從平行公設探討三角形相似與全等性質

本研究旨在透過國中數學領域課程內容，探討課程中所編排之三角形全等、平行線、比例線段、三角形相似等單元，並將課本教材解構再重建，重新組織另一套系統性知識。研究者以幾何原本的平行公設出發，運用四個引理：平行線間距離處處相等、兩直線距離處處相等則平行、平行線截比例線段性質及其逆敘述，來探究三角形的相似與全等性質的等價關係。研究者希望透過此研究，使學生觀察某個定理在數學發展中的關鍵角色，連結過去的經驗，進而瞭解整個數學結構，解讀更高深的數學知識，並做好數學概念連結的動作。研究結果如下：一、藉由探究國中數學之幾何證明，深化學生對數學定理之理解，培養學生多元思考的能力。二、藉由數學史敘事引導，讓學生主動探索數學概念的歷史根源，了解數學的優美。三、數學是一種語言，是訓練思考最完美的工具，在熟練後才能更深刻。

關鍵字

平行公設；相似三角形性質；全等三角形性質

並列摘要

The main purpose of my study is to discuss the equivalence of triangular congruent, parallel lines, proportional line segments, and triangular similarities. By deconstructing and rebuilding textbooks, I will reorganize another set of systematic knowledge. My research starts with parallel postulate of Euclid's Elements and uses four lemmas: parallel lines are always the same equidistant, if the distance between two straight lines is the same, lines will be parallel, the parallel cuts two sides of triangle into proportional line segments, and converse theorem, to explore the equivalence of the similarities and congruence properties of triangles. Through my study, students will focus on the key role of certain theorem in the development of math, link the past experience, and then understand the entire math structure. And interpret more advanced math knowledge. Then finally make math conceptual connections. The results are as follows： First, by exploring the geometric proof, we can strengthen students' understanding of math theorems and develop their ability to think in multiple ways. Second, with the guiding of math history, students can learn actively to explore the roots of the concept and understand the beauty of math. Third, math is a language, the most perfect training and thinking tool, and students can understand deeply after proficiency.