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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
...
3

418

␪

␪

538
(a)

448

Problem 12
...
If ∠AOC is 43◦ , find ∠AOD,
∠DOB and ∠BOC

(b)

Figure 20
...

Thus, 360◦ = 58◦ + 108◦ + 64◦ + 39◦ + θ
from which, θ = 360◦ − 58◦ − 108◦ − 64◦ − 39◦ = 91◦

The symbol shown in Figure 20
...


From Figure 20
...

Hence, ∠ AOD = 180◦ − 43◦ = 137◦
...

Hence,
∠ DOB = 43◦ and ∠BOC 137◦

D

A

438

Figure 20
...
3(a),
θ + 41◦ = 90◦
from which,
θ = 90◦ − 41◦ = 49◦

Figure 20
...
Determine angle β in Figure 20
...
7

α = 180◦ − 133◦ = 47◦ (i
...
supplementary angles)
...

Problem 14
...
8
A

23⬚37⬘

F

␪

B
G

E

Although we may be more familiar with degrees, radians
is the SI unit of angular measurement in engineering
(1 radian ≈ 57
...

π
(a) Since 180◦ = π rad then 1◦ =
rad
...
274 rad
...
616666
...
616666
...
616666
...
447 rad
...
Convert 0
...
8

Let a straight line FG be drawn through E such that FG
is parallel to AB and CD
...

∠ECD = ∠FEC (alternate angles between parallel lines
FG and CD), hence ∠FEC = 35◦ 49
...
Determine angles c and d in
Figure 20
...
9

a = b = 46◦ (corresponding angles between parallel
lines)
...

Hence, 46◦ + c + 90◦ = 180◦ , from which, c = 44◦
...

◦
Alternatively, 90 + c = d (vertically opposite angles)
...
Convert the following angles to
radians, correct to 3 decimal places
...
57076
...
743 rad = 0
...


State the general name given to an angle of
197◦
...


State the general name given to an angle of
136◦
...


State the general name given to an angle of
49◦
...


State the general name given to an angle of
90◦
...


Determine the angles complementary to the
following
...


Determine the angles supplementary to
(a) 78◦ (b) 15◦ (c) 169◦41 11

7
...
10
...


With reference to Figure 20
...

y
56
7
8

2
1
3
4

x

Figure 20
...


(b)

vertically opposite angles
supplementary angles
corresponding angles
alternate angles

In Figure 20
...

137⬚29⬘

378

␣
␪
558

␪

16⬚49⬘

508

Figure 20
...
In Figure 20
...


708
␪

298

␪

1508
␪

a

698

608

808

b

(f)

(e)

688

578

␪

Figure 20
...
78

c

␪

11
...
14
...
14
368
␪
(i)

Figure 20
...
Convert 76◦ to radians, correct to 3 decimal
places
...
Convert 34◦ 40 to radians, correct to 3 decimal places
...
Convert 0
...


171

Angles and triangles
20
...

The sum of the three angles of a triangle is equal
to 180◦
...
3
...
e
...
An
example is shown in triangle ABC in Figure 20
...

A right-angled triangle is one which contains a right
angle; i
...
, one in which one of the angles is 90◦
...
15(b)
...
17

548
508

A

c
678

598

B

C

E

F

(a)

(b)
␪

Figure 20
...
e
...
An example is shown in triangle PQR in
Figure 20
...

An equilateral triangle is one in which all the sides and
all the angles are equal; i
...
, each is 60◦
...
16(b)
...
18

With reference to Figure 20
...

(b) Angle θ is called an exterior angle of the triangle
and is equal to the sum of the two opposite interior
angles; i
...
, θ = A + C
...


8

A
1318

608

278
Q

(a)

R

B

608
(b)

C

Figure 20
...
An example is shown in triangle
EFG in Figure 20
...

A scalene triangle is one with unequal angles and
therefore unequal sides
...
17(b)
...
19

c

a

B

172 Basic Engineering Mathematics
A right-angled triangle ABC is shown in Figure 20
...

The point of intersection of two lines is called a vertex
(plural vertices); the three vertices of the triangle are
labelled as A, B and C, respectively
...
The side opposite the right angle is given
the special name of the hypotenuse
...
19, is always the longest side of
a right-angled triangle
...

With reference to angle A, BC is the opposite side and
AC is the adjacent side
...
So, in the triangle ABC, length AB = c, length
BC = a and length AC = b
...

∠ is the symbol used for ‘angle’
...
Another way of indicating
an angle is to use all three letters
...
e
...
Similarly, ∠BAC means ∠A and ∠ACB means
∠C
...

Problem 18
...
20

Name the types of triangle shown in

(d) Obtuse-angled scalene triangle (since one of the
angles lies between 90◦ and 180◦ )
...


Problem 19
...
21, with reference to angle θ, which side
is the adjacent?
C

A

␪
B

Figure 20
...
With reference to angle θ, the opposite side
is BC
...

Problem 20
...
22, determine angle θ

2
...
1

2

398

2
...
5
␪

2
...
5
(e)

Figure 20
...


(b) Acute-angled scalene triangle (since all the
angles are less than 90◦)
...


Figure 20
...

The triangle is right-angled
...

Figure 20
...

Determine the value of θ and α in

Angles and triangles

173

Problem 23
...
25

A
62⬚
D

B

C

e

␪
15⬚
E
␣

a
558 b
628
d c

Figure 20
...
23

180◦

In triangle ABC, ∠A + ∠B + ∠C =
(the angles in
a triangle add up to 180◦)
...
Thus,
∠DCE = 28◦ (vertically opposite angles)
...

Hence, ∠θ = 28◦ + 15◦ = 43◦
...


a = 62◦ and c = 55◦ (alternate angles between parallel
lines)
...

b = d = 63◦ (alternate angles between parallel lines)
...

Check: e = a = 62◦ (corresponding angles between
parallel lines)
...
ABC is an isosceles triangle in
which the unequal angle BAC is 56◦
...
24
...
Also, calculate ∠DBC
A

Practice Exercise 78
page 348)
1
...
26
...
24

Since triangle ABC is isosceles, two sides – i
...
AB and
AC – are equal and two angles – i
...
∠ABC and ∠ACB –
are equal
...

Hence, ∠ABC + ∠ACB = 180◦ − 56◦ = 124◦
...

2
An angle of 180◦ lies on a straight line; hence,
∠ABC + ∠DBC = 180◦ from which,
∠ DBC = 180◦ − ∠ABC = 180◦ − 62◦ = 118◦
...
e
...


97⬚

45⬚
45⬚

(d)

(c)

53⬚
60⬚
5

5
37⬚

(e)

Figure 20
...


Find the angles a to f in Figure 20
...


O
658

328

578
N
838

114°

b

a
(a)

M

c

␪

(b)
P

1058
d

Figure 20
...


f

Determine ∠φ and ∠x in Figure 20
...


1058

e

E

(c)

588

Figure 20
...


B

In the triangle DEF of Figure 20
...
30

7
...
31(a) and (b), find angles w, x, y
and z
...
28

4
...
28, determine
angle D
...


MNO is an isosceles triangle in which
the unequal angle is 65◦ as shown in
Figure 20
...
Calculate angle θ
...
31

8
...
32(a) and (b)
...
32

9
...
33
...


Problem 24
...
35 are congruent and name their
sequence

f

1258 e

(b) two sides of one are equal to two sides of the other
and the angles included by these sides are equal
(SAS),

h

C

i

G
E

B

D
L

k 998

J

H
A

Figure 20
...
Triangle ABC has a right angle at B and
∠BAC is 34◦
...
If the
bisectors of ∠ABC and ∠ACD meet at E,
determine ∠BEC
...
If in Figure 20
...

A

C

978

K
(b)
F

T
S
V

M

U

E
C

B

O

N

D

Q

P

(c)

R

W

X
(d)

A
(e)

Figure 20
...
e
...

(b) Congruent GIH, JLK (side, angle, side; i
...
, SAS)
...
34

(c) Congruent MNO, RQP (right angle, hypotenuse,
side; i
...
, RHS)
...
4

Congruent triangles

Two triangles are said to be congruent if they are equal
in all respects; i
...
, three angles and three sides in one
triangle are equal to three angles and three sides in the
other triangle
...
It is not indicated that
any side coincides
...
e
...


176 Basic Engineering Mathematics
Problem 25
...
36, triangle PQR is
isosceles with Z , the mid-point of PQ
...
Determine
the values of angles RPZ and RXZ

L

F

A
E

G

K

M

I

O

H
C
N

R

J

D

P

(a)

678

X

(b)

(c)

Y

288
P

B

V

288
Z

U

Q
R

Q

Figure 20
...

∠RXZ = ∠QPR + 28◦ and ∠RYZ = ∠RQP + 28◦ (exterior angles of a triangle equal the sum of the two interior
opposite angles)
...


2
...

Hence, XZ = YZ
...

∠QRZ = 67◦ and thus ∠PRQ = 67◦ + 67◦ = 134◦
...

2
∠ RXZ = 23◦ + 28◦ = 51◦ (external angle of a triangle equals the sum of the two interior opposite
angles)
...
37

∠PXZ = 180◦ − ∠RXZ and ∠QYZ = 180◦ − ∠RYZ
...


Triangles PRZ and QRZ are congruent since
PR = RQ, ∠RPZ = ∠RQZ and PZ = ZQ (SAS)
...


Z

T
(d)

20
...
Show that triangles AEB and
CDB are congruent
...
With
reference to Figure 20
...
e
...


State which of the pairs of triangles in
Figure 20
...


588
a

C

Q

658 588
p
R

Figure 20
...

side a

In Figure 20
...
82
p
=
=
12
...
44 10
...
82
= 4
...
44
10
...
e
...
0 cm

D
f 5 5
...
42 cm

C

p q
=
z
y

p
6
...
97 10
...
82
p = 12
...
32 cm
10
...
39

In triangle ABC, 50◦ + 70◦ + ∠C = 180◦ , from which
∠C = 60◦
...

Hence, triangles ABC and DEF are similar, since their
angles are the same
...
0
a
=
i
...

=
d
f
4
...
0
Hence, side, a =

Hence,

A

12
...
42) = 10
...

5
...
42

z5

q 5 6
...
63

y51

R

cm

X

Figure 20
...

In triangle XYZ, ∠X = 180◦ − 90◦ − 55◦ = 35◦
...
The triangles may he redrawn as
shown in Figure 20
...


x 5 7
...
82 cm

Figure 20
...
97

cm

Also,
358

R

Z

Since BD is parallel to AE then ∠CBD = ∠CAE and
∠CDB = ∠CEA (corresponding angles between parallel lines)
...

Since the angles in triangle CBD are the same as in
triangle CAE, the triangles a

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