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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2019
Geometry Essentials For Dummies (9781119590446) was previously published as Geometry Essentials For Dummies (9781118068755). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product. Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams. * Get down to the basics -- get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals * Conquer proofs with confidence -- follow easy-to-grasp instructions for understanding the components of a formal geometry proof * Take triangles in strides -- learn how to take in a triangle's sides, analyze its angles, work through an SAS proof, and apply the Pythagorean Theorem * Polish up on polygons -- get the lowdown on quadrilaterals and other polygons: their angles, areas, properties, perimeters, and much more
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Geometry Essentials For Dummies®
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2019 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Control Number: 2019937506
ISBN 978-1-119-59044-6 (pbk); ISBN 978-1-119-59047-7 (ebk); ISBN 978-1-119-59046-0 (ebk)
Geometry Essentials For Dummies®
Table of Contents
Cover
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
Geometry Proofs
Am I Ever Going to Use This?
Getting Down with Definitions
A Few Points on Points
Lines, Segments, and Rays
Investigating the Plane Facts
Everybody’s Got an Angle
Bisection and Trisection
Chapter 2: Geometry Proof Starter Kit
The Lay of the (Proof) Land
Reasoning with If-Then Logic
Complementary and Supplementary Angles
Addition and Subtraction
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
Triangle Classification by Angles
The Triangle Inequality Principle
Sizing up Triangle Area
Regarding Right Triangles
The Pythagorean Theorem
Pythagorean Triple Triangles
Two Special Right Triangles
Chapter 5: Congruent Triangle Proofs
Proving Triangles Congruent
Taking the Next Step with CPCTC
The Isosceles Triangle Theorems
The Two Equidistance Theorems
Chapter 6: Quadrilaterals
Parallel Line Properties
The Seven Special Quadrilaterals
Working with Auxiliary Lines
The Properties of Quadrilaterals
Proving That You’ve Got a Particular Quadrilateral
Chapter 7: Polygon Formulas
The Area of Quadrilaterals
The Area of Regular Polygons
Angle and Diagonal Formulas
Chapter 8: Similarity
Similar Figures
Proving Triangles Similar
Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem
More Proportionality Theorems
Chapter 9: Circle Basics
Radii, Chords, and Diameters
Arcs and Central Angles
Tangents
The Pizza Slice Formulas
The Angle-Arc Formulas
The Power Theorems
Chapter 10: 3-D Geometry
Flat-Top Figures
Pointy-Top Figures
Spheres
Chapter 11: Coordinate Geometry
The Coordinate Plane
Slope, Distance, and Midpoint
Equations for Lines and Circles
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
Index
About the Author
Connect with Dummies
End User License Agreement
List of Tables
Chapter 6
TABLE 6-1 Questions about Sides of Parallelograms
TABLE 6-2 Questions about Angles of Parallelograms
TABLE 6-3 Questions about Parallelogram Diagonals
List of Illustrations
Chapter 1
FIGURE 1-1:
PS
and
WZ
, each made up of three pieces.
FIGURE 1-2: Some points, lines, and segments.
FIGURE 1-3: Catching a few rays.
FIGURE 1-4: Some angles and their parts.
FIGURE 1-5: Coplanar and non-coplanar points.
FIGURE 1-6: Perpendicular and oblique lines, rays, and segments.
FIGURE 1-7: Parallel and intersecting planes.
FIGURE 1-8: Examining all the angles.
FIGURE 1-9: Adjacent and non-adjacent angles.
FIGURE 1-10: Complementary angles can join forces to form a right angle.
FIGURE 1-11: Together, supplementary angles can form a straight line.
FIGURE 1-12: A three-way SPLIT.
Chapter 2
FIGURE 2-1: Anatomy of a geometry proof.
FIGURE 2-2: Proving that Spot is a mammal.
FIGURE 2-3: Double duty — using both versions of the midpoint definition in the ...
FIGURE 2-4: Follow the arrows from bubble to bubble.
FIGURE 2-5: Adding one thing to two congruent things.
FIGURE 2-6: Adding congruent things to congruent things.
FIGURE 2-7: Congruent angles divided into congruent parts.
Chapter 3
FIGURE 3-1: The first two lines of the proof.
FIGURE 3-2: The first three lines of the proof.
FIGURE 3-3: The first five lines of the proof (minus reason 5).
FIGURE 3-4: The proof’s last two lines.
FIGURE 3-5: The end of the proof (so far).
FIGURE 3-6: The finished proof.
Chapter 4
FIGURE 4-1: Two run-of-the-mill isosceles triangles.
FIGURE 4-2: The triangle inequality principle lets you find the possible lengths...
FIGURE 4-3: Triangle ABC changes as side
grows.
FIGURE 4-4:
is one of the altitudes of
.
FIGURE 4-5:
and
are the other two altitudes of
.
FIGURE 4-6: Triangle ABC with its three altitudes.
FIGURE 4-7: A triangle takes up half the area of a rectangle.
FIGURE 4-8: Right triangle WXR with its three altitudes.
FIGURE 4-9: The Pythagorean Theorem is as easy as
.
FIGURE 4-10: A funny-looking hexagon made up of right triangles.
FIGURE 4-11: Two triangles from famous families.
FIGURE 4-12: Use a ratio to figure out what family this triangle belongs to.
FIGURE 4-13: The
triangle.
FIGURE 4-14: Find the missing lengths.
FIGURE 4-15: The
triangle.
FIGURE 4-16: Find the missing lengths.
Chapter 5
FIGURE 5-1: Triangles with congruent sides are congruent.
FIGURE 5-2: Two sides and the angle between them make these triangles congruent.
FIGURE 5-3: An amicable separation of triangles.
FIGURE 5-4: Two angles and their shared side make these triangles congruent.
FIGURE 5-5: Two congruent angles and a side not between them make these triangle...
FIGURE 5-6: A critical pair of proof lines: Congruent triangles and CPCTC.
FIGURE 5-7: The congruent sides tell you that the angles are congruent.
FIGURE 5-8: The congruent angles tell you that the sides are congruent.
FIGURE 5-9: The first equidistance theorem.
FIGURE 5-10: The second equidistance theorem.
Chapter 6
FIGURE 6-1: Two parallel lines, one transversal, and eight angles.
FIGURE 6-2: The royal family tree of quadrilaterals.
FIGURE 6-3: Connecting two points creates triangles you can use.
FIGURE 6-4: Four pairs of Z-angles.
FIGURE 6-5: The two pairs of Z-angles from the preceding proof — a backward Z an...
FIGURE 6-6: A run-of-the-mill parallelogram.
FIGURE 6-7: The two kids and one grandkid (the square) of the parallelogram.
FIGURE 6-8: A mathematical kite that looks ready for flying.
FIGURE 6-9: A trapezoid (on the left) and an isosceles trapezoid (on the right).
Chapter 7
FIGURE 7-1: The relationship between a parallelogram and a rectangle.
FIGURE 7-2: The kite takes up half of each of the four small rectangles and thus...
FIGURE 7-3: The relationship between a trapezoid and a rectangle.
FIGURE 7-4: Use a
triangle to find the area of this parallelogram.
FIGURE 7-5: Drawing in the height creates a right triangle.
FIGURE 7-6: Find the area of this rhombus.
FIGURE 7-7: A kite with a funky side length.
FIGURE 7-8: A regular hexagon cut into six congruent, equilateral triangles.
FIGURE 7-9: Interior and exterior angles.
Chapter 8
FIGURE 8-1: These quadrilaterals are
similar
because they’re exactly the same sh...
FIGURE 8-2: Similar quadrilaterals that aren’t lined up.
FIGURE 8-3: Flipping
PQRS
over to make
SRQP
lines it up nicely with
ABCD
— pure ...
FIGURE 8-4: Three similar right triangles: small, medium, and large.
FIGURE 8-5: Altitude
lets you apply the Altitude-on-Hypotenuse Theorem.
FIGURE 8-6: A line parallel to a side cuts the other two sides proportionally.
FIGURE 8-7: Because the angle is bisected, segments
c
and
d
are proportional to ...
Chapter 9
FIGURE 9-1: A
central angle cuts out a
arc.
FIGURE 9-2: Arc
is
of the circle’s circumference.
FIGURE 9-3: A pizza-slice sector and a segment of a circle.
FIGURE 9-4: Angles with vertices on a circle.
FIGURE 9-5: Chord-chord angles are inside a circle.
FIGURE 9-6: Three kinds of angles outside a circle.
FIGURE 9-7: As the angle gets farther from the center of the circle, it gets sma...
FIGURE 9-8: The Tangent-Secant Power Theorem:
.
FIGURE 9-9: The Secant-Secant Power Theorem:
.
Chapter 10
FIGURE 10-1: A prism and a cylinder with their bases and lateral rectangles.
FIGURE 10-2: A pyramid and a cone with their heights and slant heights.
Chapter 11
FIGURE 11-1: The
x-y
coordinate system.
FIGURE 11-2: Slope is the ratio of the rise to the run.
FIGURE 11-3: The slope tells you how steep a line is.
FIGURE 11-4: The distance between two points is also the length of the hypotenus...
Guide
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Table of Contents
Begin Reading
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Introduction
Geometry is a subject full of mathematical richness and beauty. The ancient Greeks were into it big time, and it’s been a mainstay in secondary education for centuries. Today, no education is 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 math that they’ve done before. Geometry requires you to use deductive logic in formal proofs. This process involves a special type of verbal and mathematical reasoning that’s new to many students. The subject also involves working with two- and three-dimensional shapes. The spatial reasoning 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. Or if you’re a first-time student of geometry, it can prevent you from hitting the wall in the first place. When the world of geometry opens up to you and things start to click, you may come to 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 write geometry proofs.
My approach throughout is to explain geometry in plain English with a minimum of technical jargon. Plain English suffices for geometry because its principles, for the most part, are accessible with your common sense. I see no reason to obscure geometry concepts behind a lot of fancy-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 that the chapters stand on their own as much as possible. So you don’t have to read this book cover to 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 use bold for the instructions in many-step processes.
As in most geometry books, figures are not necessarily drawn to scale — though most of them are.
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 standard high school–level geometry course, or …
You’re a parent of a geometry student, and you’d like to understand the fundamentals of geometry so you can help your child do his or her homework and prepare for quizzes and tests, or …
You’re anyone who wants to refresh your recollection of the geometry you studied years ago or wants to explore geometry for the first time.
You remember some basic algebra. The good news is that you need very little algebra for doing geometry — but you do need some. In the problems that do involve algebra, I try to lay out all the solutions step by step.
Icons Used in This Book
Next to this icon are definitions of geometry terms, explanations of geometry principles, and a few other things you should remember as you work through the book.
This icon highlights shortcuts, memory devices, strategies, and so on.
Ignore these icons, and you may end up doing lots of extra work or getting the wrong answer or both. Read carefully when you see the bomb with the burning fuse!
This icon identifies the theorems and postulates — little mathematical truths — that you use to form the logical arguments in geometry proofs.
Where to Go from Here
If you’re a geometry beginner, you should probably start with Chapter 1 and work your way through the book in order, but if you already know a fair amount of the subject, feel free to skip around. For instance, if you need to know about quadrilaterals, check out Chapter 6. Or if you already have a good handle on geometry proof basics, you may want to dive into the more advanced proofs in Chapter 5.
And from there, naturally, you can go
To the head of the class
To Go to collect $200
To chill out
To explore strange new worlds, to seek out new life and new civilizations, to boldly go where no man (or woman) has gone before
If you’re still reading this, what are you waiting for? Go take your first steps into the wonderful world of geometry!
Chapter 1
An Overview of Geometry
IN THIS CHAPTER
Surveying the geometric landscape: Shapes and proofs
Understanding points, lines, rays, segments, angles, and planes
Cutting segments and angles in two or three congruent pieces
Studying geometry is sort of a Dr. Jekyll-and-Mr. Hyde thing. You have the ordinary geometry of shapes (the Dr. Jekyll part) and the strange world of geometry proofs (the Mr. Hyde part).
Every day, you see various shapes all around you (triangles, rectangles, boxes, circles, balls, and so on), and you’re probably already familiar with some of their properties: area, perimeter, and volume, for example. In this book, you discover much more about these basic properties and then explore more advanced geometric ideas about shapes.
Geometry proofs are an entirely different sort of animal. They involve shapes, but instead of doing something straightforward like calculating the area of a shape, you have to come up with a mathematical argument that proves something about a shape. This process requires not only mathematical skills but verbal skills and logical deduction skills as well, and for this reason, proofs trip up many, many students. If you’re one of these people and have already started singing the geometry-proof blues, you might even describe proofs — like Mr. Hyde — as monstrous. But I’m confident that, with the help of this book, you’ll have no trouble taming them.
The Geometry of Shapes
Have you ever reflected on the fact that you’re literally surrounded by shapes? Look around. The rays of the sun are — what else? — rays. The book in your hands has a shape, every table and chair has a shape, every wall has an area, and every container has a shape and a volume; most picture frames are rectangles, DVDs are circles, soup cans are cylinders, and so on.
One-dimensional shapes
There aren’t many shapes you can make if you’re limited to one dimension. You’ve got your lines, your segments, and your rays. That’s about it. On to something more interesting.
Two-dimensional shapes
As you probably know, two-dimensional shapes are flat things like triangles, circles, squares, rectangles, and pentagons. The two most common characteristics you study about 2-D shapes are their area and perimeter. I devote many chapters in this book to triangles and quadrilaterals (shapes with four sides); I give less space to shapes that have more sides, like pentagons and hexagons. Then there are the shapes with curved sides: The only curved 2-D shape I discuss is the circle.
Three-dimensional shapes
In this book, you work with prisms (a box is one example), cylinders, pyramids, cones, and spheres. The two major characteristics of these 3-D shapes are their surface area and volume. These two concepts come up frequently in the real world; examples include the amount of wrapping paper you need to wrap a gift box (a surface area problem) and the volume of water in a backyard pool (a volume problem).
Geometry Proofs
A geometry proof — like any mathematical proof — is an argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the thing you’re trying to prove. Here’s a very simple example using the line segments in Figure 1-1.
FIGURE 1-1:PS and WZ, each made up of three pieces.
For this proof, you’re told that segment is congruent to (the same length as) segment , that is congruent to , and that is congruent to . You have to prove that is congruent to .
Now, you may be thinking, “That’s obvious — if is the same length as and both segments contain these equal short pieces and the equal medium pieces, then the longer third pieces have to be equal as well.” And you’d be right. But that’s not how the proof game is played. You have to spell out every little step in your thinking. Here’s the whole chain of logical deductions:
(this is given).
and
(these facts are also given).
Therefore,
(because if you add equal things to equal things, you get equal totals).
Therefore,
(because if you start with equal segments, the whole segments
and
,
and take away equal parts of them,
and
,
the parts that are left must be equal).
Am I Ever Going to Use This?
You’ll likely have plenty of opportunities to use your knowledge about the geometry of shapes. What about geometry proofs? Not so much.
