Geometry for dummies

411 Pages·2016·10.814 MB·English

by  Mark, Ryan

Geometry
3rd edition
by Mark Ryan
Geometry For Dummies®, 3rd Edition
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2016 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Control Number: 2016936127
ISBN 978-1-119-18155-2 (pbk); ISBN 978-1-119-18164-4 (ebk); ISBN 978-1-119-18156-9 (ebk)
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents at a Glance
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Part 1: Getting Started with Geometry Basics . . . . . . . . . . . . . . . .5
CHAPTER 1: Introducing Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
CHAPTER 2: Building Your Geometric Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
CHAPTER 3: Sizing Up Segments and Analyzing Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Part 2: Introducing Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
CHAPTER 4: Prelude to Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
CHAPTER 5: Your Starter Kit of Easy Theorems and Short Proofs . . . . . . . . . . . . . . . . .55
CHAPTER 6: The Ultimate Guide to Tackling a Longer Proof . . . . . . . . . . . . . . . . . . . . . 75
Part 3: Triangles: Polygons of the Three-Sided Variety . . . . . 87
CHAPTER 7: Grasping Triangle Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
CHAPTER 8: Regarding Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
CHAPTER 9: Completing Congruent Triangle Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Part 4: Polygons of the Four-or-More-Sided Variety . . . . . . . 153
CHAPTER 10: The Seven Wonders of the Quadrilateral World . . . . . . . . . . . . . . . . . . . 155
CHAPTER 11: Proving That You Have a Particular Quadrilateral . . . . . . . . . . . . . . . . . . 177
CHAPTER 12: Polygon Formulas: Area, Angles, and Diagonals . . . . . . . . . . . . . . . . . . . 193
CHAPTER 13: Similarity: Same Shape, Different Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 .11
Part 5: Working with Not-So-Vicious Circles . . . . . . . . . . . . . . . . 235
CHAPTER 14: Coming Around to Circle Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
CHAPTER 15: Circle Formulas and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 .55
Part 6: Going Deep with 3-D Geometry . . . . . . . . . . . . . . . . . . . . . . 277
CHAPTER 16: 3-D Space: Proofs in a Higher Plane of Existence . . . . . . . . . . . . . . . . . . 279
CHAPTER 17: Getting a Grip on Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Part 7: Placement, Points, and Pictures:
Alternative Geometry Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303
CHAPTER 18: Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
CHAPTER 19: Changing the Scene with Geometric Transformations . . . . . . . . . . . . . 323
CHAPTER 20: Locating Loci and Constructing Constructions . . . . . . . . . . . . . . . . . . . . 343
Part 8: The Part of Tens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
CHAPTER 21: Ten Things to Use as Reasons in Geometry Proofs . . . . . . . . . . . . . . . . 363
CHAPTER 22: Ten Cool Geometry Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Table of Contents
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
About This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Conventions Used in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
What You’re Not to Read . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Foolish Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Icons Used in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Beyond the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Where to Go from Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
PART 1: GETTING STARTED WITH GEOMETRY BASICS . . . . . . 5
CHAPTER 1: Introducing Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Studying the Geometry of Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
One-dimensional shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Two-dimensional shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Three-dimensional shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Getting Acquainted with Geometry Proofs . . . . . . . . . . . . . . . . . . . . . . . 10
Easing into proofs with an everyday example . . . . . . . . . . . . . . . . . 11
Turning everyday logic into a proof . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Sampling a simple geometrical proof . . . . . . . . . . . . . . . . . . . . . . . . . 13
When Am I Ever Going to Use This? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
When you’ll use your knowledge of shapes . . . . . . . . . . . . . . . . . . . 14
When you’ll use your knowledge of proofs . . . . . . . . . . . . . . . . . . . . 15
Why You Won’t Have Any Trouble with Geometry . . . . . . . . . . . . . . . . . 16
CHAPTER 2: Building Your Geometric Foundation . . . . . . . . . . . . . . . . 17
Getting Down with Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A Few Points on Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Lines, Segments, and Rays Pointing Every Which Way . . . . . . . . . . . . . 22
Singling out horizontal and vertical lines . . . . . . . . . . . . . . . . . . . . . . 22
Doubling up with pairs of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Investigating the Plane Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Everybody’s Got an Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Goldilocks and the three angles: Small, large, and just “right” . . . . 26
Angle pairs: Often joined at the hip . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Table of Contents v
CHAPTER 3: Sizing Up Segments and Analyzing Angles . . . . . . . . . . 31
Measuring Segments and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Measuring segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Measuring angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Adding and Subtracting Segments and Angles . . . . . . . . . . . . . . . . . . . . 36
Cutting in Two or Three: Bisection and Trisection . . . . . . . . . . . . . . . . . 37
Bisecting and trisecting segments . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Bisecting and trisecting angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Proving (Not Jumping to) Conclusions about Figures . . . . . . . . . . . . . . . 40
PART 2: INTRODUCING PROOFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
CHAPTER 4: Prelude to Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Getting the Lay of the Land: The Components of a Formal
Geometry Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Reasoning with If-Then Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
If-then chains of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
You’ve got your reasons: Definitions, t heorems,
and postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bubble logic for two-column proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Horsing Around with a Two-Column Proof . . . . . . . . . . . . . . . . . . . . . . . 52
Your Starter Kit of Easy Theorems
CHAPTER 5:
and Short Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Doing Right and Going Straight: Complementary and
Supplementary Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Addition and Subtraction: Eight No-Big-Deal Theorems . . . . . . . . . . . . 59
Addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
Subtraction theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Like Multiples and Like Divisions? Then These Theorems
Are for You! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
The X-Files: Congruent Vertical Angles Are Out There . . . . . . . . . . . . . . 69
Pulling the Switch with the Transitive and Substitution Properties . . . . 71
CHAPTER 6: The Ultimate Guide to Tackling a Longer Proof . . . . 75
Making a Game Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Using All the Givens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
Making Sure You Use If-Then Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chipping Away at the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Jumping Ahead and Working Backward . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Filling In the Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Writing Out the Finished Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
vi Geometry For Dummies
PART 3: TRIANGLES: POLYGONS OF THE
THREE-SIDED VARIETY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87
CHAPTER 7: Grasping Triangle Fundamentals . . . . . . . . . . . . . . . . . . . . .89
Taking In a Triangle’s Sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Scalene triangles: Akilter, awry, and askew . . . . . . . . . . . . . . . . . . . . 90
Isosceles triangles: Nice pair o’ legs . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Equilateral triangles: All parts are created equal . . . . . . . . . . . . . . . 92
Introducing the Triangle Inequality Principle . . . . . . . . . . . . . . . . . . . . . 92
Getting to Know Triangles by Their Angles . . . . . . . . . . . . . . . . . . . . . . . 94
Sizing Up Triangle Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Scaling altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Determining a triangle’s area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96
Locating the “Centers” of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Balancing on the centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Finding three more “centers” of a triangle . . . . . . . . . . . . . . . . . . . .103
CHAPTER 8: Regarding Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Applying the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Perusing Pythagorean Triple Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 113
The Fab Four Pythagorean triple triangles . . . . . . . . . . . . . . . . . . . 114
Families of Pythagorean triple triangles . . . . . . . . . . . . . . . . . . . . . .116
Getting to Know Two Special Right Triangles . . . . . . . . . . . . . . . . . . . . 118
The 45°- 45°- 90° triangle — half a square . . . . . . . . . . . . . . . . . . . 119
The 30°- 60°- 90° triangle — half of an equilateral triangle . . . . . 120
CHAPTER 9: Completing Congruent Triangle Proofs . . . . . . . . . . . . . 125
Introducing Three Ways to Prove Triangles Congruent . . . . . . . . . . . . 126
SSS: Using the side-side-side method . . . . . . . . . . . . . . . . . . . . . . . 127
SAS: Taking the side-angle-side approach . . . . . . . . . . . . . . . . . . . . 128
ASA: Taking the angle-side-angle tack . . . . . . . . . . . . . . . . . . . . . . . 131
CPCTC: Taking Congruent Triangle Proofs a Step Further . . . . . . . . . .133
Defining CPCTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Tackling a CPCTC proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Eying the Isosceles Triangle Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 137
Trying Out Two More Ways to Prove Triangles Congruent . . . . . . . . . 139
AAS: Using the angle-angle-side theorem . . . . . . . . . . . . . . . . . . . . 139
HLR: The right approach for right triangles . . . . . . . . . . . . . . . . . . .142
Going the Distance with the Two Equidistance Theorems . . . . . . . . . 143
Determining a perpendicular bisector . . . . . . . . . . . . . . . . . . . . . . .144
Using a perpendicular bisector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Making a Game Plan for a Longer Proof . . . . . . . . . . . . . . . . . . . . . . . . 147
Running a Reverse with Indirect Proofs . . . . . . . . . . . . . . . . . . . . . . . . . 149
Table of Contents vii
PART 4: POLYGONS OF THE FOUR-OR-MORE-SIDED
VARIETY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153
CHAPTER 10: The Seven Wonders of the Quadrilateral World . . . . 155
Getting Started with Parallel-Line Properties . . . . . . . . . . . . . . . . . . . . 156
Crossing the line with transversals: D efinitions
and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Applying the transversal theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Working with more than one transversal . . . . . . . . . . . . . . . . . . . . 160
Meeting the Seven Members of the Quadrilateral Family . . . . . . . . . . 161
Looking at quadrilateral relationships . . . . . . . . . . . . . . . . . . . . . . .163
Working with auxiliary lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
Giving Props to Quads: The Properties of Quadrilaterals . . . . . . . . . . 166
Properties of the parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Properties of the three special p arallelograms . . . . . . . . . . . . . . . . 170
Properties of the kite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Properties of the trapezoid and the isosceles trapezoid . . . . . . . . 175
Proving That You Have a Particular
CHAPTER 11:
Quadrilateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177
Putting Properties and Proof Methods Together . . . . . . . . . . . . . . . . . 178
Proving That a Quadrilateral Is a Parallelogram . . . . . . . . . . . . . . . . . .180
Surefire ways of ID-ing a parallelogram . . . . . . . . . . . . . . . . . . . . . . 180
Trying some parallelogram proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Proving That a Quadrilateral Is a Rectangle, Rhombus,
or Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Revving up for rectangle proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Waxing rhapsodic about rhombus proofs . . . . . . . . . . . . . . . . . . . . 187
Squaring off with square proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Proving That a Quadrilateral Is a Kite . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
CHAPTER 12: Polygon Formulas: Area, Angles, and Diagonals . . . . 193
Calculating the Area of Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . .193
Setting forth the quadrilateral area formulas . . . . . . . . . . . . . . . . . 194
Getting behind the scenes of the formulas . . . . . . . . . . . . . . . . . . . 194
Trying a few area problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Finding the Area of Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Presenting polygon area formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Tackling more area problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Using Polygon Angle and Diagonal Formulas . . . . . . . . . . . . . . . . . . . . 205
Interior and exterior design: Exploring polygon angles . . . . . . . . . 206
Handling the ins and outs of a polygon angle problem . . . . . . . . .207
Criss-crossing with diagonals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
viii Geometry For Dummies