Geometry Essentials For Dummies?
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Table of Contents
About This Book Conventions Used in This Book
Icons Used in This Book
Where to Go from Here
Chapter 1: An Overview of Geometry
The Geometry of Shapes
One-dimensional shapes Two-dimensional shapes
Am I Ever Going to Use This?
When you’ll use your knowledge of shapes When you’ll use your knowledge of proofs
Getting Down with Definitions
A Few Points on Points
Lines, Segments, and Rays
Horizontal and vertical lines Doubling up with pairs of lines
Investigating the Plane Facts
Everybody’s Got an Angle
Five types of angles Angle pairs
Bisection and Trisection
Chapter 2: Geometry Proof Starter Kit
The Lay of the (Proof) Land Reasoning with If-Then Logic
If-then chains of logic Definitions, theorems, and postulates
Complementary and Supplementary Angles
Addition and Subtraction
Addition theorems Subtraction theorems
Like Multiples and Like Divisions
Congruent Vertical Angles
Transitivity and Substitution
Chapter 3: Tackling a Longer Proof
Making a Game Plan Using All the Givens
Using If-Then Logic
Chipping Away at the Problem
Filling In the Gaps
Writing Out the Finished Proof
Chapter 4: Triangle Fundamentals
Taking In a Triangle’s Sides
Scalene triangles Isosceles triangles
Triangle Classification by Angles
The Triangle Inequality Principle
Sizing Up Triangle Area
A triangle’s altitude or height Determining a triangle’s area
Regarding Right Triangles
The Pythagorean Theorem
Pythagorean Triple Triangles
The Fab Four triangles Families of Pythagorean triple triangles
Two Special Right Triangles
The 45?- 45?- 90? triangle The 30?- 60?- 90? triangle
Chapter 5: Congruent Triangle Proofs
Proving Triangles Congruent
SSS: The side-side-side method SAS: side-angle-side
ASA: The angle-side-angle tack
Last but not least: HLR
Taking the Next Step with CPCTC
Defining CPCTC Tackling a CPCTC proof
The Isosceles Triangle Theorems
The Two Equidistance Theorems
Determining a perpendicular bisector Using a perpendicular bisector
Chapter 6: Quadrilaterals
Parallel Line Properties
Parallel lines with a transversal The transversal theorems
The Seven Special Quadrilaterals
Working with Auxiliary Lines
The Properties of Quadrilaterals
Properties of the parallelogram Properties of the three special parallelograms
Properties of the kite
Properties of the trapezoid and the isosceles trapezoid
Proving That You’ve Got a Particular Quadrilateral
Proving you’ve got a parallelogram Proving that you’ve got a rectangle, rhombus, or square
Proving that you’ve got a kite
Chapter 7: Polygon Formulas
The Area of Quadrilaterals
Quadrilateral area formulas Why the formulas work
Trying a few area problems
The Area of Regular Polygons
The polygon area formulas Tackling an area problem
Angle and Diagonal Formulas
Interior and exterior angles A polygon angle problem
Criss-crossing with diagonals
Chapter 8: Similarity
Defining similar polygons How similar figures line up
Solving a similarity problem
Proving Triangles Similar
Tackling an AA proof Using SSS~
An SAS~ proof
Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem
More Proportionality Theorems
The Side-Splitter Theorem The Angle-Bisector Theorem
Chapter 9: Circle Basics
Radii, Chords, and Diameters
Five circle theorems Using extra radii
Arcs and Central Angles
The Pizza Slice Formulas
Determining arc length Sector and segment area
The Angle-Arc Formulas
Angles on a circle Angles inside a circle
Angles outside a circle
Keeping the formulas straight
The Power Theorems
The Chord-Chord Theorem The Tangent-Secant Theorem
The Secant-Secant Theorem
Condensing the power theorems into a single idea
Chapter 10: 3-D Geometry
Flat-Top Figures Pointy-Top Figures
Chapter 11: Coordinate Geometry
The Coordinate Plane Slope, Distance, and Midpoint
The slope dope The distance formula
The midpoint formula
Trying out the formulas
Equations for Lines and Circles
Line equations The circle equation
Chapter 12: Ten Big Reasons to Use in Proofs
The Reflexive Property Vertical Angles Are Congruent
The Parallel-Line Theorems
Two Points Determine a Line
All Radii Are Congruent
If Sides, Then Angles
If Angles, Then Sides
Geometry Essentials For Dummies ?
by Mark Ryan
Geometry Essentials For Dummies ?
Published byWiley Publishing, Inc.
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Hoboken, NJ 07030-5774www.wiley.com
Copyright ? 2011 by Wiley Publishing, Inc., Indianapolis, Indiana
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted inany form or by any means, electronic, mechanical, photocopying, recording, scanning, orotherwise, except as permitted under Sections 107 or 108 of the 1976 United States CopyrightAct, without either the prior written permission of the Publisher, or authorization throughpayment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive,Danvers, MA 01923, 978-750-8400, fax 978-646-8600. Requests to the Publisher for permissionshould be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street,Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at
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About the Author
A graduate of Brown University and the University of Wisconsin Law School, Mark Ryan has beenteaching math since 1989. He runs The Math Center (www.themathcenter.com) in Winnetka,Illinois, where he teaches high school math courses, including an introduction to geometry anda workshop for parents based on a program he developed, The 10 Habits of Highly Successful Math
. In high school, he twice scored a perfect 800 on the math portion of the SAT, and heStudents
not only knows mathematics but also has a gift for explaining it in plain English. He practicedlaw for four years before deciding he should do?something he enjoys and use his natural talentfor mathematics. Ryan is a member of the Authors Guild and the National Council of Teachers ofMathematics.
Geometry Essentials For Dummies is Ryan’s sixth book. Everyday Math for Everyday Life (Grand
Central Publishing) was published in 2002; Calculus For Dummies (Wiley), in 2003; Calculus
(Wiley), in 2005; Geometry Workbook For Dummies (Wiley), in 2006; andWorkbook For Dummies
Geometry For Dummies, 2nd Edition (Wiley) in 2008. His math books have sold over a quarter of amillion copies.
Also a tournament backgammon player and a skier and tennis player, Ryan lives in Chicago.
We’re proud of this book; please send us your comments through our online registration formlocated at http://dummies.custhelp.com. For other comments, please contact our Customer CareDepartment within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002.
Some of the people who helped bring this book to market include the following:
Acquisitions, Editorial, and Media Development
Project Editor: Joan Friedman
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Indexer: Potomac Indexing, LLC Publishing and Editorial for Consumer Dummies Diane Graves Steele, Vice President and Publisher, Consumer Dummies Kristin Ferguson-Wagstaffe, Product Development Director, Consumer Dummies Ensley Eikenburg, Associate Publisher, Travel Kelly Regan, Editorial Director, Travel Publishing for Technology Dummies Andy Cummings, Vice President and Publisher, Dummies Technology/General User Composition Services Debbie Stailey, Director of Composition Services
Geometry is a subject full of mathematical richness and beauty. The ancient Greeks were into itbig time, and it’s been a mainstay in secondary education for centuries. Today, no educationis complete without at least some familiarity with the fundamental principles of geometry.
But geometry is also a subject that bewilders many students because it’s so unlike the maththat they’ve done before. Geometry requires you to use deductive logic in formal proofs. Thisprocess involves a special type of verbal and mathematical reasoning that’s new to manystudents. The subject also involves working with two- and three-dimensional shapes. The spatialreasoning required for this is another thing that makes geometry different and challenging.
Geometry Essentials For Dummies can be a big help to you if you’ve hit the geometry wall. Orif you’re a first-time student of geometry, it can prevent you from hitting the wall in thefirst place. When the world of geometry opens up to you and things start to click, you may cometo really appreciate this topic, which has fascinated people for millennia.
About This Book
Geometry Essentials For Dummies covers all the principles and formulas you need to analyze two-and three-dimensional shapes, and it gives you the skills and strategies you need to writegeometry proofs.
My approach throughout is to explain geometry in plain English with a minimum of technicaljargon. Plain English suffices for geometry because its principles, for the most part, areaccessible with your common sense. I see no reason to obscure geometry concepts behind a lot offancy-pants mathematical mumbo-jumbo. I prefer a street-smart approach.
This book, like all For Dummies books, is a reference, not a tutorial. The basic idea is thatthe chapters stand on their own as much as possible. So you don’t have to read this book coverto cover — although, of course, you might want to.
Conventions Used in This Book
Geometry Essentials For Dummies follows certain conventions that keep the text consistent:
Variables and names of points are in italics.
Important math terms are often in italics and are defined when necessary. Italics are also
sometimes used for emphasis.
Important terms may be bolded when they appear as keywords within a bulleted list. I also usebold for the instructions in many-step processes.
As in most geometry books, figures are not necessarily drawn to scale — though most of themare.
As I wrote this book, here’s what I assumed about you:
You’re a high school student (or perhaps a junior high student) currently taking a standardhigh school–level geometry course, or . . .