Download 1.2 Postulates of Euclidean Geometry

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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.
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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.
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• A plane is a flat surface. It has length and width but without thickness.
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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.
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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?
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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