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4 Essential Concepts of Geometry 1.2 Postulates of Euclidean Geometry According to the Elements written by Euclid, the followings are the basic postulates of plane geometry: 1. A straight line can be drawn between any two points. 2.A line can be extended indefinitely in both directions. 3. A circle can be drawn with a center and a radius. 4. All right angles are equal to each other. 5.(The parallel postulate) If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines will meet on that side if they are produced indefinitely. The parallel postulate can also be stated as follows: Given a line L and a point P not on the line, there is only one line which can be drawn through P and parallel to L. Chapter 1 Basic Concepts of Euclidean Geometry 1.3 Axioms (Common Notions) Apart from the above five postulates, the Elements also states the following axioms of plane geometry: 1. Things that equal to the same thing also equal to one another. (Transitive property) 2.If equals are added to equals, then the wholes are equal. 3. If equals are subtracted from equals, then the remainders are equal. 4.Things that coincide with one another equal to one another. (Reflexive property) 5. The whole is greater than the part. f i n i ti on De 1.4 Basic Definitions of Euclidean Geometry 1.4.1 • A point is that which has no part. It has no length, width or thickness. • A line is a set of continuous points that can extend indefinitely in either of its direction. It has length but without width and thickness. f i n i ti on De • A plane is a flat surface. It has length and width but without thickness. f i n i ti on De 1.4.2 1.4.3 A line segment is a part of a line consisting of two end points and the set of all points between them. A ray is a part of a line consisting of an end point, and the set of all points on one side of the end point. 5 14 Essential Concepts of Geometry Problems 1. Determine the truth values of the following statements. (a) For any two points on the plane, we can draw only one line passing through them. (b) For any two points A and B on a line, we can always find another point C such that B lies between A and C. (c) If we extend the sides of a triangle ∆ABC, then we can obtain 6 rays. (d) If we extend any two coplanar lines indefinitely, then we can obtain at least one intersection point. (e) For any three non-collinear points, there is only one plane which can pass through all these points. (f) If two points of a line lie on a plane, then the whole line lies on the same plane. (g) If any two non-overlapping planes intersect, then their intersection is a line. (h) If a line and a plane intersect, then there is at most one intersection point. Chapter 1 Basic Concepts of Euclidean Geometry 2. Complete the following table. Number of end-points line 0 ray 1 line segment 3. Extensibility Extendable to 2 opposite directions 2 In the diagram below, ∠AOE is an acute angle. How many acute angles are there? 15 136 Essential Concepts of Geometry Relevance to the Hong Kong Mathematics Curriculum The following tables list out the topics in this book which are relevant to some learning units of the Hong Kong Primary and Secondary Mathematics Curriculum. Key Stage 1 (Primary 1 – 3) Learning Units in Shape and Space Dimension Relevant topics in this book 3-D shapes (I) Prisms, pyramids and spheres Chapter 3 3-D shapes (II) Prisms, cylinders, pyramids and cones Chapter 3 Straight lines and curves Chapter 1 2-D shapes Polygons and circles Chapter 2 Quadrilaterals (I) Rectangles, squares, trapeziums, rhombuses, etc. Chapter 2 Quadrilaterals (II) Characteristics of parallelograms Chapter 2 Triangles Chapter 2 Angles (I) Angles and right angles Chapter 1 Angles (II) Acute and obtuse angles Chapter 1 Parallel and perpendicular Chapter 1 Learning Units in Measures Dimension Length and distance (I) Basic concept Relevant topics in this book Chapter 5 Relevance to the Hong Kong Mathematics Curriculum Key Stage 2 (Primary 4 – 6) Learning Units in Shape and Space Dimension Relevant topics in this book Quadrilaterals (III) Characteristics of quadrilaterals Chapter 2 Symmetry Chapter 8 3-D shapes (III) Characteristics of prisms, pyramids and spheres Chapter 3 3-D shapes (IV) Vertices, edges, faces Chapter 3 Circles Chapter 1 Suggested enhanced topic — Tessellation Chapter 9 Suggested enhanced topic — Rotational symmetry Chapter 8 Learning Units in Measures Dimension Relevant topics in this book Perimeter (I) Irregular shapes, squares and rectangles Chapter 5 Perimeter (II) Circumference Chapter 5 Area (I) Squares, rectangles Chapter 5 Area (II) Parallelograms, triangles, trapeziums and polygons Chapter 5 Volume (I) Cuboids, cubes Chapter 5 Volume (II) Capacity and volume Chapter 5 Relevance to the Hong Kong Mathematics Curriculum 137