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CHAPTER 1 Lines, Angles, and Triangles
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1.1 Historical Background of Geometry 1.2 Undefined Terms of Geometry: Point, Line, and Plane 1.3 Line Segments 1.4 Circles 1.5 Angles 1.6 Triangles 1.7 Pairs of Angles
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Methods of Proof
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2.1 Proof By Deductive Reasoning 2.2 Postulates (Assumptions) 2.3 Basic Angle Theorems 2.4 Determining the Hypothesis and Conclusion 2.5 Proving a Theorem
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Congruent Triangles
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3.1 Congruent Triangles 3.2 Isosceles and Equilateral Triangles
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Parallel Lines, Distances, and Angle Sums
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4.1 Parallel Lines 4.2 Distances 4.3 Sum of the Measures of the Angles of a Triangle 4.4 Sum of the Measures of the Angles of a Polygon 4.5 Two New Congruency Theorems
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Parallelograms,Trapezoids, Medians, and Midpoints
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5.1 Trapezoids 5.2 Parallelograms 5.3 Special Parallelograms: Rectangle, Rhombus, and Square 5.4 Three or More Parallels; Medians and Midpoints
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Circles
6.1 The Circle; Circle Relationships 6.2 Tangents 6.3 Measurement of Angles and Arcs in a Circle
Similarity
7.1 Ratios 7.2 Proportions 7.3 Proportional Segments 7.4 Similar Triangles 7.8 Mean Proportionals in a Right Triangle 7.9 Pythagorean Theorem 7.10 Special Right Triangles
Trigonometry
8.1 Trigonometric Ratios 8.2 Angles of Elevation and Depression
Areas
9.1 Area of a Rectangle and of a Square 9.2 Area of a Parallelogram 9.3 Area of a Triangle 9.4 Area of a Trapezoid 9.5 Area of a Rhombus 9.6 Polygons of the Same Size or Shape 9.7 Comparing Areas of Similar Polygons
Contents 179
CHAPTER 10 Regular Polygons and the Circle
10.1 Regular Polygons 10.2 Relationships of Segments in Regular Polygons of 3, 4, and 6 Sides 10.3 Area of a Regular Polygon 10.4 Ratios of Segments and Areas of Regular Polygons 10.5 Circumference and Area of a Circle 10.6 Length of an Arc; Area of a Sector and a Segment 10.7 Areas of Combination Figures
CHAPTER 11 Locus
11.1 Determining a Locus 11.2 Locating Points by Means of Intersecting Loci 11.3 Proving a Locus
CHAPTER 12 Analytic Geometry
12.1 Graphs 12.2 Midpoint of a Segment 12.3 Distance Between Two Points 12.4 Slope of a Line 12.5 Locus in Analytic Geometry 12.6 Areas in Analytic Geometry 12.7 Proving Theorems with Analytic Geometry
CHAPTER 13 Inequalities and Indirect Reasoning
13.1 Inequalities 13.2 Indirect Reasoning
CHAPTER 14 Improvement of Reasoning
14.1 Definitions 14.2 Deductive Reasoning in Geometry 14.3 Converse, Inverse, and Contrapositive of a Statement 14.4 Partial Converse and Partial Inverse of a Theorem 14.5 Necessary and Sufficient Conditions
CHAPTER 15 Constructions
15.1 Introduction 15.2 Duplicating Segments and Angles 15.3 Constructing Bisectors and Perpendiculars 15.4 Constructing a Triangle 15.5 Constructing Parallel Lines 15.6 Circle Constructions 15.7 Inscribing and Circumscribing Regular Polygons 15.8 Constructing Similar Triangles
CHAPTER 16 Proofs of Important Theorems
16.1 Introduction 16.2 The Proofs
CHAPTER 17 Extending Plane Geometry into Solid Geometry
17.1 Solids 17.2 Extensions to Solid Geometry 17.3 Areas of Solids: Square Measure 17.4 Volumes of Solids: Cubic Measure
CHAPTER 18 Transformations
18.1 Introduction to Transformations 18.2 Transformation Notation 18.3 Translations 18.4 Reflections 18.5 Rotations 18.6 Rigid Motions 18.7 Dihilations
CHAPTER 19 Non-Euclidean Geometry
19.1 The Foundations of Geometry 19.2 The Postulates of Euclidean Geometry 19.3 The Fifth Postulate Problem 19.4 Different Geometries
Formulas for Reference Answers to Supplementary Problems Index
302 306 323
Lines, Angles, and Triangles
1.1 Historical Background of Geometry
The word geometry is derived from the Greek words geos (meaning earth) and metron (meaning measure). The ancient Egyptians, Chinese, Babylonians, Romans, and Greeks used geometry for surveying, navigation, astronomy, and other practical occupations. The Greeks sought to systematize the geometric facts they knew by establishing logical reasons for them and relationships among them. The work of men such as Thales (600 B.C.), Pythagoras (540 B.C.), Plato (390 B.C.), and Aristotle (350 B.C.) in systematizing geometric facts and principles culminated in the geometry text Elements, written in approximately 325 B.C. by Euclid. This most remarkable text has been in use for over 2000 years.
1.2 Undefined Terms of Geometry: Point, Line, and Plane
1.2A Point, Line, and Plane are Undefined Terms
These undefined terms underlie the definitions of all geometric terms. They can be given meanings by way of descriptions. However, these descriptions, which follow, are not to be thought of as definitions.
1.2B Point
A point has position only. It has no length, width, or thickness. A point is represented by a dot. Keep in mind, however, that the dot represents a point but is not a point, just as a dot on a map may represent a locality but is not the locality. A dot, unlike a point, has size. A point is designated by a capital letter next to the dot, thus point A is represented: A.
1.2C Line
A line has length but has no width or thickness. A line may be represented by the path of a piece of chalk on the blackboard or by a stretched rubber band. A line is designated by the capital letters of any two of its points or by a small letter, thus:
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