Revision exercise on deductive geometry.

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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