Download 1.5 Relations between Angles with a Common Vertex

Contents
Chapter 1
Basic Concepts of Euclidean Geometry............................................................... 1
1.1Introduction.......................................................................................................2
1.2 Postulates of Euclidean Geometry.................................................................4
1.3 Axioms (Common Notions)..............................................................................5
1.4 Basic Definitions of Euclidean Geometry.......................................................5
1.5 Relations between Angles with a Common Vertex......................................8
1.6 Relations between Angles formed by Two Lines and a Transversal.........10
1.7 Parallel Lines and Perpendicular Lines.........................................................11
1.8 Drawing Parallel Lines by Straightedge and Compasses Only.................12
1.9 Drawing Perpendicular Lines by Straightedge and Compasses Only.....13
Problems...................................................................................................................14
References...............................................................................................................18
Chapter 2
Plane Figures.......................................................................................................... 19
2.1 Concepts of Polygons....................................................................................20
2.2 Some Common Polygons..............................................................................23
Problems...................................................................................................................25
References...............................................................................................................28
Chapter 3
Solids....................................................................................................................... 29
3.1Polyhedron......................................................................................................30
3.2 Prism, Pyramid and Frustum of Pyramid.......................................................33
3.3 Regular Polyhedron and Archimedean Solid.............................................35
3.4 Cylinder, Cone, Frustum of Cone and Sphere............................................37
Problems...................................................................................................................39
References...............................................................................................................44
Chapter 4
Construction of Regular Polygon and Regular Polyhedron............................... 45
4.1 Construction of an Equilateral Triangle by Straightedge and
Compasses Only.............................................................................................46
4.2 Construction of a Square by Straightedge and Compasses Only...........47
4.3 Construction of a Regular Pentagon by Straightedge and
Compasses Only.............................................................................................48
4.4 Construction of Regular Polyhedron by Paper-folding..............................51
Problems...................................................................................................................54
References...............................................................................................................56
Chapter 5
Perimeter, Area and Volume................................................................................ 57
5.1 Length and Perimeter....................................................................................58
5.2Area..................................................................................................................59
5.3Volume.............................................................................................................61
Problems...................................................................................................................62
References...............................................................................................................66
Chapter 6
Pick’s Formula........................................................................................................ 67
6.1Introduction.....................................................................................................68
6.2Pick’s Formula..................................................................................................72
Problems...................................................................................................................75
References...............................................................................................................78
Chapter 7
Cavalieri’s Principle............................................................................................... 79
7.1Introduction.....................................................................................................80
7.2 The Cavalieri’s First Principle..........................................................................80
7.3 The Cavalieri’s Second Principle...................................................................83
Problems...................................................................................................................87
References...............................................................................................................90
Chapter 8
Geometric Transformations................................................................................... 91
8.1Introduction.....................................................................................................92
8.2 Isometric Transformations..............................................................................93
8.3 Examples of Applications..............................................................................96
8.4Symmetry.......................................................................................................101
Problems.................................................................................................................103
References.............................................................................................................108
Chapter 9
Tessellations.......................................................................................................... 109
9.1Introduction...................................................................................................110
9.2 Tessellations by Polygons.............................................................................112
9.3Rep-tiles.........................................................................................................113
9.4Escher’s Tessellations.....................................................................................114
9.5 Penrose Tilings...............................................................................................115
Problems.................................................................................................................116
References.............................................................................................................118
Chapter 10 The Euler Polyhedral Formula............................................................................. 119
10.1Introduction...................................................................................................120
10.2 The Euler Polyhedral Formula......................................................................121
10.3 The Euler-Poincare Polyhedral Formula.....................................................125
Problems.................................................................................................................126
References.............................................................................................................128
Revision Exercises......................................................................................................................... 129
Relevance to the Hong Kong Mathematics Curriculum.......................................................... 135
Answers......................................................................................................................................... 143
Index.............................................................................................................................................. 149
Contents
Chapter 1
Basic Concepts of Euclidean Geometry
Chapter 2
Plane Figures
1
17
Chapter 3Solids
27
Chapter 4
Construction of Regular Polygon and Regular Polyhedron
41
Chapter 5
Perimeter, Area and Volume
55
Chapter 6
Pick’s Formula
67
Chapter 7
Cavalieri’s Principle
79
Chapter 8
Geometric Transformations
91
Chapter 9Tessellations
109
Chapter 10 The Euler Polyhedral Formula
119
Challenging Questions
129
Syllabus Summary
135
Answers145
Index150
Chapter 1
Basic Concepts of Euclidean Geometry
Euclid of Alexandria
(about 325 – 265 B.C.)
Greek mathematician, author of the
Elements
In this chapter, you will learn:
• the Postulates and Axioms of Euclidean Geometry
• the basic definitions in Euclidean Geometry
• the relationship between angles with a common vertex
• the relationship between angles formed by two lines and a transversal
• parallel lines and perpendicular lines
• construction of parallel lines and perpendicular lines by straightedge and
compasses only
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Essential Concepts of Geometry
1.1Introduction
The origin of geometry was motivated by practical problems of measurement, such as land
measurement, fields surveying and temple constructions, etc. In fact, the word geometry
has its roots in the Greek word geometrein, which means ‘to measure the land’. Its further
development into a rigorous abstract mathematical subject was largely due to the efforts and
contributions of ancient Greek mathematicians, including Thales, Pythagoras, Hippocrates,
Eudoxus, Euclid, Archimedes, Appollonius, Heron, Menelaus, Ptolemy and Pappus, etc.
HKEP
Pythagoras (570 – 495 B.C.)
Thales (624 – 546 B.C.)
The most famous and influential geometry textbook ever written in ancient Greek is the Elements
（幾何原本）. The author was Euclid of Alexandria (about 325 – 265 B.C.). The whole book
consists of a collection of definitions, postulates, axioms (common notions), propositions
(theorems and constructions) and proofs. The propositions have progressed in small steps,
and they are introduced and proved based on the already known definitions, postulates, axioms
and other propositions that have been proved previously. This book is one of the oldest extant
Greek mathematical works that provide an axiomatic deductive treatment of geometry. Almost
all important theorems of geometry and arithmetic developed at around 300 B.C. by the Greek
mathematicians were collected there and presented with a rigor which remained unparalleled
for the following two thousand years.
Chapter 1 Basic Concepts of Euclidean Geometry
The Elements starts with the basic definitions, postulates and axioms. The definitions explain
the basic terms of geometry, such as point, line and angle, etc. The axioms or common notions
are self-evident statements that any sensible person would accept them as true without proof,
which are the usual rules for equations or inequalities. Similar to axioms, postulates are self-
evident statements or unproved assertions about geometry. The distinction between an axiom
and a postulate is that an axiom refers to a self-evident assumption common to many areas of
inquiry1, while a postulate refers to a hypothesis specific to a certain line of inquiry, that is
accepted to be true without proof.
The English version of the Elements
In this chapter, we will introduce the famous postulates and axioms of Euclidean geometry,
and some basic definitions of geometry which are often taught at the primary or secondary
level. Also, we will describe how to construct parallel lines and perpendicular lines by using
straightedge and compasses only.
That is why axioms are also known as common notions.
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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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Essential Concepts of Geometry
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1.4.4
1.4.5
Collinear points are points lying on the same line.
A broken line is composed of a series of line segments joining end to end,
and they do not lie on a single straight line.
Note: In a broken line, any two adjacent segments share the same end
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point.
1.4.6
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1.4.7
Coplanar points are points lying on the same plane.
An angle is the union of two rays having a common end point. An angle
can be represented by ∠AOB (or ∠BOA), where O is the vertex of the
angle and OA, OB are the sides of the angle.
Note: The lengths of the sides of an angle do not affect the measure of
the angle and the common measurment unit of an angle is either in
degree or in radian1.
An angle can be formed by rotating a ray at its endpoint. For example, we can form an angle by
rotating OA at the vertex O to become OB. Then, OA is called the initial side and OB is called
the terminating side2.
Radian measure is included in Module 2 (Extended Part) of the Hong Kong Senior Secondary
Mathematics Curriculum. It is covered in the learning unit ʻMore about Trigonometric Functions’.
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The term ‘terminal side’ is often used in Hong Kong Mathematics textbooks.
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Chapter 1 Basic Concepts of Euclidean Geometry
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1.4.8
1.4.10
An acute angle is an angle q such that 0° < q < 90°.
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A right angle is an angle of 90°.
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1.4.9
1.4.11
An obtuse angle is an angle q such that 90° < q < 180°.
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A complete angle1 is an angle of 360°.
1.4.12
A reflex angle is an angle q such that 180° < q < 360°.
The term ‘round angle’ is often used in Hong Kong Mathematics textbooks.
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Essential Concepts of Geometry
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1.5 Relations between Angles with a Common Vertex
1.5.1
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1.5.2
Two angles are called adjacent angles if they have a common vertex and a
common side.
Two angles are called complementary angles 1 if the sum of their
measures is 90°.
This term is often introduced in the topic ‘Trigonometric Identities of Complementary Angles’ in Hong
Kong Mathematics textbooks. For example, the angles q and (90° – q ) in the trigonometric identity
cos (90° – q ) = sin q are complementary angles.
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Chapter 1 Basic Concepts of Euclidean Geometry
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1.5.3
Two angles are called supplementary angles if the sum of their measures
is 180°.
When angles share a common vertex, they are called angles at a point.
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1.5.4
1.5.5
When two straight lines intersect, the non-adjacent angles formed are
called vertically opposite angles.
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10 Essential Concepts of Geometry
1.6 Relations between Angles formed by Two Lines
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and a Transversal
1.6.1
The pair of angles on the same side of the transversal but inside the two
lines are called interior angles on the same side of the transversal.
e.g.1. ∠c and ∠h are interior angles on the same side of the transversal.
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∠b and ∠e are interior angles on the same side of the transversal.
1.6.2
The pair of angles on the same side of the transversal but outside the two
lines are called exterior angles on the same side of the transversal1.
e.g.1. ∠d and ∠g are exterior angles on the same side of the transversal.
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2.
∠a and ∠f are exterior angles on the same side of the transversal.
1.6.3
Corresponding angles are a pair of angles in matching corners.
e.g.1. ∠a and ∠e are corresponding angles.
2.
∠b and ∠f , ∠c and ∠g, ∠d and ∠h are also corresponding angles.
This term is seldom introduced in Hong Kong Mathematics textbooks.
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