G E O M E T R Y CHAPTER 2 REASONING AND PROOF. Notes & Study Guide CHAPTER 2 NOTES

Transcription

1 G E O M E T R Y CHAPTER 2 REASONING AND PROOF Notes & Study Guide

2 2 TABLE OF CONTENTS CONDITIONAL STATEMENTS... 3 DEFINTIONS & BICONDITIONAL STATEMENTS... 6 DEDUCTIVE REASONING... 9 REASONING WITH PROPERTIES OF ALGEBRA HOMEWORK EXAMPLES... 15

3 CONDITIONAL STATEMENTS 3 INTRODUCTION Chapter 2 may be the most difficult chapter in the course because it deals with logic and reasoning. These topics can be difficult to describe to students who may not have welldeveloped logic/common sense skills. What we will discuss are the different types of statements (conditionals) that are used in Geometry that help establish the rules and laws we ll use. We will also examine the step-by-step process (deductive reasoning) that mathematicians use to prove the statements that they make are true (proofs). CONDITIONAL STATEMENTS Almost all the rules and laws that we use in Geometry are written as conditional statements. In general, all conditional statements state that if A happens, then B is true. This is why we often call them if then statements. In other words, if all the requirements in A are met, then we accept whatever B says as true. THE TWO PARTS All conditional statements contain two parts Hypothesis the first part of the statement with the if. Conclusion the second part of the statement with the then. CONDITIONAL STATEMENT EXAMPLES If an animal has stripes, then the animal is a tiger. If the temperature drops below 70, then the pool will close. If a trapezoid is isosceles, then each pair of base angles is congruent. If a quadrilateral is a parallelogram, then its opposite sides are congruent.

4 4 CONDITIONAL STATEMENTS HOW CONDITIONAL STATEMENTS WORK When everything here happens everything here can be accepted. If a geometric shape has 3 sides, then the shape is a triangle. If today is Wednesday, then tomorrow is Thursday. If Tommy s grade is higher than 90%, then he will receive an A. Conclusions are not accepted unless all of the hypothesis requirements are met! TYPES OF CONDITIONAL STATEMENTS There are four types of conditional statements. The conditional itself is the first type. The other three are created directly from the conditional. They are called converses, inverses and contrapositives. IMPORTANT! When we write these statements, we DO NOT care if they are true or if they even make sense. We ll deal with that later. CONVERSE The converse of a conditional statement is formed by simply switching the hypothesis and the conclusion. Conditional If the wind is blowing, then the kite will fly. Converse If the kite will fly, then the wind is blowing. < end of page>

5 CONDITIONAL STATEMENTS 5 INVERSE The inverse of a conditional statement is formed by writing its negation. This means the negative (or opposite) of something. To do it, we use the word not. Conditional If the wind is blowing, then the kite will fly. Inverse If the wind is NOT blowing, then the kite will NOT fly. CONTRAPOSITIVE The contrapositive of a conditional statement is formed by writing its negation AND switching its hypothesis and conclusion. It is a combo of a converse and an inverse. Conditional If the wind is blowing, then the kite will fly. Contrapositive If the kite will NOT fly, then the wind is NOT blowing. SUMMARY OF CONDITIONAL STATEMENTS Standard Notation (written or spoken) Symbol Notation CONDITIONAL If p, then q. p q CONVERSE If q, then p. q p INVERSE If not p, then not q. ~p ~q CONTRAPOSITIVE If not q, then not p. ~q ~p The symbol notation is used as a more convenient way to write conditional statements. Assign a variable (p and q) to each part and use a squiggle.

6 6 DEFINITIONS AND BICONDITIONALS BICONDITIONAL STATEMENTS A conditional statement and its converse can be combined into one larger statement that is known as a biconditional. Biconditionals can be identified by their use of the phrase if and only if. Conditional If the wind is blowing, then the kite will fly. Converse If the kite will fly, then the wind is blowing. Biconditional The wind is blowing if and only if the kite will fly. STEPS TO CREATE A BICONDITIONAL Start with the original statement Drop the IF Replace the THEN with IF AND ONLY IF EXAMPLES If you see lightning, then you hear thunder. You see lightning if and only if you hear thunder. If today is Saturday, then Troy is working at the restaurant. If the measure of angle A is 30, then angle A is acute. If two lines intersect, then their intersection is exactly one point. The phrase if and only if can be shortened using IFF

7 DEFINITIONS AND BICONDITIONALS 7 PROVING CONDITIONAL STATEMENTS TRUE Conditional statements can either be true or false. To prove one is true, you must present an argument that shows the conclusion is true for ALL possible hypotheses (proof). To prove one is false, you must come up with a single example hat shows the statement is not true (counterexample). This process is the same as the counterexample definition in Chapter 1. EXAMPLES Show that each statement is false by providing a counterexample. If Calvin s hair is wet, then he went swimming in the pool If a geometric shape has four sides, then it is a rectangle. If Tracy s car won t start, then the car is out of gas. DEFINITIONS A biconditional statement that is true in both directions (conditional and converse) is classified as a definition. If either direction is false, then it cannot be a definition. It is still a biconditional though. Definitions are important. Since they are true forwards and backwards, they can be counted on to work all the time. There is no chance of finding a counterexample that makes them fail.

8 8 DEFINITIONS AND BICONDITIONALS EXAMPLES An animal is a tiger IFF it has stripes. COND: If an animal is a tiger, then it has stripes. TRUE CONV: If an animal has stripes, then it is a tiger. FALSE The converse of this example is false, so the statement is NOT a definition. Two lines are perpendicular IFF they intersect at 90. COND: If 2 lines are perpendicular, then they intersect at 90. TRUE CONV: If 2 lines intersect at 90, then they are perpendicular. TRUE Both statements in this example as true, so the statement IS a definition. Three points are collinear IFF they lie on the same line. You can become President of the U.S. IFF you are 35 years old.

9 DEDUCTIVE REASONING 9 INTRODUCTION In this section we cover the concept of deductive reasoning and how we use it to reach conclusions when presented with a collection of statements. We also will cover the symbol notation used for conditional statements. SYMBOL NOTATION In symbol notation for conditionals, the letters p and q are used to represent the hypothesis and the conclusion in a conditional statement. An arrow represents the if-then. A double arrow means if and only if. Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p Biconditional p q DEDUCTIVE REASONING Deductive reasoning is the process of using facts, definitions and accepted truths, written in a logical order (argument), to reach a conclusion. Inductive: conclusions made by observing patterns Deductive: conclusions made by using facts or definitions There are two methods (laws) of applying deductive reasoning. The Law of Detachment The Law of Syllogism

10 10 DEDUCTIVE REASONING THE LAW OF DETACHMENT The Law of Detachment occurs when the hypothesis of a given conditional statement has been confirmed by a second statement (fact/definition). When this confirmation happens, the conclusion is accepted to be true. EXAMPLES: What can be concluded from each pair of statements? (1) If Mike visits New York, then he will spend a day with his aunt & uncle. (2) Mike visited New York last month. (C) Mike spent a day with his aunt & uncle. (1) If Bill gets a C on his test, then he will get an A for the quarter. (2) Bill got a 68% on his test. (C) THE LAW OF SYLLOGISM The Law of Syllogism uses two or more conditional statements at once so that the conclusion of one statement matches the hypothesis of the other. These matching pieces cancel each other out and create one final conditional. EXAMPLES: What can be concluded from each pair of statements? (1) If Mike visits New York, then he will spend a day with his aunt & uncle. (2) If Mike spends a day with his aunt & uncle, then he will go to bed early. (C) If Mike visits New York, then he will go to bed early. (1) If the cat catches the mouse, then the dog will bark loudly. (2) If the dog barks loudly, then the parrot will sneeze. (C) Think of the matching pieces as the links that chain the statements together.

11 DEDUCTIVE REASONING 11 MORE EXAMPLES The Shady Oaks apartments do not allow dogs. Gloria lives in a Shady Oaks apartment. If Jack s wife cheats on him, then he will step in front of a bus. If Jack steps in front of a bus, then Bonnie will be late for work. > > If Alberto finds a summer job, then he will buy a car. If Alberto buys a car, then his girlfriend will dump him. LOGIC CHAINS When the Law of Syllogism involves 3 or more statements it creates what is called a logic chain. Each matching if-then pair will cancel each other out. In the example shown, the B s, C s and D s do this. After all the canceling happens, whatever s left is the final conclusion. In this case that s If A, then E. If A, then B. If B, then C. If C, then D. If D, then E. EXAMPLE If Charlie finishes his homework, then he can play video games. If Charlie can play video games, then he will stare at the TV too long. If he stares at the TV too long, then his eyeballs will fall out. If his eyeballs fall out, then Charlie will fail his driver s test. If Charlie finishes his homework, then he will fail his driver s test.

12 12 REASONING WITH PROPERTIES FROM ALGEBRA ALGEBRAIC PROPERTIES OF EQUALITY Surprisingly, geometry makes regular use of the standard properties of Algebra. They are useful in building logical arguments (proofs) to back up our statements. The properties of Equality all do the same thing they say whatever you do to one side, you do to the other. There is a property for each operation. Addition: Subtraction: Multiplication: Division: THE FOUR PROPERTIES OF ALGEBRA If A = B, then A + C = B + C If A = B, then A C = B C If A = B, then AC = BC If A = B, then A/C = B/C There are three other important properties that are used in Geometry also REFLEXIVE A = A This one just means that all values are equal to themselves. SYMMETRIC If A = B, then B = A This one says that any equation is reversible. TRANSITIVE If A = B and B = C, then A = C This one says that if two things are equal to the same third thing, then those two things are equal to each other. EXAMPLES: Identify the Property shown. If x = 5, then x + 4 = If AB = BC, then BC = AB. If x = 12, then 5x = 60. If x 3 = 12 and x 3 = c + 6, then x 3 = c + 6.

13 REASONING WITH PROPERTIES FROM ALGEBRA 13 LOGICAL ARGUMENTS (PROOFS) When you put a series of statements together, in a logical order, that s called a logical argument. In math, we call them proofs. The final conclusion of the proof is the thing you re trying to prove is true. TYPES OF PROOFS Paragraph Proof written using full sentences as if you were verbally trying to prove something to someone. (Like a lawyer in a courtroom does) Two column Proofs written as a very organized small table that lists the statements made and the reasons that support them STATEMENTS REASONS 1. Given Info Given 2. Statement 2 Reason 2 3. Statement 3 Reason Final Conclusion Final Reason ALGEBRAIC PROOFS Anytime you have solved an algebra equation and shown your work, you have done a proof. The steps of the work are the statements. The only thing you don t usually do is write down the reasons (the Properties that allow you to do your work). EXAMPLE: Solve 5x 18 = 3x + 2 STATEMENTS REASONS 1. 5x 18 = 3x + 2 Given 2. 2x 18 = 2 Subtraction prop of = 3. 2x = 20 Addition prop of = 4. x = 10 Division prop of =

14 14 REASONING WITH PROPERTIES FROM ALGEBRA MORE EXAMPLES Write an algebraic proof (in 2-column format) for each of the following Solve: 2(3x + 1) = 5x + 14 STATEMENTS REASONS Solve: 8x 5 = -2x 15 STATEMENTS REASONS <>

15 ADDITIONAL NOTES & EXAMPLES 15 <>

16 16 ADDITIONAL NOTES & EXAMPLES <>