum of
△
9. In the figure, CDA is a straight line. Prove
that x + y = 90°.
Sol x + (
)=(
10. In the figure, BCDF is a straight line.
Prove that AC // ED.
B
A
x y
C
D
E
44°
40°
50°
__________
)
A
B
52°
96°
C
89
D
F
11. Refer to the figure. Prove that ACD is a
straight line.
B
12. In the figure, EDB is a straight line.
C
E
130°
68°
E
66°
53°
A
55°
D
D
40°
63°
27°
A
B
(a) Prove that EB ⊥ CB.
(b) Prove that EDBA is a straight line.
C
○→ Ex 9 14–19
Level Up Question
13. In the figure, AB // ED. Prove that z = x + y.
B
Add a suitable
line to relate x, y
and z.
x
A
90
D
z
C
y
E
New Century Mathematics (2nd Edition) 2B
9
Introduction to Deductive Geometry
Consolidation Exercise 9
Level 1
1.
In the figure, RO ⊥ PO. If x = y, prove that SO ⊥ QO.
2.
In the figure, AOE is a straight line, ∠BOC = 60° and
∠DOE = 30°. Prove that p + q = 90°.
3.
In the figure, KOL and MON are straight lines.
If a + b + c = 180°, prove that POQ is a straight line.
4.
In the figure, WO ⊥ OV. If a + b = 180°, prove that TO ⊥ UO.
5.
Refer to the figure.
(a) Find y.
(b) Prove that AOC is a straight line.
6.
In the figure, ABCD is a straight line and AD // EF.
If x = y, prove that a = b.
7.
In the figure, PUQ, RVS and TUVW are straight lines, and
PQ // RS. Prove that a = b.
91
8.
Refer to the figure. Prove that AB // DC.
9.
In the figure, PQRS is a straight line. If q + r = 180°, prove TQ //
UR.
10. In the figure, KL // MN. Prove that LP // QM.
11. In the figure, ABC is a straight line and AC // DE. Prove that AC
// FG.
12. In the figure, JKL and MKN are straight lines.
If m + k = 90°, prove that JL ⊥ ML.
13. In the figure, PSQ and PTR are straight lines.
Prove that a = b + 10°.
14. In the figure, KLM is a straight line. Prove that JK // NM.
15. In the figure, BCD is a straight line. If d = a + b, prove that AC //
DE.
92
Level 2
16. In the figure, AOD is a straight line. If ∠AOC = 129° and
∠BOD = 141°, prove that BO ⊥ CO.
17. In the figure, AOE, BOF and COG are straight lines,
∠AOB = ∠HOG and ∠BOC = ∠DOE. If GO ⊥ EO,
prove that HOD is a straight line.
18. In the figure, ∠NOM = 60°. Prove that JOL is a straight line.
19. In the figure, ABC and EBF are straight lines. AC // FD and EF //
CD. Prove that ∠ABE = ∠FDC.
20. In the figure, PQR and UVW are straight lines. It is given that
PR // ST // UW and QS // TV. Prove that a = b.
21. In the figure, BA // CE. If a = b, prove that BF // CD.
93
22. In the figure, BCDE is a straight line. If a + b + e + f = 180°, prove
that AC // FD.
23. Refer to the figure. If e + f = g, prove that DE // HG.
24. Refer to the figure. Prove that QP // ST.
25. In the figure, KJ // MN. If k + m = 270°, prove that KL ⊥ ML.
26. In the figure, BA // CD. BE and CE are the angle bisectors of
∠ABC and ∠BCD respectively. Prove that BE ⊥ CE.
27. In the figure, QOT, POS and RST are straight lines.
(a) Prove that QP // RT.
(b) If QR // PS and ∠PQR = 148°, is ∠POT a right angle?
Explain your answer.
94
Consolidation Exercise 9 (Answer)
5.
(a) 36°
27. (b) no
95