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:

Remember 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.

Example This icon is next to all example problems — duh.

Tip This icon gives you shortcuts, memory devices, strategies, and so on.

Warning 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?

theoremsandpostulates 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

Bullet Defining geometry

Bullet Examining theorems and if-then logic

Bullet 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.)

Warning 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 math 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 math ] 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.

Example Q. What can you assume and what can’t you assume about SIMON?

A. You can assume that

math (line segment MN) is straight; in other words, there’s no bend at point O.

Another way of saying the same thing is that math is a straight angle or a math angle.

math and math are also straight as opposed to curvy.

Therefore, SIMON is a quadrilateral because it has four straight sides.

(If you couldn’t assume that math 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

math and math are right angles.

math is an obtuse angle (an angle greater than math ).

math is an acute angle (an angle less than math ).

math is greater than math or math or math and ditto for the relative sizes of other angles.

math is shorter than math or math and ditto for the relative lengths of the other segments.

O is the midpoint of math

math is parallel to math

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 math is acute? Obtuse? Right? Isosceles (with at least two equal sides)? Equilateral (with three equal sides)?

Can you conclude that math is acute? Obtuse? Right? Isosceles? Equilateral?

What can you conclude about the length of math ?

Might math be a right angle?

Might math be a right angle?

4 Can you assume or conclude

math is isosceles?

D is the midpoint of math ?

Z is the midpoint of math ?

math is an altitude (height) of math ?

math is supplementary to math

math 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.

Warning 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.

Example 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.

Example 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.

Tip 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.

Remember 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