Chapter 5
Introduction to Deductive Geometry
Multiple Choice Section
1. In the figure, AOB, COD and EOF are straight lines, EOB  90. Which of the following must
be correct?
A
F
x
C
y
D
O
E
B
A. x  y
B. x  y  45
C. x  y  90
D. None of the above
2. In the figure, AOB is a straight line. OE and OC are the angle bisectors of AOD and BOD
respectively. Which of the following must be a right angle?
E
D
C
A
B
O
A. AOD
B. EOC
C. DOB
D. None of the above
3. In the figure, which of the following must be correct?
B
C
x  30
130  2x
A
x  20
O
D
A. x  20
B. AOB  BOD
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C. AOB is a right angle.
D. AOD is a straight line.
4. In the figure, AOE, BOF, COG and DOH are straight lines. Which of the following must be a
right angle?
C
D
B
b
b
E
A
O
a
a
H
F
G
I.
AOC
II. BOH
III. COE
A. I and II only
B. II and III only
C. I and III only
D. I, II and III
5. In the figure, AE and BD intersect at C, AC  EC and BAC  DEC. Which of the following
must be correct?
A
D
C
B
E
A. AB  CD
B. CD  DE
C. DE  BC
D. BC  DC
6. In the figure, CA and DB intersect at E, CA  DB and CD  AD. Which of the following must be
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correct?
C
B
E
D
A
I.
CDE  ADE
II. DEA  BEC
III. BD bisects ABC.
IV. AC bisects DCB.
A. I and II only
B. II and III only
C. I, II and III only
D. I, II, III and IV
7. In the fi gure, ABCD, GFC and EFB are strai ght lines, GAC  EDB, AB  DC and AG  DE.
What type of triangle is BCF?
G
E
F
A
B
C
D
A. An isosceles triangle
B. An equilateral triangle
C. An irregular triangle
D. It cannot be determined.
8. In the figure, BCDEF is a straight line, BC  FE, AC  AE and AD  BF. Which of the following
must be correct?
A
B
I.
C
D
E
F
ACD  AED
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II. ACB  AEF
III. ADB  ADF
A. I only
B. II only
C. I and II only
D. I, II and III
9. The figure shows ABC. Which of the following must be correct ?
C
b
a
A
c
I.
a2  b2  c2
B
II. The perimeter of ABC is less than 2(a  b).
III. The perimeter of ABC is greater than 2c.
A. I only
B. II only
C. II and III only
D. I, II and III
10. In the figure, ABC is a triangle and BC is produced to D. Which of the following must be
correct?
A
x
c
b
z
y
B
I.
a
C
D
zx
II. a  b  c
III. If a  b, then x  y.
A. I only
B. II only
C. I and II only
D. I, II and III
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Section A(1)
1. In the figure, AOB is a straight line. Express COB in terms of x.
C
B
x
O
A
2. In the figure, AOB and COD are straight lines. Express COB in terms of y.
B
C
O
y
D
A
3. In the figure, AOD  90 and BOC  80. Prove that a  b  190.
A
O
a
D
b
80
B
C
4. In the figure, AOB is a straight line. Prove that a  b  100.
E
D
F
C
20
30
b
A
a
30
O
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B
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5. In the figure, AOE, BOF, COG and DOH are straight lines.
Prove that a  b  c  d  180.
C
D
B
b
a
c
E
O
A
d
F
H
G
6. In the figure, BDC is a straight line and AB  AC. Prove that ABD  ACD.
A
B
C
D
7. In the figure, AE and BD intersect at C, AC  DC and BC  EC. Prove that AB  DE.
B
A
C
E
D
8. In the figure, BAC  BDC  90 and AC  DC. Prove that BC bisects ABD.
A
C
B
D
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9. In the figure, AB  AC and BD  CD. Prove that ABD  ACD.
A
D
C
B
10. In the figure, ACD and BCE are straight lines, ABC  CED  30 and BC  CE  5 cm. Pr o ve t hat
AB  DE.
A
B
30
5 cm
C
5 cm
E
30
D
11. In the figure, prove that ABC is an isosceles triangle.
A
78
3x 12
2x 9
B
C
12. In the figure, AB  AC and BP bisects ABC. Prove that y  3x.
A
y
B
P
x
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C
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13. In the figure, BDC is a straight line. BAD  CAD  a and b  c.
Prove that AD  BC.
A
a
a
b
c
B
C
D
14. In the figure, BDEC is a straight line. AB  AC and a 1  a 2 . Prove that BE CD.
A
b1
B
b2
a1 a2
D
C
E
15. In the figure, AC bisects BCD and BC  DC. Prove that BCA  DCA.
A
D
B
C
16. In the figure, BCDE, AGD and FGC are straight lines. ABE  FEB  90, a  f and BC  ED.
Prove that AD  FC.
A
F
a
f
G
B
C
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D
E
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17. In ABD, C is the mid-point of BD and AB  BC. Prove that 3AB  AD.
A
B
D
C
18. Construct a right-angled triangle with a as its hypotenuse and b as an adjacent side with proof.
a
b
19. Construct a right-angled triangle with a and b as the adjacent sides with proof.
a
b
Section A(2)
20. In the figure, AOB is a straight line. Express y in terms of x.
C
3x  70
A
y
B
O
21. In the figure, prove that AOD is a straight line.
C
B
130
A
x  20
O
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30  x
D
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22. In the figure, AOB, COD and EOF are straight lines. Prove that y  120  3x.
A
C
E
3x  50
10
O
y
F
D
B
23. In the figure, AOD  AOC and COE  90.
C
A
D
72
 2 x 2x
E
O
3x
B
(a) Find x.
(b) Prove that AOB is a straight line.
24. In the figure, CDE, CBA, DFA and BFE are straight lines, DC  BC and a 1  a 2 . Prove that ADC
 EBC.
C
B
D
a2
a1
F
E
A
25. In the figure, BPQC is a straight line, BP  CQ and AP  AQ.
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A
B
P
C
Q
(a) Prove that APB  AQC.
(b) Prove that APB  AQC.
26. In the figure, AEC and DEB are straight lines, AB  DC and AC  DB.
D
A
E
C
B
(a) Prove that ABC  DCB.
(b) Prove that EBC is an isosceles triangle.
27. In the figure, AMN is a straight line, AB  AC and BM  CM.
C
A
a1
a2
N
M
B
(a) Prove that a 1  a 2 .
(b) Prove that CN  BN.
28. In the figure, ABCD and AEFG are two identical squares. EQPB and DQRG are straight lines.
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E
D
Q
F
R
C
P
A
G
B
(a) Prove that EAB  DAG.
(b) Prove that EB  DG.
Section B
29. In the figure, ABCD is a quadrilateral. The diagonals AC and BD intersect at M. AB  AD and BC 
DC.
A
x
B
a
y
b
M
D
C
(a) Prove that x  a.
(b) Prove that x  y  90.
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