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1. Find the sum of the interior angles of each of the following polygons.
(a) Quadrilateral
(b) Heptagon
(c) Decagon
2. Find the sum of the interior angles of each of the following polygons.
(a)
(b)
A
A
C
D
B
A
(c)
E
G
B
H
C
B
C
D
D
F
E
3. Find the unknown in each of the following figures.
(a)
A
A
(b)
D
60
70
A
E
130
B
F
(c)
z
105
100
150
130
D
140
x
100
E
y
C
B
95
C
100
D
B
C
4. Find the unknown in each of the following figures.
A
(a)
F
(b)
70
(c)
F
50
y
A
A
D
y
30
G
D
30
105
B
B
300
G
z
z
z
x
C
H
50
z
y
E
50
C
D
50
300
105
B
E
C
12.7
 2009 Chung Tai Educational Press. All rights reserved.
5. Find the unknown in each of the following figures.
(a)
F
2x
A
(b)
A
8y
3x
B
B
4x
(c)
A
95
2z
99
D
5z  10
E
4x
120
3x
E
2x
5z  5
4y  10
C
B
D
55
C
D
C
6. If each of the following is the sum of interior angles of a polygon, find the number of sides of the
polygon.
(a) 720
(b) 1 080
(c) 1 800
7. Find the size of an interior angle of each of the following regular polygons.
(a) Regular pentagon
(b) Regular decagon
(c) Regular 12-gon
8. If each of the following is the size of an interior angle of a regular polygon, find the number of sides
of the regular polygon.
(a) 120
(b) 140
(c) 165
A
9. In the figure, ABC is a straight line.
(a) Find x.
85
B
(b) Find y.
x
E
y
C
88
D
A
10. In the figure, BC // ED.
(a) Find x.
y
B
(b) Find y.
6x
115
E
5x
3x
D
C
11. If the interior angles of a pentagon are in the ratio of 2 : 3 : 4 : 5 : 6, find the size of the smallest
interior angle.
12.8
 2009 Chung Tai Educational Press. All rights reserved.
12. If each interior angle of a regular polygon is 16 more than that of a regular nonagon, find the number
of sides of the regular polygon.
13. In the figure, AE // BC.
E
A
x
D
128
115
C
B
(a) Prove that EAB  ABC  180  .
(b) Hence, find x.
14. The following figure shows a regular pentagon ABCDE.
A
B
E
C
D
(a) Find the size of each interior angle of a regular pentagon.
(b) Find BCA.
(c) Prove that BCA : ACD  1 : 2.
15. Can the size of an interior angle of a regular polygon be 166 ? Explain briefly.
16. In the figure, ABCDEF and APQRF are regular hexagon and regular pentagon respectively. APS is a
straight line.
B
S
A
P
C
F
Q
R
D
E
(a) Find BAS.
(b) Find PSC.
12.9
 2009 Chung Tai Educational Press. All rights reserved.
17. In each of the following figures, which angles are the exterior angles of the polygon?
(a)
(b)
(c)
d
d
a
a
a
d
e
c
b
c
b
c
b
18. Find the sum of the exterior angles of each of the following polygons.
(a) Triangle
(b) Regular octagon
(c) Regular 14-gon
19. Find the unknown in each of the following figures.
U
Q
(a)
(b)
A
P
89 D
110
75
A
65
y
C
88
B
x
C
D
S
E
R
PADT, ABR, QBCS and UDC
are straight lines.
(c)
R
P
35
B
z
S
T
P
Q
B
45
T
55
U
PABR, AEU, EDT, DCS and
QBC are straight lines.
Q
A
F
42
T
C
R
82
U
99
E
D
63
S
PAF, QBA, RCB, CDS, DET
and UFE are straight lines.
20. Find the size of an exterior angle of each of the following polygon s.
(a) Regular octagon
(b) Regular 18-gon
(c) Regular 20-gon
21. If each of the following is the size of an exterior angle of a regular polygon, find the number of sides
of the regular polygon.
(a) 120
(b) 36
12.10
 2009 Chung Tai Educational Press. All rights reserved.
(c) 14.4
22. Find the unknown in each of the following figures.
(a)
T
(b)
P
6x
D
S
T
A
8y
A
P
70
A
(c)
B
B
P
80
B
10z S
10z
65
2y
C
Q
76
E
E
9y
Q
8x
C
C
S
8y D
R
R
D
7z  30
Q
PAB, QBC, DCR and ADS
are straight lines.
R
PBAT, BCQ, CDR and AES
are straight lines.
PAE, ABQ, BCR, CDS and
TED are straight lines.
23. Find the unknown in each of the following figures.
(a)
P
S
A
(b)
P
U
y
101
68
D
E
A
(c)
T
R
D
130
P
Q
38
A
B
x
C
S
S
R
Q
PAD, ABQ, BCR and CDS
are straight lines.
B
95
C
R
Q
PAB, QCDT, RCB and SDEU
are straight lines.
E W
D
C
T
72
U
Y
95 X
82
100
B
125
V
z
PAEW, RBAY, QBCT,
SCDV and UDEX
are straight lines.
24. If the exterior angles of a pentagon are in the ratio of 2 : 2 : 3 : 3 : 5, find the size of the largest exterior
angle.
25. If each interior angle of a regular polygon is 3 times its exterior angle, find the number of sides of the
regular polygon.
26. If each interior angle of a regular polygon is 132 more than its exterior angle, find the number of
sides of the regular polygon.
27. Is there any regular polygon with exterior angles of 27 each? Explain briefly.
12.11
 2009 Chung Tai Educational Press. All rights reserved.
28. In the figure, AB, BC and CD are three sides of a regular polygon where BAC  18.
B
A
C
D
18
(a) Find the size of an exterior angle of this regular polygon.
(b) Find the number of sides of this regular polygon.
29. In the figure, PQ, QR and RS are three sides of a regular n-gon. PQ and SR are produced to meet at T.
T
Q
P
R
S
(a) Express TQR in terms of n.
(n  4)  180
(b) Prove that QTR 
.
n
[ In this exercise, only straight edges and compasses are allowed to be used for construction. ]
30. Copy the angle.
A
B
C
31. In the figure, construct the angle bisector of reflex angle ABC.
[ A copy of the figure is provided in the Appendix. ]
A
B
C
12.12
 2009 Chung Tai Educational Press. All rights reserved.
32. In the figure, construct the perpendicular bisector of XY, and mark the mid-point of XY as M.
[ A copy of the figure is provided in the Appendix. ]
X
Y
33. In the figure, construct a straight line passing through P and perpendicular to AB.
[ A copy of the figure is provided in the Appendix. ]
A
B
P
34. (a) Construct an angle of 60.
(b) Construct an angle of 22.5.
(c) Construct an angle of 82.5.
35. The figure shows an angle with the size of x. Construct an angle with the size of 3x.
x
36. The figures show a line segment with the length of x cm and an angle with the size of y. Construct a
triangle with a base of x cm and two base angles with the sizes of y each.
y
x cm
12.13
 2009 Chung Tai Educational Press. All rights reserved.
37. In the figure, construct PB, where ABP : PBC  1 : 3.
[ A copy of the figure is provided in the Appendix. ]
A
B
C
38. In the figure, mark a point P such that AP : PB  3 : 5.
[ A copy of the figure is provided in the Appendix. ]
B
A
P
39. In the figure, construct a straight line passing through P
and parallel to AB. Explain your method.
[ A copy of the figure is provided in the Appendix. ]
A
B
40. (a) Construct a figure according to the following steps.
Step 1: Mark a point O on a piece of paper. Construct a circle with O as the centre.
Step 2: Construct a straight line which intersects the circumference at two points. Mark the two
points as A and B.
Step 3: Construct another straight line which intersects the circumference at two points. Mark the
two points as C and D.
Step 4: Construct the perpendicular bisector of AB.
Step 5: Construct the perpendicular bisector of CD.
(b) Do the lines constructed in Step 4 and Step 5 intersect at O?
41. In the figure, ABC is a straight line. AB  1, BC  x and AD  y.
[ A copy of the figure is provided in the Appendix. ]
1
B
x
C
A
y
D
(a) Construct a straight line passing through C and parallel to BD. Mark the point of intersection with
AD as E.
(b) What is the length of DE? Explain briefly.
12.14
 2009 Chung Tai Educational Press. All rights reserved.