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6 Introduction to Geometry
6 Introduction to Geometry
Classwork (p. 6.13)
1. (a) 35

(b) 90
Quick Review
Review Exercise 6 (p. 6.4)
1. (a) (i) s, x, y
(ii) a, b
(iii) c, d, e, r
(c) 130
(d) 310

2.
(b) s, x, y
2.
(a)
(a) C
(b) D, E
(c) B
3.
(a) A
(b)
(b) D
(c) B
Classwork (p. 6.15)
1. (a)
Activity
Activity 6.1 (p. 6.23)
1. straight angle
2.
The sum of the three interior angles of a triangle is 180.
(b) EF // CD (or AF // ED // BC)
Activity 6.2 (p. 6.41)
1.
Polyhedron
V
F
E
V+F–E
Triangular prism
6
5
9
2
Cube
8
6
12
2
Regular tetrahedron
4
4
6
2
Quadrilateral prism
8
6
12
2
Regular octahedron
6
8
12
2
2.
2.
(a)
(b) FA⊥EA (or FB⊥GB or GC⊥HC or HD⊥ED or
EF⊥GF or FG⊥HG or GH⊥EH or HE⊥FE)
Classwork (p. 6.22)
V F E 2
Classified by sides
Classwork
Classified by angles
Classwork (p. 6.6)
(a) BC, CD
(c) 6 cm
Classwork (p. 6.7)
1. (a) 
(b) 
120  40  50  210  180
∴ The set of angles cannot be the interior angles of a
triangle.
(b) 
(c) 
(d) 
b=
c=
x=
y=
C
A, C, D
B, E
A, C
D, E
B
Classwork (p. 6.23)
(a) 
60  60  60  180
∴ The set of angles can be the interior angles of a
triangle.
(b) BD
2.
equilateral triangle
isosceles triangle
scalene triangle
acute-angled triangle
right-angled triangle
obtuse-angled triangle
(c) 
85  15  80  180
∴ The set of angles can be the interior angles of a
triangle.
∠ABC or ∠CBA
∠BCA, ∠ACB, ∠ECA, ∠ACE or ∠C
∠DBA or ∠ABD
∠EBD or ∠DBE
125
Mathematics in Action (3nd Edition) 1A Full Solutions
(d) 
72  88  30  190  180
∴ The set of angles cannot be the interior angles of a
triangle.
Classwork (p. 6.36)
(a)
(e) 
24  52  84  160  180
∴ The set of angles cannot be the interior angles of a
triangle.
(b)
(f) 
Since one of the angles is 0, the set of angles cannot be
the interior angles of a triangle.
Classwork (p. 6.26)
1. (a) convex polygon, equilateral polygon, equiangular
polygon
(c)
(b) convex polygon
(c) convex polygon, equilateral polygon
(d)
(d) equilateral polygon
2.
(a) (a)
(b) equilateral triangle
3.
Classwork (p. 6.40)
(a) no
(a)
(b) yes
(c) no
Classwork (p. 6.45)
1. (a)
(b)
(b)
Classwork (p. 6.32)
(a)
(c)
(b)
2.
(a)
Classwork (p. 6.33)
1.
(b)
2.
126
6 Introduction to Geometry
Quick Practice
(c)
Quick Practice 6.1 (p. 6.9)
a is an obtuse angle.
b is a reflex angle.
c is a right angle.
d is an acute angle.
Classwork (p. 6.47)
Quick Practice 6.2 (p. 6.10)
(a) 2 right angles  2  90
 180
(b)
(c)
Classwork (p. 6.49)
1.
1
1
round angle   360
5
5
 72
3
3
straight angle   180
4
4
 135
Quick Practice 6.3 (p. 6.24)
In △ABC,
∵ A  B  C  180 ( sum of △)
∴ A  43  37  180
A  180  43  37
 100
Quick Practice 6.4 (p. 6.42)
Let V, F and E be the numbers of vertices, faces and edges of
the polyhedron respectively.
∵ V = 25 and F = 15
By the Euler’s formula,
V FE 2
∴ 25  15  E  2
E  38
∴ The polyhedron has 38 edges.
2.
Classwork (p. 6.50)
1.
Consolidation Corner
Consolidation Corner (p. 6.51)
1.
(a) no. of vertices = 4, no. of edges = 6,
no. of faces = 4
(b) no. of vertices = 8, no. of edges = 12,
no. of faces = 6
2.
2.
(a) no
(b) yes
(c) no
3.
(a) yes
(b) no
(c) yes
127
Mathematics in Action (3nd Edition) 1A Full Solutions
4.
(d) round angle
(a)
(e) obtuse angle
7.
(f)
straight angle
(a)
2 straight angles  2  180
 360
(b)
(b)
(c)
1
1
round angle   360
3
3
 120
4
4
right angle   90
5
5
 72
2
5
(d) 1 right angles   90
3
3
 150
Exercise
8.
Exercise 6A (p. 6.15)
Level 1
1. (a) line segment: AD, AC, AB, BC, CD
(or DA, CA, BA, CB, DC)
marked angle: ∠BAC (or ∠CAB)
(a) The marked angle  360 
 150
(b) The marked angle  360 
2
12
 60
(b) line segment: PS, PQ, PR, QR, QS
(or SP, QP, RP, RQ, SQ)
marked angle: ∠RQS (or ∠SQR)
9.
2.
5
12
(a) The required degree  360 
(a) right angle
15
60
 90
(b) round angle
(b) The required degree  360 
(c) straight angle
 15
(d) acute angle
(e) obtuse angle
(f)
3.
10. (a)
reflex angle
(c)
(b) PQS, TSQ
a: obtuse angle; b: right angle; c: reflex angle;
d: acute angle; e: obtuse angle
5.
(a) a, h
BPC  APC  APB
 130  60
 70
11. (a) a = 50°, b = 40°, c = 125°, d = 295°
(b) b < a < c < d
(b) b, e, f
12. (a)
(c) d, j
(d) c, g, i
6.
APB  60
(b) APC  130
(a) QRS (or PRT), RQS, RSQ
4.
2.5
60
(a) right angle
(b)
(b) reflex angle
(c) acute angle
128
6 Introduction to Geometry
(c)
20. (a)
AOB  BOC  COD  118
2AOB  62  118
2AOB  56
AOB  28
(d)
(b)
AOC  AOB  BOC
 28  62
 90
∴ AOC is a right angle.
13.
21. (a)
x  153
y  82
z  125
(b)
14. (a) DC
x  y  z  153  82  125
 360
(b) AB and DC
22. (a)
Level 2
15.
Angle
2 right
angles
2  90
Size
 180
Type of
the angle
straight
angle
1
round
4
angle
1
 360
4
 90
5
right
3
angles
5
 90
3
 150
1
1 straight
2
angles
1
1  180
2
 270
right
angle
obtuse
angle
reflex
angle
A  20
B  30
C  60
D  40
E  30
(b)
A  B  C  D  E
 20  30  60  40  30
 180
1
16. Size of angle A  1  90  108
5
2
Size of angle B   360  144
5
2
Size of angle C   180  120
3
∵ Size of angle B > size of angle C > size of angle A
∴ Angle C is not the greatest angle among the three
angles.
∴ Mary’s claim is disagreed.
23. (a) (i)
(ii)
x  60
(b) (i)
17. ∠AOE, ∠COE, ∠COB
18. (a)
AEC  AEB  BEC
 40  25
 65
(ii)
(b)
 65  25
Exercise 6B (p. 6.27)
Level 1
1. 2.6 cm
 40
19. (a)
x  130
CED  BED  BEC
BXC  BXD  CXD
2.
(a) PQR
 134  56
(b) WXYZ
 78
(b)
(c) ABCDE
AXC  AXB  BXC
3.
 38  78
 116
(a) ABCDEF
(b) AB, BC, CD, DE, EF, FA
129
Mathematics in Action (3nd Edition) 1A Full Solutions
4.
(a) equilateral triangle
8.
(b) scalene triangle
(a) (i)
(ii)
(iii)
(iv)
D, E
A, B, C
A, C, E
C
(c) isosceles triangle
(b) C
5.
(a) acute-angled triangle
9.
(a) convex polygon
(b) right-angled triangle
(b) concave polygon
(c) obtuse-angled triangle
(c) convex polygon
6.
(a) (i) C
(ii) B, C, E
(iii) A, D
(d) concave polygon
(e) concave polygon
(b) (i) C, E
(ii) B, D
(iii) A
(f)
convex polygon
10. (a)
7.
(a) In △XYZ,
∵ X + Y + Z = 180 ( sum of △)
∴
60  a  80  180
a  180  60  80
This polygon has 2 diagonals.
 40
(b)
(b) In △OPQ,
∵ O + P + Q = 180 ( sum of △)
∴
90  c  65  180
c  180  90  65
 25
This polygon has 5 diagonals.
(c) In △ABC,
∵ A + B + C = 180 ( sum of △)
∴
w  65  65  180
(c)
w  180  65  65
 50
(d) In △PQR,
∵ P + Q + R = 180 ( sum of △)
x  60  x  180
∴
2 x  180  60
This polygon has 9 diagonals.
11. The length of the sides of the regular octagon
 104  8 cm
 13 cm
2 x  120
x  60
Level 2
12. ∵
∴
i.e.
∴
(e) In △XYZ,
∵ X + Y + Z = 180 ( sum of △)
90  y  3 y  180
∴
4 y  180  90
4 y  90
y  22.5
(f)
13. (a)
In △UVW,
∵ U + V + W = 180 ( sum of △)
∴ ( z  10)  (3 z  20)  z  180
5 z  30  180
5 z  180  30
5 z  150
z  30
(b)
130
O is the centre of the circle.
OA, OB and OC are the radii of the circle.
OA = OB = OC
△OAB, △OAC and △OBC are the isosceles
triangles in the figure.
6 Introduction to Geometry
(c)
(b) ∵ The sum of an obtuse angle and the right angle
is greater than 180°.
∴ An obtuse-angled triangle cannot contain a right
angle.
18. (a) ∵ The hexagon ABCDEF is made up of 6 identical
equilateral triangles.
∴ AB = BC = CD = DE = EF = FA
Size of each interior angle of an equilateral triangle
180

3
 60
ABC  2  60
 120
By the similar argument,
BCD = CDE = DEF = EFA = FAB = 120°.
∴ ABCDEF is a regular hexagon.
(d)
14. (a) 5
(b) (i) △ABD and △ADE
(ii) △BCD and △ACE
(iii) △ACD
(b) Length of a side of each equilateral triangle
15
 cm
3
 5 cm
Perimeter of the hexagon
 5  6 cm
15. (a) In △PRS,
∵ RPS + PRS + RSP = 180 ( sum of △)
∴ m  58  (65  15)  180
 30 cm
m  180  58  65  15
 42
19. In △UVW,
∵ VUW + UVW + UWV = 180 ( sum of △)
∴ (a  10)  70  2a  180
3a  180  10  70
In △PQS,
∵ QPS + PQS + QSP = 180 ( sum of △)
∴ 42  n  15  180
n  180  42  15
 120
a  40
 123
In △UVY,
∵ VUY  UVY  UYV  180 ( sum of △)
∴ [(40  10)  45]  70  b  180
145  b  180
(b) In △ABD,
∵ BAD + ABD + ADB = 180 ( sum of △)
∴ (20  y )  40  90  180
y  180  20  40  90
 30
b  180  145
 35
In △ACD,
∵ CAD  ACD  ADC  180 ( sum of △)
∴
30  z  90  180
In △UXY,
∵ XUY  UXY  UYX  180 ( sum of △)
∴ 45  c  (40  35)  180
120  c  180
z  180  30  90
 60
c  180  120
 60
16. In △ABC,
∵ BAC  ABC  ACB  180 ( sum of △)
∴ 40  r  80  180
Exercise 6C (p. 6.37)
Level 1
1. (a), (b)
r  180  40  80
 60
In △BDF,
∵ DBF + BDF + BFD = 180 ( sum of △)
∴ 60  30  s  180
s  180  60  30
 90
17. (a) ∵ If a triangle contains two right angles, the
remaining angle is 0°. It is impossible.
∴ A triangle cannot contain two right angles.
131
Mathematics in Action (3nd Edition) 1A Full Solutions
2.
10.
(a), (b)
Level 2
11.
3.
12. (a) (i), (ii), (iii)
4.
(a)
(b) yes
(b)
13. (a)
(b)
5.
6.
14.
7.
15. (a)
8.
(b) A = 26, B = 34, C = 120
16. (a)
9.
132
6 Introduction to Geometry
(b) MN = 6.4 cm
6.
(a) yes
17. (a)
(b) no
(c) no
(b) (i) QR = 5.0 cm, RP = 5.0 cm
(ii) △PQR is an equilateral triangle.
(d) yes
Exercise 6D (p. 6.52)
Level 1
1. (a) no
(b) yes
(c) no
7.
(a) no
(d) no
(b) yes
(e) yes
2.
(a) no. of vertices = 5, no. of edges = 8, no. of faces = 5
(b) no. of vertices = 8, no. of edges = 12, no. of faces = 6
8.
(a)
(c) no. of vertices = 6, no. of edges = 9, no. of faces = 5
3.
4.
Let V, F and E be the numbers of vertices, faces and edges
of the polyhedron respectively.
∵ F = 10 and E = 18
By the Euler’s formula,
V FE 2
V
 10  18  2
∴
V  10
∴ The polyhedron has 10 vertices.
Let V, F and E be the numbers of vertices, faces and edges
of the polyhedron respectively.
∵ V = 13 and F = 17
By the Euler’s formula,
V F E 2
∴ 13  17  E  2
∴
5.
(b)
E  28
The polyhedron has 28 edges.
(c)
(a)
(b)
133
Mathematics in Action (3nd Edition) 1A Full Solutions
9.
13. (a) Isometric grid paper:
(a)
Oblique grid paper:
(b)
(b) Isometric grid paper:
10. (a)
Oblique grid paper:
(b)
14. (a) Isometric grid paper:
Level 2
11. (a) no
Oblique grid paper:
(b) yes
(c) no
(d) yes
12. (a)
(b) Isometric grid paper:
(b)
134
6 Introduction to Geometry
Oblique grid paper:
Revision Exercise 6 (p. 6.61)
Level 1
1. a: BDE (or CDE), b: BED (or MED), c: BME,
d: reflex DBE (or reflex DBM or reflex CBE or
reflex CBM), e: CAE (or MAE)
Check Yourself (p. 6.59)
1. (a) a: 90°, b: 132°, c: 42°, d: 236°, e: 40°
2.
a: acute angle, b: reflex angle, c: right angle,
d: acute angle, e: reflex angle, f: obtuse angle
3.
(a)
 57  33

 90
(b) a: right angle, b: obtuse angle, c: acute angle,
d: reflex angle, e: acute angle
2.
BOD  BOC  COD
AOD  AOB  BOD
 28  90
(a) a scalene
 118
(b) an isosceles
AOE  AOD  DOE
 118  62
(c) an acute-angled
 180
(d) a right-angled
(b) BOD: right angle, AOD: obtuse angle,
AOE: straight angle
3.
Polygon
A
B
C
D
4.
Equilateral
polygon
Equiangular
polygon
Convex
polygon





Concave
polygon

4.
(a) (i) concave
(ii) no
(b) (i) convex
(ii) yes

(a) 
(c) (i) concave
(ii) yes
(b) 
(c) 
5.
The angle the hour hand rotates  360 
 120
(d) 
5.
4
12
In △ABC,
∵ CAB + ABC + ACB = 180 ( sum of △)
∴ 46  ( x  18)  88  180
152  x  180
x  28
In △ABE,
∵ BAE  ABE  AEB  180
∴ ( y  46)  28  62  180
y  136  180
6.
The reflex angle formed  360 
10
12
 300
7.
( sum of △)
(a) ∵ The sum of all the interior angles of a triangle is
180.
∴ 60  45  x  180
x  180  60  45
 75
y  44
(b) ∵ The sum of all the interior angles of a triangle is
180.
∴ 90  y  42  180
6.
y  180  90  42
 48
8.
7.
(a)
(a) In △ABC,
∵ A + B + C = 180 ( sum of △)
∴ 20  70  C  180
C  180  20  70
 90
(b) yes
135
Mathematics in Action (3nd Edition) 1A Full Solutions
(b) In △ABC,
∵ A + B + C = 180 ( sum of △)
∴ 35  B  63  180
13. (a)
B  180  35  63
 82
(c) In △ABC,
∵ A + B + C = 180 ( sum of △)
∴ A  18  47  180
A  180  18  47
 115
(b)
9.
10.
14. (a)
11. (a) (i), (ii)
(b)
(b) 3 cm
12. (a) (i) no
(ii) no
(c)
(b) (i) no
(ii) no
(c) (i) no
(ii) yes
15. Isometric grid paper:
(d) (i) yes
(ii) yes
Oblique grid paper:
(e) (i) yes
(ii) no
136
6 Introduction to Geometry
Level 2
16. (a) 8
20. (a) In △PQR,
∵ P  Q  R  180 ( sum of △)
∴ 3x  4 x  2 x  180
(b) (i) △ABF, △BCF and △ACF
(ii) △ABF, △BCF, △ACF and △DEF
(iii) △BCD
9 x  180
x  20
17. There are 3 hours 15 minutes from 2:15 p.m. to 5:30 p.m.
1
15 hours
∵ 15 minutes 
60
1
 hours
4
 0.25 hours
∴ 3 hours 15 minutes = 3.25 hours
The angle the hour hand rotates
1
 360   3.25
12
 97.5
(b) P  3 x  3(20)  60
Q  4 x  4(20)  80
R  2 x  2( 20)  40
∴ △PQR is an acute-angled triangle.
21. (a) In △PQR,
∵ P  Q  R  180 ( sum of △)
∴ 3x  ( x  10)  ( x  15)  180
5 x  25  180
5 x  155
x  31
18. (a) In △ABC,
∵ BAC + ABC + ACB = 180 ( sum of △)
∴ 52  32  (38  x)  180
x  180  52  32  38
 58
(b) P  3x  3(31)  93
Q  x  10  31  10  41
R  x  15  31  15  46
∴ △PQR is an obtuse-angled triangle.
In △BCD,
∵ CBD + BCD + CDB = 180 ( sum of △)
∴ 32  58  y  180
22. (a)
y  180  32  58
 90
(b) In △ABD,
∵ BAD + ABD + ADB = 180 ( sum of △)
∴ 36  p  90  180
p  180  36  90
 54
In △ABC,
∵ BAC + ABC + ACB = 180 ( sum of △)
∴ 36  (54  46)  q  180
136  q  180
(b) 13 cm
23. (a)
q  180  136
 44
19. In △XYZ,
∵ YXZ + XYZ + XZY = 180
∴ 50  x  70  180
(b) obtuse-angled triangle
( sum of △)
24. (a), (b)
x  180  50  70
 60
In △RST,
∵ SRT + RST + RTS = 180
∴ 61  55  y  180
( sum of △)
y  180  61  55
 64
In △MYT,
∵ YMT + MYT + MTY = 180 ( sum of △)
∴ z  60  64  180
z  180  60  64
 56
137
Mathematics in Action (3nd Edition) 1A Full Solutions
25. (a) (i)
(ii)
2.
Answer: D
∵ A  reflex A  360
∴ The sum of A and reflex A is always equal to a
round angle.
∴ The answer is D.
3.
Answer: B
∵ All sides of an equilateral polygon are of equal
length.
∴ The answer is B.
4.
Answer: C
For I:
∵ The sum of two obtuse angles is greater than 180°.
∴ It is impossible to form a triangle with two obtuse
angles.
∴ There is at most one obtuse angle in any triangle.
∴ I must be true.
For II:
∵ In an acute-angled triangle, all interior angles are
acute.
∴ Interior angles of a triangle can be all acute.
∴ II must be true.
For III:
∵ The sum of two right angles is equal to 180°.
∴ It is impossible to form a triangle with two right
angles.
∴ A triangle cannot have more than one right angle.
∴ III must be false.
∴ The answer is C.
5.
Answer: D
In △ABC,
∵ A + B + C = 180 ( sum of △)
∴
a  b  68  180
(b)
Yes, the circle passes through D.
△ACD is an isosceles triangle.
26. (a), (b)
27. Isometric grid paper:
a  b  112
6.
Oblique grid paper:
Answer: B
With the notation in the figure,
in △ADE,
∵ DAE + ADE + AED = 180 ( sum of △)
∴
DAE  73  56  180
DAE  180  73  56
DAE  x  90
Multiple Choice Questions (p. 6.65)
1. Answer: C
ABE is an acute angle.
∵ ABD  ABC  DBC
 51
x  90  51
 39
 180  90
 90
∴ ABD is a right angle.
CBE is an obtuse angle.
DBE is an acute angle.
∴ The answer is C.
138
7.
Answer: C
∵ Sides of cuboids can be different in length.
∴ Cuboids may not be regular polyhedra.
8.
Answer: A
9.
Answer: A
6 Introduction to Geometry
Exam Corner
Exam-type Questions (p. 6.67)
1. In △ACG,
GAC + ACG + AGC = 180 ( sum of △)
54  ACG  70  180
ACG  180  54  70
 56
In △BDE,
EBD + EDB + BED = 180 ( sum of △)
EBD  65  55  180
EBD  180  65  55
 60
In △BCF,
FBC + BCF + BFC = 180 ( sum of △)
60  56  x  180
x  180  60  56
 64
2.
3.
4.
Answer: D
∵ All sides of an equilateral polygon are of equal
length.
∴ The answer is D.
5.
Answer: D
∵ All interior angles of a convex polygon are less than
180°.
∴ The answer is D.
139