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):

1. Can you assume that the triangles are congruent?
2. Can you conclude that is acute? Obtuse? Right? Isosceles (with at least two equal sides)? Equilateral (with three equal sides)?
3. Can you conclude that is acute? Obtuse? Right? Isosceles? Equilateral?
4. What can you conclude about the length of ?
5. Might be a right angle?
6. Might be a right angle?

4 Can you assume or conclude

1. is isosceles?
2. D is the midpoint of ?
3. Z is the midpoint of ?
4. is an altitude (height) of ?
5. is supplementary to
6. 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.

1. Give an example of a theorem that’s not reversible and explain why the reverse is false.
2. 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:

1. “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...