Geometry 2 2 Continued Deductive Reasoning Essential Question

Geometry 2. 2 Continued: Deductive Reasoning Essential Question: How is deductive reasoning different than inductive reasoning?

Topic/Objectives n n n Use symbolic notation to represent logical statements. Form conclusions using logic. To use deductive reasoning. 13 January 2022 2. 2 Continued Deductive Reasoning 2

Deductive Reasoning Uses facts, definitions and properties in a logical order to make a logical argument. Geometric proofs always use deductive reasoning. 13 January 2022 2. 2 Continued Deductive Reasoning 3

Rules of Deduction 1. Law of Detachment 2. Law of Syllogism 13 January 2022 2. 2 Continued Deductive Reasoning 4

Law of Detachment If P Q is true, and P is true, then Q is true. Example: If I get an A in Geometry, then I will get a new bike. I get an A in Geometry. Conclude: I get a new bike. 13 January 2022 2. 2 Continued Deductive Reasoning 5

Example Judy’s parents tell her that if she wants to go to the movies Saturday night, she has to wash the windows. Judy gets up Saturday and washes all the windows. What do we conclude? Judy can go to the movies that night. 13 January 2022 2. 2 Continued Deductive Reasoning 6

What about this? If two angles form a linear pair, then they are supplementary. A and B are supplementary. Conclusion: A and B form a linear pair. WRONG! This is saying P Q, and Q is true. This doesn’t make P true. 1 and 2 are supp, but 1 not a linear pair. 2 13 January 2022 2. 2 Continued Deductive Reasoning 7

Commonly seen errors. “If there is a storm, then the power will go out”, said Marco. Suddenly the power goes out, leaving everyone in the dark. “Gee, I guess there is a storm!”, exclaims Marco. “No, you dummy: you forgot to pay the electric bill”, said his sister. Just because the conclusion is true, does not mean that the hypothesis is true. 13 January 2022 2. 2 Continued Deductive Reasoning 8

Law of Detachment (again) P Q is true. P Q P is true. P Conclusion: Q is true. Q 13 January 2022 2. 2 Continued Deductive Reasoning 9

Law of Syllogism In symbols: If P Q is true, and Q R is true, then P R is also true. Example: If Sam gets an A in math, then he can buy a car. If he can buy a car, then he will be popular. Conclusion: If Sam gets an A in math, then he will be popular. 13 January 2022 2. 2 Continued Deductive Reasoning 10

Example Write the conclusion that we can make based on the law of syllogism. If I watch too much TV, then I will be lazy. If I am lazy, then I won’t get an A in math. Conclusion: If I watch too much TV, then I won’t get an A in math. 13 January 2022 2. 2 Continued Deductive Reasoning 11

Extending a Syllogism If I wake up, then I can get dressed. If I get dressed, then I can go to school. If I go to school, then I can walk to class. If I walk to class, then I will meet the school bully. If I meet the school bully, then I will get in a fight. If I get in a fight, then I will be suspended. If I get suspended, then I will stay home. If I stay home, then I will go to sleep. Conclusion: If I wake up, then I will go to sleep. 13 January 2022 2. 2 Continued Deductive Reasoning 12

So where’s the math? If angles form a linear pair, then they are supplementary. If angles are supplementary, then their sum is 180°. Conclusion: If angles form a linear pair, then their sum is 180°. We have just proved a theorem! 13 January 2022 2. 2 Continued Deductive Reasoning 13

Syllogism (again) Jump from beginning to the end. If P Q, and Q R, then P R These are the same means… 13 January 2022 2. 2 Continued Deductive Reasoning 14

Putting Two Laws Together If Jake gets a job, then he can pay for football camp. If he goes to football camp, then he will make the varsity team. If he makes varsity, then he will be Homecoming King. Jake gets a job. Conclusion: Jake will be Homecoming King. 13 January 2022 2. 2 Continued Deductive Reasoning 15

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Practice 4. Determine if statement (3) follows from statements (1) and (2). If so, what Law was used? (1) If you eat sushi, then you will get sick. (2) You eat sushi. (3) You get sick. 13 January 2022 Yes this follows, by Law of Detachment. 2. 2 Continued Deductive Reasoning 17

Practice 5. Determine if statement (3) follows from statements (1) and (2). If so, what Law was used? (1) If x = 10, then y = 20 (2) If y = 20, then z = 25 (3) If x = 10, then z = 25 Yes this follows by the Law of Syllogism. 13 January 2022 2. 2 Continued Deductive Reasoning 18

Practice 6. Determine if statement (3) follows from statements (1) and (2). If so, what Law was used? (1) If you are a lawyer, then you will be rich. (2) You are rich. (3) Then you are a lawyer No, this doesn’t follow. The argument is invalid. You could get rich another way. 13 January 2022 2. 2 Continued Deductive Reasoning 19

Last Practice Q P 7. If Jim talks back to his boss, then he will lose his job. If he loses. Qhis job, he will have to sell his R car. Jim talks back to the boss. P What’s the conclusion? Jim has to sell his car. P Q, Q R P R means P R P is True (Syllogism) R is True 13 January 2022 2. 2 Continued Deductive Reasoning (Detachment) 20