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Transcript
Key Stage 3
Measures, Shape and Space Dimension
Learning Unit:Simple Introduction to Deductive Geometry
Learning Objectives:
• develop a deductive approach to study geometric properties
through studying the story of Euclid and his book - Elements
• develop an intuitive idea of deductive reasoning by presenting
proofs of geometric problems relating with angles and lines
• understand and use the conditions for congruent and similar
triangles to perform simple proofs
Programme Title:
Simple Introduction to Deductive Geometry
Programme Objectives
1. Use daily life examples and geometrical problems to illustrate that a
conclusion deduced from the process of deductive reasoning is more
reliable than a result obtained by an intuitive method.
2. Introduce Euclid’s framework of geometry.
3. Introduce the converse theorem of a geometrical theorem.
4. Use examples to illustrate the methods of proving geometrical theorems.
Programme Content
The programme uses several examples to illustrate that: a result obtained by an
intuitive method may not be correct. Based on established principles, an
accurate conclusion can be deduced through the process of deductive reasoning.
The programme introduces Euclid and his contribution to the study of geometry.
-1-
What Euclid did was just like the idea of building a wall – he used definitions
and axioms to build up the foundation layers, then on top of those and layer by
layer, he developed various theorems to form an organized framework of
geometry.
The programme introduces the idea of a converse theorem of a geometrical
theorem. It also uses an example to illustrate two thinking processes of
formulating the proof of a geometrical problem:
 Forward deduction – to start with given conditions, use relevant axioms
and theorems, deduce forward step by step to reach the required
conclusion.
 Backward analysis – to start from the required conclusion, deduce
backward the preceding steps to meet with the established facts.
Worksheet Answers
1. corresponding angles,ABCD;
alternate angles,ABCD;
adjacent angles on a straight line.
2. EDB = 90a;
ACB = a;
DCF = a;FDC = 90a.
3. DAB = 55;
BAC = 1802ABC = 180255 = 70;
since vertically opposite angles are equal, therefore BAC = FAG = 70
-2-
Key Stage 3 ETV Programme
《Simple Introduction to Deductive Geometry》
Worksheet
1. The following uses the properties of parallel lines to prove that: interior
angle sum of a triangle is 180. State the reason for each step.
C
x
c
y
D
a
b
A
To prove: a  b  c  180
Proof:
ax
(
B
)
by
(
abc
xyc
 180
(
)
)
A
E
2. Given that:
AB = AC
ABC = a
DEAB and
DFAF
Prove that:EDB = FDB
a
B
C
F
D
【Forward Deduction】It is known that DEAB and ABC = a. In
BDE, what results can be deduced?
It is known that AB = AC and ABC = a. In ABC, what
results can be deduced?
Furthermore, in CDF, what results can be deduced?
-3-
Write down the proof of the problem:
F
3. Given that : AB=AC,DAB=55,FAG =70
Prove that:DAEC
【Backward Analysis】
In order to prove that DA // EC, we can first
show that their alternate angles are equal. That is
(1)
to show that ABC =
G
70
D
A
55
E
B
C
Since AB=AC, therefore ABC is an isosceles triangle. When statement (1)
(2)
is true, the vertex, BAC =
From the relation between BAC and FAG, is statement (2) also true?
(3)
Based on the above analysis and results obtained in (1) to (3), write down
the proof of the problem systematically:
-4-