Geometry for Dummies

Description

Hit the geometry wall? Get up and running with this no-nonsense guide! Does the thought of geometry make you jittery? You're not alone. Fortunately, this down-to-earth guide helps you approach it from a new angle, making it easier than ever to conquer your fears and score your highest in geometry. From getting started with geometry basics to making friends with lines and angles, you'll be proving triangles congruent, calculating circumference, using formulas, and serving up pi in no time. Geometry is a subject full of mathematical richness and beauty. But it's a subject that bewilders many students because it's so unlike the math they've done before—it requires the use of deductive logic in formal proofs. If you're having a hard time wrapping your mind around what that even means, you've come to the right place! Inside, you'll find out how a proof's chain of logic works and even discover some secrets for getting past rough spots along the way. You don't have to be a math genius to grasp geometry, and this book helps you get un-stumped in a hurry! Find out how to decode complex geometry proofsLearn to reason deductively and inductivelyMake sense of angles, arcs, area, and moreImprove your chances of scoring higher in your geometry class There's no reason to let your nerves get jangled over geometry—your understanding will take new shape with the help of Geometry For Dummies. Title Page Copyright Page Table of Contents Introduction About This Book Conventions Used in This Book What You’re Not to Read Foolish Assumptions Icons Used in This Book Beyond the Book Where to Go from Here Part 1 Getting Started with Geometry Basics Chapter 1 Introducing Geometry Studying the Geometry of Shapes One-dimensional shapes Two-dimensional shapes Three-dimensional shapes Getting Acquainted with Geometry Proofs Easing into proofs with an everyday example Turning everyday logic into a proof Sampling a simple geometrical proof When Am I Ever Going to Use This? When you’ll use your knowledge of shapes When you’ll use your knowledge of proofs Why You Won’t Have Any Trouble with Geometry Chapter 2 Building Your Geometric Foundation Getting Down with Definitions A Few Points on Points Lines, Segments, and Rays Pointing Every Which Way Singling out horizontal and vertical lines Doubling up with pairs of lines Investigating the Plane Facts Everybody’s Got an Angle Goldilocks and the three angles: Small, large, and just “right” Angle pairs: Often joined at the hip Chapter 3 Sizing Up Segments and Analyzing Angles Measuring Segments and Angles Measuring segments Measuring angles Adding and Subtracting Segments and Angles Cutting in Two or Three: Bisection and Trisection Bisecting and trisecting segments Bisecting and trisecting angles Proving (Not Jumping to) Conclusions about Figures Part 2 Introducing Proofs Chapter 4 Prelude to Proofs Getting the Lay of the Land: The Components of a Formal Geometry Proof Reasoning with If-Then Logic If-then chains of logic You’ve got your reasons: Definitions, theorems, and postulates Bubble logic for two-column proofs Horsing Around with a Two-Column Proof Chapter 5 Your Starter Kit of Easy Theorems and Short Proofs Doing Right and Going Straight: Complementary and Supplementary Angles Addition and Subtraction: Eight No-Big-Deal Theorems Addition theorems Subtraction theorems Like Multiples and Like Divisions? Then These Theorems Are for You! The X-Files: Congruent Vertical Angles Are Out There Pulling the Switch with the Transitive and Substitution Properties Chapter 6 The Ultimate Guide to Tackling a Longer Proof Making a Game Plan Using All the Givens Making Sure You Use If-Then Logic Chipping Away at the Problem Jumping Ahead and Working Backward Filling In the Gaps Writing Out the Finished Proof Part 3 Triangles: Polygons of the Three-Sided Variety Chapter 7 Grasping Triangle Fundamentals Taking In a Triangle’s Sides Scalene triangles: Akilter, awry, and askew Isosceles triangles: Nice pair o’ legs Equilateral triangles: All parts are created equal Introducing the Triangle Inequality Principle Getting to Know Triangles by Their Angles Sizing Up Triangle Area Scaling altitudes Determining a triangle’s area Locating the “Centers” of a Triangle Balancing on the centroid Finding three more “centers” of a triangle Chapter 8 Regarding Right Triangles Applying the Pythagorean Theorem Perusing Pythagorean Triple Triangles The Fab Four Pythagorean triple triangles Families of Pythagorean triple triangles Getting to Know Two Special Right Triangles The 45°- 45°- 90° triangle — half a square The 30°- 60°- 90° triangle — half of an equilateral triangle Chapter 9 Completing Congruent Triangle Proofs Introducing Three Ways to Prove Triangles Congruent SSS: Using the side-side-side method SAS: Taking the side-angle-side approach ASA: Taking the angle-side-angle tack CPCTC: Taking Congruent Triangle Proofs a Step Further Defining CPCTC Tackling a CPCTC proof Eying the Isosceles Triangle Theorems Trying Out Two More Ways to Prove Triangles Congruent AAS: Using the angle-angle-side theorem HLR: The right approach for right triangles Going the Distance with the Two Equidistance Theorems Determining a perpendicular bisector Using a perpendicular bisector Making a Game Plan for a Longer Proof Running a Reverse with Indirect Proofs Part 4 Polygons of the Four-or-More-Sided Variety Chapter 10 The Seven Wonders of the Quadrilateral World Getting Started with Parallel-Line Properties Crossing the line with transversals: Definitions and theorems Applying the transversal theorems Working with more than one transversal Meeting the Seven Members of the Quadrilateral Family Looking at quadrilateral relationships Working with auxiliary lines Giving Props to Quads: 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 Chapter 11 Proving That You Have a Particular Quadrilateral Putting Properties and Proof Methods Together Proving That a Quadrilateral Is a Parallelogram Surefire ways of ID-ing a parallelogram Trying some parallelogram proofs Proving That a Quadrilateral Is a Rectangle, Rhombus, or Square Revving up for rectangle proofs Waxing rhapsodic about rhombus proofs Squaring off with square proofs Proving That a Quadrilateral Is a Kite Chapter 12 Polygon Formulas: Area, Angles, and Diagonals Calculating the Area of Quadrilaterals Setting forth the quadrilateral area formulas Getting behind the scenes of the formulas Trying a few area problems Finding the Area of Regular Polygons Presenting polygon area formulas Tackling more area problems Using Polygon Angle and Diagonal Formulas Interior and exterior design: Exploring polygon angles Handling the ins and outs of a polygon angle problem Criss-crossing with diagonals Chapter 13 Similarity: Same Shape, Different Size Getting Started with Similar Figures Defining and naming similar polygons How similar figures line up Solving a similarity problem Proving Triangles Similar Tackling an AA proof Using SSS~ to prove triangles similar Working through an SAS~ proof CASTC and CSSTP, the Cousins of CPCTC Working through a CASTC proof Taking on a CSSTP proof Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem Getting Proportional with Three More Theorems The side-splitter theorem: It’ll make you split your sides Crossroads: The side-splitter theorem extended The angle-bisector theorem Part 5 Working with Not-So-Vicious Circles Chapter 14 Coming Around to Circle Basics The Straight Talk on Circles: Radii and Chords Defining radii, chords, and diameters Introducing five circle theorems Working through a proof Using extra radii to solve a problem Pieces of the Pie: Arcs and Central Angles Three definitions for your mathematical pleasure Six scintillating circle theorems Trying your hand at some proofs Going Off on a Tangent about Tangents Introducing the tangent line The common-tangent problem Taking a walk on the wild side with a walk-around problem Chapter 15 Circle Formulas and Theorems Chewing on the Pizza Slice Formulas Determining arc length Finding sector and segment area Pulling it all together in a problem Digesting the Angle-Arc Theorems and Formulas Angles on a circle Angles inside a circle Angles outside a circle Keeping your angle-arc formulas straight Powering Up with the Power Theorems Striking a chord with the chord-chord power theorem Touching on the tangent-secant power theorem Seeking out the secant-secant power theorem Condensing the power theorems into a single idea Part 6 Going Deep with 3-D Geometry Chapter 16 3-D Space: Proofs in a Higher Plane of Existence Lines Perpendicular to Planes Parallel, Perpendicular, and Intersecting Lines and Planes The four ways to determine a plane Line and plane interactions Chapter 17 Getting a Grip on Solid Geometry Flat-Top Figures: They’re on the Level Getting to the Point of Pointy-Top Figures Rounding Things Out with Spheres Part 7 Placement, Points, and Pictures: Alternative Geometry Topics Chapter 18 Coordinate Geometry Getting Coordinated with the Coordinate Plane The Slope, Distance, and Midpoint Formulas The slope dope Going the distance with the distance formula Meeting each other halfway with the midpoint formula The whole enchilada: Putting the formulas together in a problem Proving Properties Analytically Step 1: Drawing a general figure Step 2: Solving the problem algebraically Deciphering Equations for Lines and Circles Line equations The standard circle equation Chapter 19 Changing the Scene with Geometric Transformations Some Reflections on Reflections Getting oriented with orientation Finding a reflecting line Not Getting Lost in Translations A translation equals two reflections Finding the elements of a translation Turning the Tables with Rotations A rotation equals two reflections Finding the center of rotation and the equations of two reflecting lines Third Time’s the Charm: Stepping Out with Glide Reflections A glide reflection equals three reflections Finding the main reflecting line Chapter 20 Locating Loci and Constructing Constructions Loci Problems: Getting in with the Right Set The four-step process for locus problems Two-dimensional locus problems Three-dimensional locus problems Drawing with the Bare Essentials: Constructions Three copying methods Bisecting angles and segments Two perpendicular line constructions Constructing parallel lines and using them to divide segments Part 8 The Part of Tens Chapter 21 Ten Things to Use as Reasons in Geometry Proofs The Reflexive Property Vertical Angles Are Congruent The Parallel-Line Theorems Two Points Determine a Line All Radii of a Circle Are Congruent If Sides, Then Angles If Angles, Then Sides The Triangle Congruence Postulates and Theorems CPCTC The Triangle Similarity Postulates and Theorems Chapter 22 Ten Cool Geometry Problems Eureka! Archimedes’s Bathtub Revelation Determining Pi The Golden Ratio The Circumference of the Earth The Great Pyramid of Khufu Distance to the Horizon Projectile Motion Golden Gate Bridge The Geodesic Dome A Soccer Ball Index EULA