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Chapter 20

Angles and triangles
20
...
1)
...
This chapter involves the measurement of
angles and introduces types of triangle
...
2

c

Angular measurement

h e
g f

R

An angle is the amount of rotation between two straight
lines
...

If a circle is divided into 360 equal parts, then each part
is called 1 degree and is written as 1◦
i
...

or

1 revolution = 360◦
1
1 degree is
th of a revolution
360

Some angles are given special names
...

• An angle equal to 90◦ is called a right angle
...

• Any angle greater than 180◦ and less than 360◦ is
called a reflex angle
...

• If two angles add up to 90◦ they are called complementary angles
...

• Parallel lines are straight lines which are in the same
plane and never meet
...
1
...
1016/B978-1-85617-697-2
...
1

With reference to Figure 20
...
Such pairs of
angles are called vertically opposite angles
...
Such pairs of
angles are called corresponding angles
...
Such pairs of angles are called
alternate angles
...
Such pairs of
angles are called interior angles
...
2
...

i
...


1 degree = 60 minutes

which is written as

1◦ = 60
...
41◦ 29
29◦
= 41
...

1 minute further subdivides into 60 seconds,
1 minute = 60 seconds

i
...


which is written as

1 = 60
...

43◦ 29
+ 27◦ 43
71◦ 12
1◦

(i) 29 + 43 = 72
(ii) Since 60 = 1◦, 72 = 1◦12

(Notice that for minutes, 1 dash is used and for seconds,
2 dashes are used
...


20
...
2 Radians and degrees
One radian is defined as the angle subtended at the centre
of a circle by an arc equal in length to the radius
...
)
With reference to Figure 20
...

(iv) 43◦ + 27◦ + 1◦ (carried) = 71◦
...

This answer can be obtained using the
follows
...

1
...
Press ◦ ’ ’ ’
◦
4
...
Press +
6
...
Press ◦ ’ ’ ’
8
...

10
...

Problem 2
...

(ii) 1◦ or 60 is ‘borrowed’ from the degrees column,
which leaves 83◦ in that column
...

(iv) 83◦ − 56◦ = 27◦, which is placed in the degrees
column
...
2

When s is the whole circumference, i
...
when
s = 2πr,
s
2πr
θ= =
= 2π
r
r
In one revolution, θ = 360◦
...
e
...
30◦
1 rad =
π

Here are some worked examples on angular measurement
...

3
...
Enter 84
2
...
Press ’ ’ ’
5
...

◦
7
...
Enter 39
9
...
Press =
Answer = 27◦ 34

calculator as
Enter 13
Enter 56
Press ◦ ’ ’ ’

Thus, 84◦ 13 − 56◦39 = 27◦ 34
...


Evaluate 19◦ 51 47 + 63◦ 27 34
19◦ 51 47
+ 63◦ 27 34
83◦ 19 21
1◦ 1

Angles and triangles
(i) 47 + 34 = 81
(ii) Since

60

=

1 , 81

= 1 21

(iii) The 21 is placed in the seconds column and 1
is carried in the minutes column
...
Enter 63
4
...
Press =

0
...
753 × 60 = 45
...
18 = 0
...


This answer can be obtained using the calculator as
follows
...
Enter 51
1
...
Press ◦ ’ ’ ’
◦
4
...
Enter 47
6
...
Press +
8
...
Press ◦ ’ ’ ’
◦
10
...
Press ’ ’ ’
12
...
Press ◦ ’ ’ ’
14
...

Problem 4
...
45◦ by calculator
60
27◦
39◦27 = 39
= 39
...

3
...
Enter 39
2
...
Press =
6
...
Press ’ ’ ’
◦
Answer = 39
...
Convert 63◦ 26 51 to degrees in
decimal form, correct to 3 decimal places
51
= 63◦ 26
...
85◦
= 63
...
85 = 63
60
63◦ 26 51 = 63
...

This answer can be obtained using the calculator as
follows
...
Enter 26
6
...
4475◦

Problem 6
...
753◦ to degrees, minutes
and seconds

(v) Since 60 = 1◦, 79 = 1◦19

(vii) 19◦ + 63◦ + 1◦ (carried) = 83◦
...


2
...
Enter 51
8
...
753◦ = 53◦ 45 11
This answer can be obtained using the calculator as
follows
...
Enter 53
...
Press =
Answer = 53◦45 10
...
Press ◦ ’ ’ ’
Now try the following Practice Exercise
Practice Exercise 76 Angular measurement
(answers on page 348)
1
...


Evaluate 76◦ 31 − 48◦37

3
...


Evaluate 41◦ 37 16 + 58◦ 29 36

5
...


Evaluate 79◦26 19 − 45◦58 56 + 53◦ 21 38

7
...


8
...


9
...
952◦ to degrees and minutes
...
Convert 58
...

Here are some further worked examples on angular
measurement
...
State the general name given to the
following angles: (a) 157◦ (b) 49◦ (c) 90◦ (d) 245◦
(a) Any angle between 90◦ and 180◦ is called an
obtuse angle
...

(b) Any angle between 0◦ and 90◦ is called an acute
angle
...

(c)

An angle equal to 90◦ is called a right angle
...

Thus, 245◦ is a reflex angle
...

48◦ 39

Find the angle complementary to

(b) An angle of 180◦ lies on a straight line
...
3(b),
180◦ = 53◦ + θ + 44◦
from which,

θ = 180◦ − 53◦ − 44◦ = 83◦

Problem 11
...
5

If two angles add up to 90◦ they are called complementary angles
...

74◦ 25

108⬚

Find the angle supplementary to

39⬚
␪

58⬚

If two angles add up to 180◦ they are called supplementary angles
...
5

180◦ − 74◦ 25 = 105◦ 35
Problem 10
...
e
...
44 cm

12

Q

9

12


...
e
...
In Figure 20
...
In Figure 20
...
63 cm

X

9
CD
=
, from which
6+9
12
 
9
= 7
...
A rectangular shed 2 m wide and
3 m high stands against a perpendicular building of
height 5
...
A ladder is used to gain access to the
roof of the building
...
58 mm
25
...
43, where AF
is the minimum length of the ladder
...
Hence, triangles
BAD and EDF are similar since their angles are the
same
...
36 mm

Figure 20
...


PQR is an equilateral triangle of side 4 cm
...
If PS is 9 cm, find the length of ST
...
Find the
length of PX
...


In Figure 20
...


AB = AC − BC = AC − DE = 5
...
5 m
AB
BD
2
...
e
...
4 m = minimum distance
Hence, EF = 2
2
...


4
...
5 m

Shed
D

F

E

C

Figure 20
...

Figure 20
...
46, AF = 8 m, AB = 5 m and
BC = 3 m
...

C

B

D

Now try the following Practice Exercise
Practice Exercise 80 Similar triangles
(answers on page 349)
1
...
44, find the lengths x and y
...
46

E

F

Angles and triangles
20
...
48:

To construct any triangle, the following drawing instruments are needed:

A

(a) ruler and/or straight edge

b 5 3 cm

(b) compass
608

(c) protractor

a 5 6 cm

B

(d) pencil
...
48

Here are some worked problems to demonstrate triangle
construction
...

Problem 30
...

(iii) From C measure a length of 3 cm and label A
...


D
G

C

F

Problem 32
...
47

With reference to Figure 20
...

(ii) Set compass to 5 cm and with centre at A describe
arc DE
...

(iv) The intersection of the two curves at C is the vertex of the required triangle
...

It may be proved by measurement that the ratio of the
angles of a triangle is not equal to the ratio of the sides
(i
...
, in this problem, the angle opposite the 3 cm side is
not equal to half the angle opposite the 6 cm side)
...
Construct a triangle ABC such that
a = 6 cm, b = 3 cm and ∠C = 60◦

708
Q

448
5 cm

R

Figure 20
...
49:
(i) Draw a straight line 5 cm long and label it QR
...
Draw QQ
...
Draw RR
...

Problem 33
...
5 cm and
∠X = 90◦

180 Basic Engineering Mathematics
V
Z
P U

centred at Y and set to 6
...


S

(iv) The intersection of the arc UV with XC produced,
forms the vertex Z of the required triangle
...


C
R
B

A

Q
X

A9

Y

Figure 20
...
50:
(i) Draw a straight line 5 cm long and label it XY
...
With compass centred at X make an arc at A and A
...
) With compass centred at A
draw the arc PQ
...
Join the intersection of the arcs, C to X , and a right angle to
XY is produced at X
...
)
(iii) The hypotenuse is always opposite the right angle
...
Using a compass

Now try the following Practice Exercise
Practice Exercise 81 Construction of
triangles (answers on page 349)
In the following, construct the triangles ABC for
the given sides/angles
...


a = 8 cm, b = 6 cm and c = 5 cm
...


a = 40 mm, b = 60 mm and C = 60◦
...


a = 6 cm, C = 45◦ and B = 75◦
...


c = 4 cm, A = 130◦ and C = 15◦
...


a = 90 mm, B = 90◦ , hypotenuse = 105mm
...

The n’th term is: a + (n − 1)d
n
Sum of n terms, Sn = [2a + (n − 1)d]
2

Geometric progression:
If a = first term and r = common ratio, then the geometric progression is: a, ar, ar2 ,
...

4
...

10
...

16
...


Exercise 5 (page 11)

19 kg
2
...
479 mm
−66
5
...
−225
−2136
8
...
£10 7701
−4
11
...
5914
189 g
14
...
$15 333
89
...

3
...

5
...

9
...

9
...

17
...

(a) 8613 kg (b) 584 kg
(a) 351 mm (b) 924 mm
(a) 10 304 (b) −4433
6
...

(a) 8067 (b) 3347
10
...

4
...

8
...


(a) 48 m (b) 89 m
(a) 1648 (b) 1060
18 kg

1
...
14
7
...
88
8
...
1016/B978-1-85617-697-2
...


3 4 1 3 5
, , , ,
7 9 2 5 8
9
10
...
1
15
16
18
...


6
...

21
...
2
3
...
11
8
...

13
...

18
...

11
...

19
...

8
...

16
...


4
...
2

5
...
5

5
12
1
9
...

23
4
...
15

3
28
8
10
...


15
...
400 litres
22
...

15

1
...
59
6
...
−1

2
5

Exercise 6 (page 13)

11
...
7

(a) £1827 (b) £4158

Exercise 3 (page 6)
1
...

5
...

9
...
2

11
...
4
20
17
12
...
−

3
...
2

1
6

3
4
1
9
...
4

13
20
1
10
...


Answers to practice exercises
Exercise 14 (page 25)

Chapter 3

1
...
571
5
...
96
8
...
0871

Exercise 8 (page 17)
1
...
23
8
...


13
20
21
6
...
(a) 1
50
5
...
0
...


4
...
6

7
16

(e) 16

17
80

1
...

7
...

13
...
182
2
...
122
3
...
82
0
...
0
...
2
...
273
8
...
256
9
...
30366
6
1
...
3
...
37
...
2 × 10
14
...
767 ×10
15
...
32 ×106

12
...
6875 13
...
21875 14
...
1875

Exercise 16 (page 27)
1
...
4667

Exercise 9 (page 18)
1
...
18
5
...
297

2
...
785
3
...
38
6
...
000528

2
...
3
6
...
3

4
...
27

3
...
54
7
...
52 mm

4
...
83

13
14

3
...
458

6
...
7083

7
...
2
...
3

1
3

10
...
0776

1
...
9205
5
...
4424
9
...
6992

2
...
7314
6
...
0321
10
...
8452

3
...
9042
7
...
4232

4
...
2719
8
...
1502

Exercise 18 (page 28)

4
...
47
...
385
...
582
...
9 6
...
82
7
...
1
8
...
6
0
...
0
...
1
...
53
...
84 14
...
69
15
...
81 (b) 24
...
00639 (b) 0
...
(a) 8
...
6˙
2400

1
...
995
5
...
6977
9
...
520

Exercise 12 (page 23)
3
...
62
7
...
330

4
...
832
8
...
45

Exercise 13 (page 24)
1
...
25
2
...
0361 3
...
923 4
...
296 × 10−3
5
...
4430 6
...
197 7
...
96 8
...
0549
9
...
26 10
...
832 × 10−6

2
...
782
6
...
92
10
...
3770

3
...
72
7
...
0

4
...
42
8
...
90

Exercise 19 (page 29)
1
...

7
...


Chapter 4

2
...
1
...
12
...


Exercise 17 (page 27)

Exercise 11 (page 20)

1
...
797
5
...
42
9
...
59

1
21
9
...
567
5
...

5
...

13
...

18
...
40
3
...
13459
4
...
9
6
...
4481 7
...
36 × 10−6
9
...
625 × 10−9
10
...
70

Exercise 15 (page 25)

3
125

15
...
28
U1

Exercise 68 (page 140)
1
...
75, 0
...
75, 2
...
75;
1
Gradient =
2
2
...
(a) 6, −3 (b) −2, 4 (c) 3, 0 (d) 0, 7
3
...
(a) 2, − (b) − , −1 (c) , 2 (d) 10, −4
2
3
3
18
3
3
5
6
...
(a) and (c), (b) and (e)
8
...
(1
...
(1, 2)

11
...
4 (d) l = 2
...
P = 0
...
5

13
...
(a) 40◦ C (b) 128 
2
...
5 V

Exercise 66 (page 127)
1
...
5◦C

3
...
25 (b) 12 (c) F = 0
...
99
...
(a) 29
...
31 × 10−6 s
4
...
993 m (b) 2
...
(a) 50◦ C (b) 55
...
30
...
(a) 3
...
46 s
8
...
45 mol/cm3
10
...
(a) 7
...
966 s

(d) 89
...
−0
...
73
5
...
5 m/s (b) 6
...
7t + 15
...
m = 26
...
63
7
...
31 t (b) 22
...
09 W + 2
...
(a) 96 × 109 Pa (b) 0
...
8 × 106 Pa

348 Basic Engineering Mathematics
1
1
(b) 6 (c) E = L + 6 (d) 12 N (e) 65 N
5
5
10
...
85, b = 12, 254
...
5 kPa, 280 K
9
...
(−2
...
2), (0
...
8); x = −2
...
6
10
...
2 or 2
...
75 and −1
...
3 or −0
...
(a) y (b) x 2 (c) c (d) d

2
...
(a) (b) x (c) b (d) c
x
x
1
y
5
...
a = 1
...
4, 11
...
y = 2x 2 + 7, 5
...
x = 4, y = 8 and x = −0
...
5
2
...
5 or 3
...
24 or 3
...
5 or 3
...
(a) y (b)

8
...
a = 0
...
6 (i) 94
...
2

Exercise 75 (page 162)
1
...
0, −0
...
5
2
...
1, −4
...
8, 8
...
x = 1
4
...
0, 0
...
6
5
...
7 or 2
...
x = −2
...
0 or 1
...
x = −1
...

2
...

4
...

6
...

9
...
0012 V2 , 6
...
0, b = 0
...
7, b = 2
...
53, 3
...
0, c = 1
...
y = 0
...
24x
T0 = 35
...
27, 65
...
28 radians

Exercise 72 (page 156)
x = 2, y = 4
x = 3
...
5
x = 2
...
2
a = 0
...
6

Exercise 76 (page 167)
1
...
27◦54
3
...
100◦6 52
◦


◦


5
...
86 49 1 7
...
55◦ 8
...
754◦
9
...
58◦22 52

Exercise 77 (page 169)
1
...
obtuse 3
...
right angle
5
...

3
...

7
...
x = 1, y = 1
4
...
x = −2, y = −3

Exercise 73 (page 160)
1
...
−0
...
6
3
...
9 or 6
...
−1
...
1
5
...
8 or 2
...
x = −1
...
75, −0
...
x = −0
...
6
8
...
63 (b) 1 or −0
...
(a) 102◦ (b) 165◦ (c) 10◦ 18 49
7
...
3◦ (h) 79◦ (i) 54◦
8
...
59◦ 20
10
...
51◦
12
...
326 rad 13
...
605 rad 14
...
(a) acute-angled scalene triangle
(b) isosceles triangle (c) right-angled triangle
(d) obtuse-angled scalene triangle
(e) equilateral triangle (f ) right-angled triangle

Answers to practice exercises
2
...
DF, DE
4
...
122
...
φ = 51◦, x = 161◦
7
...
a = 18◦ 50 , b = 71◦10 , c = 68◦ , d = 90◦,
e = 22◦ , f = 49◦, g = 41◦
9

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