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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2024
Don't be a square! Strengthen your geometrical skills
Lots of students need extra practice to master geometry. Thankfully, there's Geometry Workbook For Dummies. Packed with hundreds of practice problems and easy-to-understand concept explanations, this book takes a hands-on approach to showing you the geometric ropes. Inside, you'll find a helpful review of basic terms and concepts, so you can hit the ground running when you get to the more advanced stuff. In classic Dummies style, this workbook offers easy ways to understand theorems, proofs, and other geometry fundamentals. Figure out congruent triangles, wrap your mind around angle-arc theorems, connect radii and chords, and get smart about all the core concepts of geometry.
- Work through hundreds of practice problems to solidify your geometry know-how
- Clear up any confusion with easy-to-understand explanations of all key concepts
- Get tips for avoiding common mistakes and improving your test scores
For students or parents looking for a hands-on approach to learning geometry, this is the perfect Dummies guide. It's great resource all on its own, or pair it with Geometry For Dummies for even more effective book learning.
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Table of Contents
Cover
Title Page
Copyright
Introduction
About This Book
Conventions Used in This Book
How to Use This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: Getting Started with Geometry
Chapter 1: Introducing Geometry and Geometry Proofs
What Is Geometry?
Making the Right Assumptions
If-Then Logic: If You Bought This Book, Then You Must Love Geometry!
What’s a Geometry Proof?
Solutions
Chapter 2: Points, Segments, Lines, Rays, and Angles
Hammering Out Basic Definitions
Looking at Union and Intersection Problems
Uncovering More Definitions
Division in the Ranks: Bisection and Trisection
Perfect Hilarity for Perpendicularity
You Complete Me: Complementary and Supplementary Angles
X Marks the Spot: Vertical Angles
Solutions
Chapter 3: Your First Geometry Proofs
Ready to Try Some Proofs?
Proofs Involving Complementary and Supplementary Angles
Proofs Involving Adding and Subtracting Segments and Angles
Proofs Involving Multiplying and Dividing Angles and Segments
Proofs Involving the Transitive and Substitution Properties
Solutions
Part 2: Triangles, Proof and Non-Proof Problems
Chapter 4: Triangle Fundamentals and Other Cool Stuff (No Proofs)
Triangle Types and Triangle Basics
Altitudes, Area, and the Super Hero Formula
Balancing Things Out with Medians and Centroids
Locating Three More “Centers” of a Triangle
The Pythagorean Theorem
Solving Pythagorean Triple Triangles
Unique Degrees: Two Special Right Triangles
Solutions
Chapter 5: Proofs Involving Congruent Triangles
Sizing Up Three Ways to Prove Triangles Congruent
Corresponding Parts of Congruent Triangles Are Congruent
Using Isosceles Triangle Rules: If Sides, Then Angles; If Angles, Then Sides
Exploring Two More Ways to Prove Triangles Congruent
Explaining the Two Equidistance Theorems
Solutions
Part 3: Polygons, Proof and Non-Proof Problems
Chapter 6: Quadrilaterals: Your Fine, Four-Sided Friends (Including Proofs)
Double-Crossers: Transversals and Their Parallel Lines
Quadrilaterals: It’s a Family Affair
Discovering the Properties of the Parallelogram and the Kite
Properties of Rhombuses, Rectangles, and Squares
Unearthing the Properties of Trapezoids and Isosceles Trapezoids
Proving That a Quadrilateral Is a Parallelogram or a Kite
Proving That a Quadrilateral Is a Rhombus, Rectangle, or Square
Solutions
Chapter 7: Area, Angles, and the Many Sides of Polygon Geometry (No Proofs)
Square Units: Finding the Area of Quadrilaterals
The Standard Formula for the Area of Regular Polygons
More Fantastically Fun Polygon Formulas
Solutions
Chapter 8: Similarity: Size Doesn’t Matter (Including Proofs)
Defining Similarity
Proving Triangles Similar
Corresponding Sides and CSSTP — Cats Stalk Silently Then Pounce
Similar Rights: The Altitude-on-Hypotenuse Theorem
Discovering Three More Theorems Involving Proportions
Solutions
Part 4: Circles, Proof and Non-Proof Problems
Chapter 9: Circular Reasoning (Including Proofs)
The Segments Within: Radii and Chords
Introducing Arcs and Central Angles
Touching on Radii and Tangents
Solutions
Chapter 10: Scintillating Circle Formulas (No Proofs)
Pizzas, Slices, and Crusts: Finding Area and “Perimeter” of Circles, Sectors, and Segments
Angles, Circles, and Their Connections: The Angle-Arc Theorems and Formulas
The Power Theorems That Be
Solutions
Part 5: 3-D Geometry: Proof and Non-Proof Problems
Chapter 11: 2-D Stuff Standing Up (Including Proofs)
Lines Perpendicular to Planes: They’re All Right
Parallel, Perpendicular, and Intersecting Lines and Planes
Solutions
Chapter 12: Solid Geometry: Digging into Volume and Surface Area (No Proofs)
Starting with Flat-Top Figures
Sharpening Your Skills with Pointy-Top Figures
Rounding Out Your Understanding with Spheres
Solutions
Part 6: Coordinate Geometry, Loci, and Constructions: Proof and Non-Proof Problems
Chapter 13: Coordinate Geometry, Courtesy of Descartes (Including Proofs)
Formulas, Schmormulas: Slope, Distance, and Midpoint
Mastering Coordinate Proofs with Algebra
Using the Equations of Lines and Circles
Solutions
Chapter 14: Transforming the (Geometric) World: Reflections, Rotations, and Translations (No Proofs)
Reflections on Mirror Images
Lost in Translation
So You Say You Want a … Rotation?
Working with Glide Reflections
Solutions
Chapter 15: Laboring Over Loci and Constructions (No Proofs)
Tackling Locus Problems
Compass and Straightedge Constructions
Solutions
Chapter 16: Ten Things You Better Know (for Geometry), or Your Name Is Mudd
The Pythagorean Theorem (the Queen of All Geometry Theorems)
Special Right Triangles
Area Formulas
Sum of Angles
Circle Formulas
Angle-Arc Theorems
Power Theorems
Coordinate Geometry Formulas
Volume Formulas
Surface Area Formulas
Index
About the Author
Connect with Dummies
End User License Agreement
List of Illustrations
Chapter 1
FIGURE 1-1: A standard two-column proof listing statements and reasons.
FIGURE 1-2: A proof with the reasons written in if-then form.
Chapter 4
FIGURE 4-1: An altitude inside a triangle and outside a triangle.
FIGURE 4-2: Nine little squares plus 16 little squares equals 25 little squares...
FIGURE 4-3: Two special right triangles.
Chapter 5
FIGURE 5-1: The first equidistance theorem gives you the perpendicular bisector...
FIGURE 5-2: The second equidistance theorem lets you know that you have congrue...
Chapter 6
FIGURE 6-1: Parallel lines and a transversal — angles, angles everywhere with l...
FIGURE 6-2: The quadrilateral family tree.
FIGURE 6-3:
ABCD
is a parallelogram.
FIGURE 6-4:
PQRS
is a kite.
FIGURE 6-5:
RHOM
is a rhombus.
Chapter 7
FIGURE 7-1: is an exterior angle of quadrilateral
MILY.
Vertical angle is
n
...
Chapter 8
FIGURE 8-1: Three similar right triangles in one: Triple the pleasure, triple t...
FIGURE 8-2: Multiple parallel lines with transversals.
Chapter 9
FIGURE 9-1: Two wheels that are tangent to the ground — take a break from geome...
FIGURE 9-2: Both sides of a dunce cap are the same length.
Chapter 10
FIGURE 10-1: A sector and an arc that make up one-twelfth of the circle.
FIGURE 10-2: For a given arc (like ), no matter where you move the vertex of a...
FIGURE 10-3: Angles with a vertex
on
a circle.
FIGURE 10-4: An angle with a vertex
inside
a circle.
FIGURE 10-5: Angles with a vertex
outside
a circle.
FIGURE 10-6: The Chord-Chord Power Theorem.
FIGURE 10-7: The Tangent-Secant Power Theorem.
FIGURE 10-8: The Secant-Secant Power Theorem.
Chapter 14
FIGURE 14-1: A triangle and its transformations.
FIGURE 14-2: A rotation is equivalent to two reflections.
FIGURE 14-3: The third time’s the charm: With only three reflections, your figu...
Chapter 15
FIGURE 15-1: Identifying points that work.
FIGURE 15-2: The locus of points equidistant from two given points is the perpe...
FIGURE 15-3: Four points equidistant from two given lines.
FIGURE 15-4: All points on the horizontal line are equidistant from the two giv...
FIGURE 15-5: Five more points equidistant from the two given lines.
FIGURE 15-6: The locus solution.
FIGURE 15-7: Copying an angle.
FIGURE 15-8: Bisecting an angle.
FIGURE 15-9: Constructing a perpendicular bisector.
FIGURE 15-10: Constructing a perpendicular line through a point on a line.
FIGURE 15-11: Constructing a perpendicular line through a point not on a line.
FIGURE 15-12: Use this triangle for problems 7 and 8.
FIGURE 15-13: Points an inch above and an inch below
FIGURE 15-14: Points making a semi-circle to the left of point
J
.
FIGURE 15-15: The locus solution.
FIGURE 15-16: Points on the line are equidistant from
A
and
X
.
FIGURE 15-17: Four perpendicular bisectors.
FIGURE 15-18: The final result.
FIGURE 15-19: A ray in quadrant I.
FIGURE 15-20: A bow tie of sorts.
FIGURE 15-21: Five points equidistant from the
x
-axis and the point at
FIGURE 15-22: The parabola .
FIGURE 15-23: A right angle constructed at
A
and and constructed to have a ...
FIGURE 15-24: Copying a triangle.
FIGURE 15-25: The first steps in constructing the incenter of .
Chapter 16
FIGURE 16-1: Angles (a) on, (b) outside, (c) inside a circle.
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
About the Author
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Geometry Workbook For Dummies®, 2nd Edition
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
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Published simultaneously in Canada
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Library of Congress Control Number: 2024946943
ISBN 978-1-394-27612-7 (pbk); ISBN 978-1-394-27616-5 (ebk); ISBN 978-1-394-27614-1 (ebk)
Introduction
If you’ve already bought this book, then you have my undying respect and admiration (not to mention — cha ching — that with my royalty from the sale of this book, I can now afford, oh, say, half a cup of coffee). And if you’re just thinking about buying it, well, what are you waiting for? Buying this book (and its excellent companion volume, Geometry For Dummies) can be an important first step on the road to gaining a solid grasp of a subject — and now I’m being serious — that is full of mathematical richness and beauty. By studying geometry, you take part in a long tradition going back at least as far as Pythagoras (one of the early, well-known mathematicians to study geometry, but certainly not the first). There is no mathematician, great or otherwise, who has not spent some time studying geometry.
I spend a great deal of time in this book explaining how to do geometry proofs. Many students have a lot of difficulty when they attempt their first proofs. I can think of a few reasons for this. First, geometry proofs, like the rest of geometry, have a spatial aspect that many students find challenging. Second, proofs lack the cut-and-dried nature of most of the math that students are accustomed to (in other words, with geometry proofs there are way more instances where there are many correct ways to proceed, and this takes some getting used to). And third, proofs are, in a sense, only half math. The other half is deductive logic — something new for most students, and something that has a significant verbal component. The good news is that if you practice the dozen or so strategies and tips for doing proofs presented in this book, you should have little difficulty getting the hang of it. These strategies and tips work like a charm and make many proofs much easier than they initially seem.
About This Book
Geometry Workbook For Dummies, like Geometry For Dummies, is intended for three groups of readers:
High school students (and possibly junior high students) taking a standard geometry course with the traditional emphasis on geometry proofs
The parents of geometry students
Anyone of any age who is curious about this interesting subject, which has fascinated both mathematicians and laypeople for well over two thousand years
Whenever possible, I explain geometry concepts and problem solutions with a minimum of technical jargon. I take a common-sense, street-smart approach when explaining mathematics, and I try to avoid the often stiff and formal style used in too many textbooks. You get answer explanations for every practice problem. And with proofs, in addition to giving you the steps of the solutions, I show you the thought process behind the solutions. I supplement the problem explanations with tips, shortcuts, and mnemonic devices. Often, a simple tip or memory trick can make learning and retaining a new, difficult concept much easier. The pages here should contain enough blank space to allow you to write out your solutions right in the book.
Conventions Used in This Book
This book uses certain conventions:
Variables are in
italics.
Important math terms are often in
italics
and are defined when necessary. These terms may be
bolded
when they appear as keywords within a bulleted list. Italics are also used for emphasis.
As in most geometry books, figures are not necessarily drawn to scale.
Extra-hard problems are marked with an asterisk. Don’t try these problems on an empty stomach!
For all proof problems, don’t assume that the number of blank lines (where you’ll put your solutions) corresponds exactly to the number of steps needed for the proof.
How to Use This Book
Like all For Dummies books, you can use this book as a reference. You don’t need to read it cover to cover or work through all problems in order. You may need more practice in some areas than others, so you may choose to do only half of the practice problems in some sections, or none at all.
However, as you’d expect, the order of the topics in Geometry Workbook For Dummies roughly follows the order of a traditional high school geometry course. You can, therefore, go through the book in order, using it to supplement your coursework. If I do say so myself, I expect you’ll find that many of the explanations, methods, strategies, and tips in this book will make problems you found difficult or confusing in class seem much easier.
I give hints for many problems, but if you want to challenge yourself, you may want to cover them up and attempt the problem without the hint.
And if you get stuck while doing a proof, you can try reading a little bit of the “game plan” or the solution to the proof. These aids are in the solutions section at the end of every chapter. But don’t read too much at first. Read a small amount and see whether it gives you any ideas. Then, if you’re still having trouble, read a little more.
Foolish Assumptions
As William Shakespeare said, “A fool thinks himself to be wise, but a wise man knows himself to be a fool.” Here’s what I’m assuming about you — fool that I am.
You’re no slouch — and therefore, you have at least some faint glimmer of curiosity about geometry (or maybe you’re totally, stark raving mad with desire to learn the subject?). How could people possibly have no curiosity at all about geometry, assuming they’re not in a coma? You are literally surrounded by shapes, and every shape involves geometry.
You haven’t forgotten basic algebra. You need very little algebra for geometry, but you do need some. Even if your algebra is a bit rusty, I doubt you’ll have any trouble with the algebra in this book: solving simple equations, using simple formulas, doing square roots, and so on.
You’re willing to invest some time and effort in doing these practice problems. With geometry — as with anything — practice makes perfect, and practice sometimes involves struggle. But that’s a good thing. Ideally, you should give these problems your best shot before you turn to the solutions. Reading through the solutions can be a good way to learn, but you’ll usually remember more if you first push yourself to solve the problems on your own — even if that means going down a few dead ends.
Icons Used in This Book
Look for the following icons to quickly spot important information:
Next to this icon are definitions of geometry terms, explanations of geometry principles, and a few things you should know from algebra. You often use geometry definitions in the reason column of two-column proofs.
This icon is next to all example problems — duh.
This icon gives you shortcuts, memory devices, strategies, and so on.
Ignore these icons, and you may end up doing lots of extra work and maybe getting the wrong answer — and then you could fail geometry, become unpopular, and lose any hope of becoming homecoming queen or king. Better safe than sorry, right?
This icon identifies the theorems and postulates that you’ll use to form the chain of logic in geometry proofs. You use them in the reason column of two-column proofs. A theorem is an if-then statement, like “if angles are supplementary to the same angle, then they are congruent.” You use postulates basically the same way that you use theorems. The difference between them is sort of a mathematical technicality (which I wouldn’t sweat if I were you).
Beyond the Book
You have online access to hundreds of geometry practice problems to supplement what’s covered in the book. To gain access to this online practice material, all you have to do is register. Just follow these simple steps:
Register your book or e-book atDummies.comto get your personal identification number (PIN).
Go to www.dummies.com/go/getaccess.
Choose your product from the drop-down list on that page.
Follow the prompts to validate your product.
Check your email for a confirmation message that includes your PIN and instructions for logging in.
If you don’t receive this email within two hours, please check your spam folder before contacting us through our support website at http://support.wiley.com or by phone at +1 (877) 762-2974.
Where to Go from Here
You can go
To
Chapter 1
To whatever chapter contains the concepts you need to practice
To
Geometry For Dummies
for more in-depth explanations
To the movies
To the beach
Into your geometry final to kick some @#%$!
Then on to bigger and better things
Part 1
Getting Started with Geometry
IN THIS PART …
Get familiar with two-column geometry proofs.
Discover points, segments, lines, rays, and angles.
Practice your skills on lots of proof problems.
Chapter 1
Introducing Geometry and Geometry Proofs
IN THIS CHAPTER
Defining geometry
Examining theorems and if-then logic
Geometry proofs: The formal and the not-so-formal
In this chapter, you get started with some basics about geometry and shapes, a couple points about deductive logic, and a few introductory comments about the structure of geometry proofs. Time to get started!
What Is Geometry?
What is geometry?! C’mon, everyone knows what geometry is, right? Geometry is the study of shapes: circles, triangles, rectangles, pyramids, and so on. Shapes are all around you. The desk or table where you’re reading this book has a shape. You can probably see a window from where you are, and it’s probably a rectangle. The pages of this book are also rectangles. Your pen or pencil is roughly a cylinder (or maybe a right hexagonal prism — see Part 5 for more on solid figures). Your shirt may have circular buttons. The bricks of a brick house are right rectangular prisms. Shapes are ubiquitous — in our world, anyway.
For the philosophically inclined, here’s an exercise that goes way beyond the scope of this book: Try to imagine a world — some sort of different universe — where there aren’t various objects with different shapes. (If you’re into this sort of thing, check out Philosophy For Dummies.)
Making the Right Assumptions
Okay, so geometry is the study of shapes. And how can you tell one shape from another? From the way it looks, of course. But — this may seem a bit bizarre — when you’re studying geometry, you’re sort of not supposed to rely on the way shapes look. The point of this strange treatment of geometric figures is to prohibit you from claiming that something is true about a figure merely because it looks true, and to force you, instead, to prove that it’s true by airtight, mathematical logic.
When you’re working with shapes in any other area of math, or in science, or in, say, architecture or design, paying attention to the way shapes look is very important: their proportions, their angles, their orientation, how steep their sides are, and so on. Only in a geometry course are you supposed to ignore to some degree the appearance of the shapes you study. (I say “to some degree” because, in reality, even in a geometry course — or when using this book — it’s still quite useful most of the time to pay attention to the appearance of shapes.)
When you look at a diagram in this or any geometry book, you cannot assume any of the following just from the appearance of the figure.
Right angles:
Just because an angle looks like an exact angle, that doesn’t necessarily mean it is one.
Congruent angles:
Just because two angles look the same size, that doesn’t mean they really are. (As you probably know,
congruent
[symbolized by ] is a fancy word for “equal” or “same size.”)
Congruent segments:
Just like with angles, you can’t assume segments are the same length just because they appear to be.
Relative sizes of segments and angles:
Just because, say, one segment is drawn to look longer than another in some diagram, it doesn’t follow that the segment really is longer.
Sometimes size relationships are marked on the diagram. For instance, a small L-shaped mark in a corner means that you have a right angle. Tick marks can indicate congruent parts. Basically, if the tick marks match, you know the segments or angles are the same size.
You can assume pretty much anything not on this list; for example, if a line looks straight, it really is straight.
Before doing the following problems, you may want to peek ahead to Chapters 4 and 6 if you’ve forgotten or don’t know the names of various triangles and quadrilaterals.
Q. What can you assume and what can’t you assume about SIMON?
A. You can assume that
(line segment MN) is straight; in other words, there’s no bend at point O.
Another way of saying the same thing is that is a straight angle or a angle.
and are also straight as opposed to curvy.
Therefore, SIMON is a quadrilateral because it has four straight sides.
(If you couldn’t assume that is straight, there could actually be a bend at point O and then SIMON would be a pentagon, but that’s not possible.)
That’s about it for what you can assume. If this figure were anywhere else other than a geometry book, you could safely assume all sorts of other things — including that SIMON is a trapezoid. But this is a geometry book, so you can’t assume that. You also can’t assume that
and are right angles.
is an obtuse angle (an angle greater than ).
is an acute angle (an angle less than ).
is greater than or or and ditto for the relative sizes of other angles.
is shorter than or and ditto for the relative lengths of the other segments.
O
is the midpoint of
is parallel to
The “real” SIMON — weird as it seems — could actually look like this:
1 What type of quadrilateral is AMER? Note: See Chapter 6 for types of quadrilaterals.
2 What type of quadrilateral is IDOL?
3 Use the figure to answer the following questions (Chapter 4 can fill you in on triangles):
Can you assume that the triangles are congruent?
Can you conclude that is acute? Obtuse? Right? Isosceles (with at least two equal sides)? Equilateral (with three equal sides)?
Can you conclude that is acute? Obtuse? Right? Isosceles? Equilateral?
What can you conclude about the length of ?
Might be a right angle?
Might be a right angle?
4 Can you assume or conclude
is isosceles?
D
is the midpoint of ?
Z
is the midpoint of ?
is an altitude (height) of ?
is supplementary to
is a right triangle?
If-Then Logic: If You Bought This Book, Then You Must Love Geometry!
Geometry theorems (and their cousins, postulates) are basically statements of geometrical truth, like “All radii of a circle are congruent.” As you can see in this section and in the rest of the book, theorems (and postulates) are the building blocks of proofs. (I may get hauled over by the geometry police for saying this, but if I were you, I’d just glom theorems and postulates together into a single group because, for the purposes of doing proofs, they work the same way. Whenever I refer to theorems, you can safely read it as “theorems and postulates.”)
Geometry theorems can all be expressed in the form, “If blah blah blah, then blah blah blah,” like “If two angles are right angles, then they are congruent” (although theorems are often written in some shorter way, like “All right angles are congruent”). You may want to flip through the book looking for theorem icons to get a feel for what theorems look like.
An important thing to note here is that the reverse of a theorem is not necessarily true. For example, the statement, “If two angles are congruent, then they are right angles,” is false. When a theorem does work in both directions, you get two separate theorems, one the reverse of the other.
The fact that theorems are not generally reversible should come as no surprise. Many ordinary statements in if-then form are, like theorems, not reversible: “If something’s a ship, then it’s a boat” is true, but “If something’s a boat, then it’s a ship” is false, right? (It might be a canoe.)
Geometry definitions (like all definitions), however, are reversible. Consider the definition of perpendicular: perpendicular lines are lines that intersect at right angles. Both if-then statements are true: 1) “If lines are perpendicular, then they intersect at right angles,” and 2) “If lines intersect at right angles, then they are perpendicular.” When doing proofs, you’ll have the occasion to use both forms of many definitions.
Q. Read through some theorems.
Give an example of a theorem that’s not reversible and explain why the reverse is false.
Give an example of a theorem whose reverse is another true theorem.
A. A number of responses work, but here’s how you could answer:
“If two angles are vertical angles, then they are congruent.” The reverse of this theorem is obviously false. Just because two angles are the same size, it does not follow that they must be vertical angles. (When two lines intersect and form an X, vertical angles are the angles straight across from each other — turn to
Chapter 2
for more info.)
Two of the most important geometry theorems are a reversible pair: “If two sides of a triangle are congruent, then the angles opposite those sides are congruent” and “If two angles of a triangle are congruent, then the sides opposite those angles are congruent.” (For more on these isosceles triangle theorems, check out
Chapter 5
.)
5 Give two examples of theorems that are not reversible and explain why the reverse of each is false. Hint: Flip through this book or your geometry textbook and look at various theorems. Try reversing them and ask yourself whether they still work.
6 Give two examples of theorems that work in both directions. Hint: See the hint for question 5.
What’s a Geometry Proof?
Many students find two-column geometry proofs difficult, but they’re really no big deal once you get the hang of them. Basically, they’re just arguments like the following, in which you brilliantly establish that your Labradoodle, Fifi, will not lay any eggs on the Fourth of July:
Fifi is a Labradoodle.
Therefore, Fifi is a dog, because all Labradoodles are dogs.
Therefore, Fifi is a mammal, because all dogs are mammals.
Therefore, Fifi will never lay any eggs, because mammals don’t lay eggs (okay, okay … except for platypuses and spiny anteaters, for you monotreme-loving nitpickers out there).
Therefore, Fifi will not lay any eggs on the Fourth of July, because if she will never lay any eggs, she can’t lay eggs on the Fourth of July.
In a nutshell: Labradoodle → dog → mammal → no eggs → no eggs on July 4. It’s sort of a domino effect. Each statement knocks over the next till you get to your final conclusion.
Check out Figure 1-1 to see what this argument or proof looks like in the standard two-column geometry proof format.
FIGURE 1-1: A standard two-column proof listing statements and reasons.
Note that the left-hand column contains specific facts (about one particular dog, Fifi), while the right-hand column contains general principles (about dogs in general or mammals in general). This format is true of all geometry proofs.
Now look at the very same proof in Figure 1-2; this time, the reasons appear in if-then form. When reasons are written this way, you can see how the chain of logic flows.
In a two-column proof, the idea or ideas in the if part of each reason must come from the statement column somewhere above the reason; and the single idea in the then part of the reason must match the idea in the statement on the same line as the reason. This incredibly important flow-of-logic structure is shown with arrows in the following proof.
FIGURE 1-2: A proof with the reasons written in if-then form.
In the preceding proof, each if clause uses only a single idea from the statement column. However, as you can see in the following practice problem, you often have to use more than one idea from the statement column in an if clause.
7 In the following facetious and somewhat fishy proof, fill in the missing reasons in if-then form and show the flow of logic as I illustrate in Figure 1-2.
Given: You forgot to set your alarm last night.
You’ve already been late for school twice this term.
Prove: You will get a detention at school today.
Note: To complete this “proof,” you need to know the school’s late policy: A student who is late for school three times in one term will be given a detention.
Statements (or Conclusions)
Reasons (or Justifications)
1) I forgot to set my alarm last night.
1) Given.
2) I will wake up late.
2)
3) I will miss the bus.
3)
4) I will be late for school.
4)
5) I’ve already been late for school twice this term.
5) Given.
6) This will be the third time this term I’ll have been late.
6)
7) I’ll get a detention at school today.
7)
Solutions
1AMER looks like a square, but you can’t conclude that because you can’t assume the sides are equal. You do know, however, that the figure is a rectangle because it has four sides and four right angles.
2IDOL also looks like a square, and again, like with question 1, you can’t conclude that, but this time you can’t conclude that because you can’t assume that the angles are right angles. But because you do know that IDOL has four equal sides, you know that it’s a rhombus.
3 Here are the answers (flip to Chapter 4 if you need to go over triangle classification):
No. The triangles look congruent, but you’re not allowed to assume that.
The tick marks tell you that is equilateral. It is, therefore, an acute triangle and an isosceles triangle. It is neither a right triangle nor an obtuse triangle.
The tick marks tell you that is isosceles and that, therefore, it is not scalene. That’s all you can conclude. It may or may not be any of the other types of triangles.
Nothing. could be the longest side of the triangle, or the shortest, or equal to the other two sides. And it may or may not have the same length as
Yes. might be a right angle, though you can’t assume that it is.
No. (If you got this question right, give yourself a pat on the back.) If were a right angle, would be a right triangle with its hypotenuse. But is the same length as and the hypotenuse of a right triangle has to be the triangle’s longest side.
4 Here are the answers:
No. The triangles might not be congruent in any number of ways. For example, you know nothing about the length of and if were, say, a mile long, the triangles would obviously not be congruent.
No. The triangles would be congruent only if and were right angles, but you don’t know that. Point
B
is free to move left or right, changing the measures of and
No. You don’t know that is a right angle.
No. The figure
looks
isosceles, but you’re not allowed to assume that
Yes. The tick marks show it.
No. Like with part
a,
you know nothing about the length of
No. You can’t assume that (the upside-down
T
means “is perpendicular to”).
Yes. You
can
assume that is straight and that is therefore, and must add up to
Yes. is and is so must also be
5 Answers vary. One example is “If angles are complementary to the same angle, then they’re congruent.” The reverse of this is false because many angles, like obtuse angles, do not have complements (obtuse angles are already bigger than so you can’t add another angle to them to get a right angle).
6Answers vary. Any of the parallel line theorems in Chapter 2 makes a good answer. For example, “If two parallel lines are cut by a transversal, then alternate interior angles are congruent.” Both this theorem and its reverse are true. To wit (in abbreviated form): “If lines are parallel, then alternate interior angles are congruent,” and “If alternate interior angles are congruent, then lines are parallel.”
I hope it goes without saying that this is not an airtight, mathematical proof.
Chapter 2
Points, Segments, Lines, Rays, and Angles
IN THIS CHAPTER
Walking a fine line: Semi-precise definitions of geometry terms
Working with union and intersection problems
Looking at supplementary and complementary angles (free stuff!)
Turning to right angles
Spotting vertical angles
In this chapter, you first review the building blocks of geometry: points, segments, lines, rays, and angles. Then I go over some terms related to those objects: midpoint, bisection, and trisection; parallel and perpendicular lines; right, acute, and obtuse angles; complementary and supplementary angles; and vertical angles. You’ll get the hang of these things working through the practice problems.
Hammering Out Basic Definitions
You probably already know what the following things are, but here are their definitions and undefinitions anyway. That’s right — I said undefinitions. Technically, point and line are undefined terms, so the first two “definitions” that follow aren’t technically definitions. But if I were you, I wouldn’t sweat this technicality.
Point:
You know, like a dot except that it actually has no size at all. Or, you could say that it’s infinitely small. (That’s pretty small, eh? But even “
infinitely
small” makes a point sound larger than it really is.)
Line:
A line’s like a thin, straight wire. (Actually, it’s infinitely thin or, even better, it has
no width at all
