Geometry Essentials for Dummies

266 Pages·2011·12.46 MB·English

by  Mark Ryan

®
Geometry Essentials For Dummies
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Table of Contents
Introduction
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
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
All Radii Are Congruent
If Sides, Then Angles
If Angles, Then Sides
Triangle Congruence
CPCTC
Triangle Similarity
Cheat Sheet