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六年級 1) 在學習了教學大綱中的主題“平行線公理及其推論”以後;因之在學生們知道了歐幾里得第五公設的表述(附錄1)以後,必須在做習題的課上,考察關於平行線公理各種表述的等價證明的習題,在貝斯金(H.M.BeckИH)的幾何教學法(?)第115頁中可以找到證明。(附錄2) 2) 歐幾里得的第五公設無異於下列命題:同一直線的垂直線和斜線恒相交。說明這一點是有好處的,其證明需要用到一個定理,即所有三角形中,任意二內角之和小於二直角。 3) 學習到教學大綱中的主題“三角形諾角之和的定理”時,必須讓學生來分析這定理的證明,說明我們在證明中用到了平行性的反定理。顯然平行性的反定理可根據關於平行線的公設來證明,因此“三角形諸角之和等於二直角”的定理的正確性可從歐幾里得第五公設推出來。
Grade 6 1) After studying the subject of the syllabus “Parallel Line Axioms and Inferences”, students must be doing exercises after they know the fifth public expression of Euclid (Appendix 1). In the above, the examination of the equivalent proofs of various expressions of the axioms of parallelism can be found in the HBMeck HH Geometry Teaching Method (?) page 115. (Appendix 2) 2) The fifth design of Euclideis is tantamount to the following proposition: The vertical line and the diagonal line of the same line always intersect. Explaining this point is beneficial. It proves that a theorem is needed. That is, in all triangles, the sum of any two internal angles is less than two right angles. 3) When learning the topic “Theorem of the sum of triangles and horns” in the syllabus, students must analyze the proof of the theorem to show that we use the inverse theorem of parallelism in the proof. Obviously the inverse theorem of parallelism can be proved based on the arbitrariness of parallel lines. Therefore, the correctness of the theorem of “the sum of the triangles’ angles is equal to two orthogonal angles” can be derived from the fifth epoch of Euclidean.