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# 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.

Introduction

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

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

Parallel Line Properties

Parallel lines with a transversal The transversal theorems

Working with Auxiliary Lines

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

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

Five circle theorems Using extra radii

Arcs and Central Angles

Tangents

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

Spheres

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

If Sides, Then Angles

If Angles, Then Sides

Triangle Congruence

CPCTC

Triangle Similarity

Cheat Sheet

Geometry Essentials For Dummies ?

by Mark Ryan

Geometry Essentials For Dummies ?

111 River St.

Hoboken, NJ 07030-5774www.wiley.com

Copyright ? 2011 by Wiley Publishing, Inc., Indianapolis, Indiana

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

http://www.wiley.com/go/permissions.

Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Referencefor the Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com, MakingEverything Easier!, and related trade dress are trademarks or registered trademarks of JohnWiley & Sons, Inc. and/or its affiliates in the United States and other countries, and may notbe used without written permission. All other trademarks are the property of their respectiveowners. Wiley Publishing, Inc., is not associated with any product or vendor mentioned in thisbook.

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.

Wiley also publishes its books in a variety of electronic formats. Some content that appears inprint may not be available in electronic books.

Library of Congress Control Number: 2011905203

ISBN: 978-1-118-06875-5

Manufactured in the United States of America

10 9 8 7 6 5 4 3 2 1

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

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

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

Introduction

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.

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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