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By Christopher Lawson,2014-08-23 01:27
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Key Stage 3

    Measures, Shape and Space Dimension

    Learning UnitSimple 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 Euclids 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. 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 anglesAB??CD

     alternate anglesAB??CD

     adjacent angles on a straight line.

2. EDB = 90,;a

     ACB = a

     DCF = aFDC = 90,;a.

3. DAB = 55

     BAC = 180,;2(,ABC = 180,;2(55 = 70

     since vertically opposite angles are equal, therefore BAC = FAG = 70

    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 D

     c y

     a b

     A B

    To prove a ; b ; c 180

     Proof a x ( )

     b y ( )

     a ; b ; c

     x ; y ; c

     180 ( )

     A

    2. Given that: AB = AC

    ABC = a

    DEAB and

    DFAF

    Prove thatEDB = FDB B ( D

    

    Forward DeductionIt is known that DEAB 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?

Write down the proof of the problem

     F G

    703. Given that : AB=ACDAB=55FAG =70 D A

    55Prove thatDA??EC

    Backward Analysis

    In order to prove that DA // EC, we can first

    show that their alternate angles are equal. That is E B C

     (1)to show that ABC =

    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:

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