This document provides 13 multi-part geometry problems involving concepts like congruence, similarity, angles, parallelograms, rectangles, and trapezoids. Each problem includes one or more figures with labeled points and geometric shapes, given information, and questions to prove properties or calculate missing angle or length values. Solutions are to be shown deductively by stating givens and using prior results to arrive at the conclusion.

1. DEDUCTIVE GEOMETRY
Mathematics
Revision Exercise
DEDUCTIVE GEOMETRY
CHAPTER 1 CONGRUENCE AND SIMILARITY
1. [2009-CE-MATH 1]
In Figure 1, C is a point lying on DE. AE and BC intersect at F. It is given that AC  AD ,
BC  DE and BCE  CAD .
(a) Prove that ABC  AED . (3 marks)
(b) If AD // BC ,
(i) prove that ABF ~ DEA ;
(ii) write down two other triangles which are similar to ABF .
(5 marks)
2. [2001-CE-MATH 1]
As shown in Figure 2, a piece of square paper ABCD of side 12 cm is folded along a line
segment PQ so that the vertex A coincides with the mid-point of the side BC. Let the new
positions of A and D be A´ and D´ respectively, and denote by R the intersection of A´D´ and
CD.
(a) Let the length of AP be x cm. By considering the triangle PBA´, find x. (3 marks)
(b) Prove that the triangles PBA´ and A´CR are similar. (3 marks)
(c) Find the length of A´R. (2 marks)
3. In Figure 3, ABCD is a parallelogram. E is a point lying on CD produced so that BE cuts AD at
F and the diagonal AC at G.
(a) (i) Prove that ABG ~ CEG .
(ii) Prove that AFG ~ CBG .
(iii) Using (a) (i) and (a) (ii), prove that BG 2  EG  FG .
(7 marks)
1
(b) If FD  BC ,
2
(i) write down a pair of congruent triangles;
(ii) find the ratio EF : FG .
(4 marks)
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2. DEDUCTIVE GEOMETRY
CHAPTER 2 ANGLES RELATED TO RECTILINEAR FIGURES
4. [2007-CE-MATH 1]
In Figure 4, ABC and DEF are straight lines. It is given that AC // DF , BC  CF ,
EBF  90  and BED  110  . Find x, y and z.
(4 marks)
5. [2008-CE-MATH 1]
In Figure 5, AB // CD . E is a point lying on AD such that AE  AC . Find x, y and z.
(5 marks)
6. In Figure 6, AB // DE . It is given that AB  BC  CD  DE , ACB  32  and
BCD  56  . Find x, y and z.
(5 marks)
7. [2005-CE-MATH 1]
In Figure 7, ABCDEF is a regular six-sided polygon. AC and BF intersect at G. Find x, y and z.
(5 marks)
8. [2002-CE-MATH 1]
In Figure 8, ABC is a triangle in which BAC  20  and AB  AC . D, E are points on AB
and F is a point on AC such that BC  CE  EF  FD .
(a) Find CEF (4 marks)
(b) Prove that AD  DF . (3 marks)
CHAPTER 3 QUADRILATERALS
9. [2006-CE-MATH 1]
In Figure 9, ABCD is a parallelogram. E is a point lying on AD such that AE  AB . It is given
that EBC  70  . Find ABE and BCD .
(3 marks)
10. In Figure 10, ABCD is a square. E is a point lying inside the square so that AD  DE  EA . BE
is produced to meet CD at F. Find x, y and z.
(5 marks)
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3. DEDUCTIVE GEOMETRY
11. In Figure 11, ABCD is a rectangle. E is the mid-point of AD and F is a point on BE such that
CF  BE .
(a) Prove that ABE ~ FCB . (3 marks)
(b) If CD  8 and DE  6 , find the length of CF . (4 marks)
12. * In Figure 12, AD // BC . It is given that M and N are the mid-points of AB and DC respectively.
AN is extended to meet BC produced at O.
(a) Prove that ADN  OCN . (3 marks)
(b) Suppose AD  4 , BC  9 and ABC  43  . Using (a), or otherwise, find
(i) AMN ;
(ii) the length of MN .
(4 marks)
13. * In figure 13, AD // BC . It is given that AB  CD  AD and BCD  60  . E is a point on BD
such that AE  BD . F is the mid-point of CD, and DG is the altitude of the trapezium ABCD.
(a) (i) Prove that AD // EF .
(ii) Using (a) (i), prove that AEFD is a parallelogram.
(5 marks)
(b) If AE  3 , find the area of quadrilateral DEGF. (4 marks)
APPENDIX FIGURES
D
A D
A
Q
D
R
C
F P
B
B A´ C
E
Figure 1 Figure 2
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4. DEDUCTIVE GEOMETRY
E
B C
A
A F
D z
110° x y
G
D E F
B C
Figure 3 Figure 4
C D
43° x E
y
y B z
E D
A x
z 32° 56°
33
C
A B
Figure 5 Figure 6
A
A B
20°
D
z° y°
F C
F
x°
E D E
B C
Figure 7 Figure 8
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5. DEDUCTIVE GEOMETRY
A D
B C
70° x°
y° F
E
A E D z°
B C
Figure 9 Figure 10
A D
B C
M
N
F
A E D
B C O
Figure 11 Figure 12
A D
F
E
B G C
Figure 13
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