WARM UP EXERCSE

1. WARM UP EXERCSE C B M A Consider the right triangle below with M the midpoint of the hypotenuse. Is MA = MC? Why or why not? 1

2. WARM UP EXERCSE C B M A Consider the right triangle below with M the midpoint of the hypotenuse. Is MA = MC? Why or why not? 2

3. §1.1 Introductory Material The student will learn about: math systems, basic terms, basic axioms, and geometric proof. 3 3

4. § 1.1 Statements and Reasoning Reasoning – Learning Geometry Requires Time, Vocabulary Development, Attention to Detail and Order, Supporting Claims, and a Lot of Thinking. The Following Types of Thinking or Reasoning Are Used to Develop Mathematical Principles

5. Types of Reasoning Intuition – an inspiration leading to the statement of a theory. Intuition – an inspiration leading to the statement of a theory. Induction – an organized effort to test the theory. Intuition – an inspiration leading to the statement of a theory. Induction – an organized effort to test the theory. Deduction – A formal argument that proves the tested theory.

6. Mathematical System. A mathematical system consist of: Undefined terms. Mathematical System. Mathematical System. Mathematical System. A mathematical system consist of: • Undefined terms. • Defined terms. A mathematical system consist of: • Undefined terms. • Defined terms. • Axioms and postulates. A mathematical system consist of: • Undefined terms. • Defined terms. • Axioms and postulates. • Theorems.

7. “With postulates, my dear, you need a gentle touch, They should not say too little, they should not say too much,And on one point above all, we must be insistent,Though postulates need not be ‘true,’ there set must be consistent.” Journey into Geometries by Marta Sved

8. Example Axiom 1: Through any two distinct points there is exactly one line. Axiom 2: Every line has at least two distinct points. Axiom 3: Not all points are on one line. Design a geometry that fits these postulates.

9. Example Axiom 1: Through any two distinct points there is exactly one line. Axiom 2: Every line has at least two distinct points. Axiom 3: Not all points are on one line. Points – Apricot, Banana and Chocolate. Lines – Apricot-Banana, Apricot-Chocolate and Banana-Chocolate.

10. Axiomatic Systems - Example Axiom 1: Every line contains at least two points. Axiom 2: Each two lines intersect in a unique point. Axiom 3: There are precisely three lines. C A B A model. 10

11. “What is thinking? I should have thought I would have known.” – Karl Gerstner

12. Conditional Statements A conditional statement is written in the form, If p then q, or p implies q, and is symbolized by p → q. The condition p is called the hypothesis and q is the conclusion. If p is a statement then ~ p is the negation of statement p. 12

13. Conditional Statements A conditional statement is in the form, p → q. You should be familiar with the converse, inverse and contrapositive of a statement. Converse: q → p. Inverse: then ~ p → ~ q. Contrapositive: ~ q → ~ p. 13

14. “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’!”Alice in Wonderland by Lewis Carroll

15. Conditional Statements Find the conditional statements and any converses, inverses, or contrapositives. “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’!” Converse: q → p. Inverse: then ~ p → ~ q. Contrapositive: ~ q → ~ p. 15

16. Conditional Statements Find the conditional statements and any converses, inverses, or contrapositives. “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’!” Converse: q → p. Inverse: then ~ p → ~ q. Contrapositive: ~ q → ~ p. 16

17. Valid Arguments An argument is valid if when all the premises are true then the conclusion is true. In a logic class truth tables are used to prove arguments valid. Can you do that with the previous statements from Alice and Wonderland? In this class we will use the historically proven methods of proof to arrive at valid conclusions. 17

18. Types of Reasoning Direct Proof A Formal Proof Consist of the Following: 1. A statement or statements of what is given. 2. A statement of what is to be proven. 3. A drawing. 4. The proof in two column or paragraph form. 18

19. Example of Direct Proof l 1 2 3 4 m Vertical Angle Theorem Given: Intersecting lines l and m. Prove: m  1 = m  4. 19

20. Types of Reasoning Indirect Proof An indirect proof should have the same four parts of a direct proof. The indirect proof assumes the conclusion is false and arrives at a contradiction to what is given. This method is sometimes referred to as “reductio ad absurdum”. 20

21. Types of Reasoning Indirect Proof Indirect proof works particularly well when: The negation of the initial premise P is easy. When Q contains a negation and denies some claim. Existence theorems. Uniqueness theorems. 21

22. Example of Indirect Proof l 1 2 3 4 m Vertical Angle Theorem Given: Intersecting lines l and m. Prove: m 1 = m 4. 22

23. Types of Reasoning Proof by Elimination/Exhaustion An elimination proof should have the same four parts of a direct proof. It is useful when there are finite possible events that occur and you can eliminate all but one of them. Then the remaining event must occur. 23

24. Example of Elimination Proof l 1 2 3 4 m Vertical Angle Theorem Given: Intersecting lines l and m. Prove: m 1 = m 4. 24 Continued

25. Example of Elimination Proof l 1 2 3 4 m Vertical Angle Theorem Given: Intersecting lines l and m. Prove:  1 =  4. 25

26. Types of Reasoning There is one more type of proof we will use and that is called induction 1. Prove for the case where n = 1. 2. Assume it is true for the case where n = k. 3. Prove for the case where n = k + 1. This idea is like a row of dominos falling after you knock over the first one. 26

27. Example of Induction 1. Prove for the case where n = 1. 2. Assume it is true for the case where n = k. Continued 27

28. Example of Induction 3. Prove for the case where n = k + 1. 28

29. Greek Proof n n + 1 29

30. Example of Induction 1. Prove for the case where n = 1. 2. Assume it is true for the case where n = k. Continued 30

31. Example of Induction 3. Prove for the case where n = k + 1. 31

32. Greek Proof n n 32

33. Summary. • We learned about several types of reasoning. • We learned about conditional statements. • We learned about valid arguments. • We learned about direct proof. • We learned about indirect proof. • We learned about proof by exhaustion. • We learned about proof by induction. 33

34. Assignment: §1.1

36. Example Through any two distinct points there is exactly one line. Every line has at least two distinct points. Not all points are on one line. Points – Apricot, Banana and Chocolate. Lines – Apricot-Banana, Apricot-Chocolate and Banana-Chocolate.