Chapter 2 Reasoning in Geometry

1. 2.2 Introduction to Logic Chapter 2 Reasoning in Geometry

2. Introduction In chapter 2 section 2, we will discuss how we use logic to develop mathematical proofs. When writing proofs, It is important to use exact and correct mathematical language. We must say what we mean!

3. Introduction Do you recognize the following conversation?

4. "Then you should say what you mean." the March Hare went on. "I do," Alice hastily replied; "at least -- at least I mean what I say -- that's the same thing, you know. " "Not the same thing a bit!" said the Hatter, "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"

5. "You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!“ "You might just as well say," added the Dormouse, who seemed to be talking in his sleep, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!“ "It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a minute.

6. Charles Dodgson Charles Dodgson lived from 1832 to 1898 Dodgson was a mathematics lecturer and author of mathematics books who is better known by the pseudonym Lewis Carroll. He is known especially for Alice's Adventures in Wonderland.

7. Conditional Statements In order to analyze statements, we will translate them into a logic statement called a conditional statement. (You will be taking notes now)

8. Essential Question: How do I recognize and analyze a conditional statement?

9. Definition • Hypothesis The if part of a conditional statement Free PowerPoint Template from www.brainybetty.com

10. Defintion • Conclusion The then part of a conditional statement Free PowerPoint Template from www.brainybetty.com

11. Definition • Conditional IF something, THEN something else If a car is a Corvette, then it is a Chevy If you are in this room right now, then you are in Geometry Free PowerPoint Template from www.brainybetty.com

12. Conditional Statements conditional statement • A _________________ is a statement that can be expressed in ________form. “if-then” 2. A conditional statement has _________. • The __________ is the ____ part. • The __________ is the ______ part. two parts hypothesis “if” conclusion “then”

13. Conditional Statements Example: (Original) I breathe when I sleep (Conditional) If I am sleeping, then I am breathing.

14. Conditional Statements Definition: A conditionalstatement is a statement that can be written in if-then form. “If_____________, then______________.” Example: Ifyour feet smell and your nose runs, thenyou're built upside down. Continued…… Lesson 2-1 Conditional Statements

15. Definition • Conditional If / then statements are conditional. The then part of the statement is depends on (is conditional to) the if part. In shorthand, the statement is “if p then q” In symbol form, p = feet smell, nose runs q = built upside down Free PowerPoint Template from www.brainybetty.com

16. Rewrite in the if-then form • All mammals breathe oxygen • If an animal is a mammal, then it breathes oxygen. • A number divisible by 9 is also divisible by 3 • If a number s divisible by 9, then it is divisible by 3. Geometry

17. Examples • If you are 13 years old, then you are a teenager. • Hypothesis: • You are 13 years old • Conclusion: • You are a teenager Geometry

18. If a car is a Corvette, then it is a Chevrolet Hypothesis Conclusion Free PowerPoint Template from www.brainybetty.com

19. Euler Diagram (Venn Diagram) Cars Chevys Corvettes Free PowerPoint Template from www.brainybetty.com

20. Euler Diagram (Venn Diagram) Chevrolets (Conclusion: then part) Corvettes If a car is a Corvette, then it is a Chevrolet (Hypothesis: If part)

21. Example: Euler Diagram Supplementary angles (Conclusion: then part) What is the conditional statement? If two angles form a linear pair, then the angles are supplementary angles Linear pairs (Hypothesis: If part)

22. Conditional Statements • The ________ of a conditional statement is formed by switching the hypothesis and the conclusion. • Example: converse (Conditional) If I am sleeping, then I am breathing. (Converse) If I am breathing, then I am sleeping.

23. Definition • Converse Changing the if and the then around • Conditional: If a car is a Corvette, then it is a Chevrolet • Converse: If a car is a Chevrolet, then it is a Corvette Free PowerPoint Template from www.brainybetty.com

24. Determine the Converse If you are wearing a skirt, then you are a female If you are a female, then you are wearing a skirt Free PowerPoint Template from www.brainybetty.com

25. Definition • Counterexample An example that proves a statement false Consider the conditional statement: If you are a female, then you are wearing a skirt Is there any females in the room that are not wearing a skirt? Free PowerPoint Template from www.brainybetty.com

26. Writing a Counterexample • Write a counterexample to show that the following conditional statement is false • If x2 = 16, then x = 4. • As a counterexample, let x = -4. • The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false. Geometry

27. Definition • Deductive Reasoning The process of drawing logically certain conclusions by using an argument Free PowerPoint Template from www.brainybetty.com

28. Euler Diagram (Venn Diagram) Chevrolets Susan’s car is a Corvette If a car is a Corvette, then it is a Chevrolet Susan’s car is a Corvette Therefore the conclusion is: Susan's car is a Chevrolet. • Corvettes • Susan’s car

29. Definition • If-Then Transitive Property • If A then B • If B then C • You can conclude: If A then C • Also known as a logic chain Free PowerPoint Template from www.brainybetty.com

30. Example Consider the following conditionals - If cats freak, then mice frisk • If sirens shriek, then dogs howl • If dogs howl, then cats freak Prove the following: If sirens shriek, then mice frisk Free PowerPoint Template from www.brainybetty.com

31. Logical Chain (Transitive property) • If cats freak, then mice frisk • If sirens shriek, then dogs howl • If dogs howl, then cats freak Using the provided statements to prove the following conclusion: If sirens shriek, then mice frisk Look for the conditional that begins with the then statement and write it down under the first Keep repeating until you get a conclusion that matches the one you’re looking for First, find the hypothesis of the conditional you are trying to prove Second, write down the conditional with that hypothesis If sirens shriek, then dogs howl Conclusion: If sirens shriek, then mice frisk If dogs howl, then cats freak If cats freak,then mice frisk Free PowerPoint Template from www.brainybetty.com

32. 1. Identify the underlined portion of the conditional statement. • hypothesis • Conclusion • neither

33. 2. Identify the underlined portion of the conditional statement. • hypothesis • Conclusion • neither

34. 4. Identify the converse for the given conditional. • If you do not like tennis, then you do not play on the tennis team. • If you play on the tennis team, then you like tennis. • If you do not play on the tennis team, then you do not like tennis. • You play tennis only if you like tennis.

35. Assignment • Read pages 90-93, Ch2 Sec 2 Complete problems on Page 95 #9-34 Due Friday Oct. 15. • This is an involved set of problems and will take some time to complete. You will be making a big mistake if you wait until Thursday evening to begin this assignment. • Suggestion: Break into small parts, complete 6 to 10 problems per day/night. Free PowerPoint Template from www.brainybetty.com