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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，ABCD； alternate angles，ABCD； adjacent angles on a straight line. 2. EDB = 90a； ACB = a； DCF = a；FDC = 90a. 3. DAB = 55； BAC = 1802ABC = 180255 = 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： ax ( B ) by ( abc xyc 180 ( ) ) A Ｅ 2. Given that: AB = AC ABC = a DEAB and DFAF Prove that：EDB = FDB ａ B Ｃ Ｆ D 【Forward Deduction】It 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? -3- Write down the proof of the problem： F 3. Given that : AB=AC，DAB=55，FAG =70 Prove that：DAEC 【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-