Geometry Essentials For Dummies

Mark Ryan is the owner of The Math Center in the Chicago area, where he teaches students in all levels of mathematics, from pre-algebra to calculus. He is the author of Calculus For Dummies and Geometry For Dummies.

• Introduction 1About This Book 1Conventions Used in This Book 2Foolish Assumptions 2Icons Used in This Book 3Where to Go from Here 3Chapter 1: An Overview of Geometry 5The Geometry of Shapes 6One-dimensional shapes 6Two-dimensional shapes 6Three-dimensional shapes 6Geometry Proofs 6Am I Ever Going to Use This? 7When you’ll use your knowledge of shapes 7When you’ll use your knowledge of proofs 8Getting Down with Definitions 9A Few Points on Points 11Lines, Segments, and Rays 12Horizontal and vertical lines 12Doubling up with pairs of lines 13Investigating the Plane Facts 14Everybody’s Got an Angle 14Five types of angles 15Angle pairs 16Bisection and Trisection 18Segments 18Angles 18Chapter 2: Geometry Proof Starter Kit 21The Lay of the (Proof) Land 21Reasoning with If-Then Logic 23If-then chains of logic 24Definitions, theorems, and postulates 25Bubble logic 26Complementary and Supplementary Angles 27Addition and Subtraction 29Addition theorems 29Subtraction theorems 33Like Multiples and Like Divisions 34Congruent Vertical Angles 36Transitivity and Substitution 37Chapter 3: Tackling a Longer Proof 41Making a Game Plan 42Using All the Givens 42Using If-Then Logic 43Chipping Away at the Problem 45Working Backward 47Filling in the Gaps 49Writing out the Finished Proof 49Chapter 4: Triangle Fundamentals 51Taking in a Triangle’s Sides 51Scalene triangles 52Isosceles triangles 52Equilateral triangles 52Triangle Classification by Angles 52The Triangle Inequality Principle 53Sizing up Triangle Area 54A triangle’s altitude or height 54Determining a triangle’s area 56Regarding Right Triangles 57The Pythagorean Theorem 58Pythagorean Triple Triangles 60The Fab Four triangles 61Families of Pythagorean triple triangles 61Two Special Right Triangles 64The 45 - 45 - 90 triangle 64The 30 - 60 - 90 triangle 66Chapter 5: Congruent Triangle Proofs 69Proving Triangles Congruent 69SSS: The side-side-side method 70SAS: Side-angle-side 72ASA: The angle-side-angle tack 74AAS: Angle-angle-side 74Last but not least: HLR 75Taking the Next Step with CPCTC 75Defining CPCTC 76Tackling a CPCTC proof 76The Isosceles Triangle Theorems 79The Two Equidistance Theorems 81Determining a perpendicular bisector 81Using a perpendicular bisector 83Chapter 6: Quadrilaterals 85Parallel Line Properties 85Parallel lines with a transversal 85The transversal theorems 87The Seven Special Quadrilaterals 89Working with Auxiliary Lines 90The Properties of Quadrilaterals 93Properties of the parallelogram 93Properties of the three special parallelograms 95Properties of the kite 98Properties of the trapezoid and the isosceles trapezoid 99Proving That You’ve Got a Particular Quadrilateral 100Proving you’ve got a parallelogram 100Proving that you’ve got a rectangle, rhombus, or square 103Proving that you’ve got a kite 104Chapter 7: Polygon Formulas 107The Area of Quadrilaterals 107Quadrilateral area formulas 108Why the formulas work 108Trying a few area problems 110The Area of Regular Polygons 113The polygon area formulas 114Tackling an area problem 114Angle and Diagonal Formulas 115Interior and exterior angles 116A polygon angle problem 117Criss-crossing with diagonals 118Chapter 8: Similarity 119Similar Figures 119Defining similar polygons 119How similar figures line up 121Solving a similarity problem 122Proving Triangles Similar 124Tackling an AA proof 125Using SSS~ 126An SAS~ proof 127Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem 128More Proportionality Theorems 130The Side-Splitter Theorem 130The Angle-Bisector Theorem 132Chapter 9: Circle Basics 135Radii, Chords, and Diameters 135Five circle theorems 136Using extra radii 136Arcs and Central Angles 138Tangents 138The Pizza Slice Formulas 140Determining arc length 140Sector and segment area 141The Angle-Arc Formulas 143Angles on a circle 144Angles inside a circle 144Angles outside a circle 145Keeping the formulas straight 146The Power Theorems 147The Chord-Chord Theorem 148The Tangent-Secant Theorem 149The Secant-Secant Theorem 149Condensing the power theorems into a single idea 150Chapter 10: 3-D Geometry 151Flat-Top Figures 151Pointy-Top Figures 154Spheres 159Chapter 11: Coordinate Geometry 161The Coordinate Plane 161Slope, Distance, and Midpoint 162The slope dope 162The distance formula 164The midpoint formula 165Trying out the formulas 166Equations for Lines and Circles 167Line equations 168The circle equation 168Chapter 12: Ten Big Reasons to Use in Proofs 171The Reflexive Property 171Vertical Angles are Congruent 171The Parallel-Line Theorems 172Two Points Determine a Line 172All Radii are Congruent 173If Sides, Then Angles 173If Angles, Then Sides 173Triangle Congruence 173CPCTC 174Triangle Similarity 174Index 175

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