Geometry Essentials For Dummies

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Geometry Essentials For Dummies

    Geometry Essentials For Dummies?

    Visit www.dummies.com/cheatsheet/geometryessentials to view this book's cheat sheet.

    Table of Contents


    About This Book Conventions Used in This Book

    Foolish Assumptions

    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

    Three-dimensional shapes

    Geometry Proofs

    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

    Segments Angles

    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

    Bubble logic

    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

    Working Backward

    Filling In the Gaps

    Writing Out the Finished Proof

    Chapter 4: Triangle Fundamentals

    Taking In a Triangle’s Sides

    Scalene triangles Isosceles triangles

    Equilateral 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

    AAS: angle-angle-side

    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

    Similar Figures

    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

    Triangle Congruence


    Triangle Similarity

Cheat Sheet

Geometry Essentials For Dummies ?

    by Mark Ryan

Geometry Essentials For Dummies ?

Published byWiley Publishing, Inc.

111 River St.

    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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    Limit of Liability/Disclaimer of Warranty: The publisher and the author make no representationsor warranties with respect to the accuracy or completeness of the contents of this work andspecifically disclaim all warranties, including without limitation warranties of fitness for aparticular purpose. No warranty may be created or extended by sales or promotional materials.The advice and strategies contained herein may not be suitable for every situation. This workis sold with the understanding that the publisher is not engaged in rendering legal,accounting, or other professional services. If professional assistance is required, theservices of a competent professional person should be sought. Neither the publisher nor theauthor shall be liable for damages arising herefrom. The fact that an organization or Websiteis referred to in this work as a citation and/or a potential source of further information doesnot mean that the author or the publisher endorses the information the organization or Websitemay provide or recommendations it may make. Further, readers should be aware that InternetWebsites listed in this work may have changed or disappeared between when this work was writtenand when it is read.

    For general information on our other products and services, 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.

For technical support, please visit www.wiley.com/techsupport.

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    Library of Congress Control Number: 2011905203

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    Manufactured in the United States of America

    10 9 8 7 6 5 4 3 2 1

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.

    Publisher’s Acknowledgments

    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

    Acquisitions Editor: Lindsay Sandman Lefevere

    Assistant Editor: David Lutton

    Technical Editors: Nancy Cozad, Amanda D. Milligan

    Senior Editorial Manager: Jennifer Ehrlich

    Editorial Supervisor and Reprint Editor: Carmen Krikorian

    Editorial Assistant: Rachelle S. Amick

    Cover Photos: ? iStockphoto.com / Deanima

    Cartoons: Rich Tennant (www.the5thwave.com)

    Composition Services

    Project Coordinator: Kristie Rees

    Layout and Graphics: Carrie A. Cesavice, Corrie Socolovitch

    Proofreader: Jacqui Brownstein

    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.

    Foolish Assumptions

    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 . . .

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