Notes — Chapter 2: Deductive Reasoning

To the Student. Chapter 1 showed geometry being used in many ways over the ages but concluded “We can’t go on like this.” Then how to proceed? The answer is: “Deductively and carefully.” Euclid showed the way around 300 B.C., and there have been many additions and improvements since. He gave us geometry as a deductive system, with clear assumptions (the postulates), definitions, and methods for proving what is true, our theorems.

You already know how to make deductions. If you look out the window and see rain, you deduce that there are clouds in the sky, even if you can’t see the sky from your window. Deductive reasoning is the stuff detectives use; it can help you solve puzzles and solve problems. We end this chapter with some famous theorems of geometry, some of which you probably know already. As we work through the course, you will see why these must be true and how they can be proved.

Lesson 1. Conditional Statements. These are the basic tools of deduction. Using Agatha Christie’s detective Hercule Poirot, we introduce such statements and the ways they are represented symbolically and by Euler diagrams. The examples and exercises give students practice in interpreting these forms and moving between them.

Lesson 2. Definitions. In formal arguments, we often need to replace a reference to some property or thing with the definition of that property or thing, so this chapter covers definitions, the converses of conditionals, and logical equivalence. The symbolic forms a — b, a <— b, and a + a, as well as the abbreviation “a iff b” are covered. The examples and exercises give practice in recognizing and shifting between these various representations.

Lesson 3. Direct Proof. This workhorse of deduction is introduced by a Rube Goldberg cartoon. The

symbolic and Euler diagram interpretations reinforce the idea behind this valid form of inference. Exercises show that direct proof is an effective tool for critical thinking and for solving puzzles and problems.

Lesson 4. Indirect Proof. If assuming the opposite (or logical negation) of your desired conclusion leads to a contradiction, then your desired conclusion must be true. Examples given of this basic method of proof range from concluding that an ammonia molecule cannot be planar, to analyzing possible poker hands, to coin-weighing problems.

Lesson 5. A Deductive System. To avoid circularity or infinite regress, some terms must be taken as basic and left undefined; certain observations must be taken as postulates. With postulates, definitions, and means for dzrect and indirect proof to derive our theorems, we have the basis for the deductive system we use in the study of geometry.

This lesson includes our first two postulates. Postulate 1: Two points determine a line. Postulate 2: Three noncollinear points determine a plane. Note that “two points” should be understood as meaning the same as the alternative “two distinct points.” Either form is acceptable, but the former is simpler and agrees more closely with common usage.

Lesson 6. Some Famous Theorems of Geometry. An opening “B.C.”cartoon makes the point that many students will already know results such as the a? + b? = c? of the Pythagorean Theorem and the fact that the sum of the angles in a triangle is 180°. Jacobs takes advantage of this head start and brings other students up to speed by introducing these results along with Circumference = 2ar and Area = mr? for the circle. He gives plausibility arguments for the Pythagorean Theorem and the Angle Sum Formula for triangles and discusses the results for the circle, leaving proofs for later.

Using these results, students can do applied problems that provide practice with algebraic methods, as the exercises show.

Looking Ahead. We prove the Angle Sum Formula for triangles using the Parallel Postulate in Chapter 6. We offer two approaches to the Pythagorean Theorem: one via areas (the formal version of the plausibility argument given in this chapter) and the other by similar triangles (Chapter 11, Lesson 2). The later approach uses the Angle Sum Theorem. For the circumference and area of the circle we use a limiting argument, like the method of exhaustion used by Archimedes (Chapter 14 of Geometry and in these notes).