1. Chapter 2 Reasoning and Introduction to Proofs Pg. 69 Conceptual Objective (CO): Hard to believe you must be taught to reason! But it is a practiced skill! The LSAT has no law questions on it…it is all reasoning. In this chapter you will learn the basic rules/vocabulary of logic; make, prove, and disprove conjectures; and begin to write proofs.

2. Chapter 2 Reasoning and Introduction to Proofs Pg. 69 Do It To It (DITI): 1.Conditional Statements (If-Then statements) Hypothesis and Conclusion 2.Converse – (Switch the If –Then ) 3.More Postulates 4.Biconditionals – (If and Only If) 5.Properties from Algebra 6.Geometric Proofs - Statements and Reasons

3. Section 2-1 Conditional Statements Pg. 71 Conceptual Objective (CO): Conditional Statements, better known as If-Then Statements, are the first basic tool of logic. IF you can begin with a known fact and make a series of logically progressive facts from there, THEN your conclusion (your proof) is irrefutable. Do It To It (DITI): 1.Identify Hypothesis-Conclusions 2.Write Converses Sec 2-1 and further discussions

4. Section 2-2 Definitions and BiConditional Statements Pg. 79 Conceptual Objective (CO): The two concepts of Definitions and TRUE BiConditional Statements go together because they share a characteristic that they must be true both forward and backward. This is the same idea as an equal sign. Do It To It (DITI): 1.Use definitions in Proofs in either direction 2.CAREFUL! Do not assume every Conditional has a true Converse! ONLY if it is a true can the Converse be used in Proofs.

5. Section 2-2 Definitions and BiConditional Statements Pg. 79 Let’s read thru Section 2-2. First, each of you read the section quietly at your table. Then we will analyze what we read together.

6. Section 2-2 Definitions and BiConditional Statements Pg. 79 It seems a bit confusing. What you need to know is that definitions and TRUE Biconditionals can be used either forwards or backwards in proofs. EX: Given: Point A lies in the interior of Angle XYZ. Angle XYA is congruent to Angle AYZ. Prove: Ray AY is an angle bisector of Angle XYZ. What is the definition of an Angle Bisector?See top of pg 36. So…we need to prove we have two adjacent angles that are congruent.

7. Section 2-2 Definitions and BiConditional Statements Pg. 79 Given: Point A lies in the interior of Angle XYZ. Angle XYA is congruent to Angle AYZ. Prove: Ray AY is an angle bisector of Angle XYZ. Definition of Angle Bisector: A ray that divides an angle into two adjacent angles that are congruent. STATEMENT REASON 1. Point A lies in the interior of Angle XYZ. 1. GIVEN Angle XYA is congruent to Angle AYZ. 2. Angle XYA is adjacent to Angle AYZ 2. They share the common Ray YA 3. Ray AY is an angle bisector of Angle XYZ 3. Definition of Angle Bisector

8. HOMEWORK HW 2-1: Pg 75: #1 – 21 ODD, #52 BRING IN MAGAZINE PAGE HW 2-2: Pg 82: #1, 13 – 23, 44, 45

9. Section 2-3 Deductive Reasoning Pg. 87 Conceptual Objective (CO): It is time to start using Conditional Statements and their knock offs for reasoning out proofs. For this section, you need to know the symbols for a Conditional not the Converse, Inverse, and Contrapositive. And for this section, we are really only interested in the two laws of logic - The Law of Detachment and the The Law of Syllogism.

10. Section 2-3 Deductive Reasoning Pg. 87 First let’s look at the symbols that represent a Conditional: If p then q can be written in symbols as p  q Do It To It (DITI): For this section we really only are interested in Two Laws: 1. The Law of Detachment - Given a true p  q, if p is true, then q is true 2. The Law of Syllogism – Given a true p  q and a true q  r, if p is true then r is true* * What Algebra Property does this remind you of?

11. Section 2-3 Deductive Reasoning Pg. 87 Do It To It (DITI): You do not need to know the names of these laws! You just need to know that they exist and they are the building blocks of logic and, in your world, Proofs! Given a true p  q, and p is true, then q is true Given a true p  q and a true q  r, then we also have a true p  r* *To answer the question from the previous slide, this should remind you of the Transitive Prop. Of Equality

12. Some Examples Given a true p  q, and p is true, then q is true Given a true p  q and a true q  r, then we also have a true p  r Let’s see how these work. First, EX # 4, pg 89. Then, EX# 5, pg 90. and then put it all together EX # 6.

13. HOMEWORK 2-3 Pg 92: #23-25, 30 – 32, 37 – 43 ODD, 49

14. Section 2-4 Reasoning with Properties from Algebra Pg. 96 Conceptual Objective (CO): You will describe the steps you do on an algebra problem using the correct property names. You will present your work in an established two-column format. THE ALGEBRA PART SHOULD BE EASY! I do not want you to get caught up in the Algebra…so the algebra is easy. Instead I want you to practice and get used to having a specific reason for every step AND for having a name for that reason. THAT SKILL will translate to your Geometric Proofs, which are not easy.

15. Section 2-4 Reasoning with Properties from Algebra Pg. 96 Do It To It (DITI): Learn the Properties! (Now you do need to memorize the names/definitions – but use the names themselves to trigger their meanings. See chart pg 96.) Read pp 96/97. We will do Pg 99 4 – 8 together. 2-4 further discussion

16. HOMEWORK 2-4 Pg 99: #15 – 25 ODD, 29-31

17. Quick Check Get into Groups of 2 or 3 Complete: Practice Worksheet 2-4 Side A The back side, Practice Worksheet 2-4 Side B will be part of your HW tonite

18. Chapter 2-5 Proving Statements about Segments Pg. 102 CO: We will now be introduced to Theorems in order to begin to construct geometric proofs. Theorems are statements that have already been PROVEN thru the process of deductive reasoning with the use of postulates and definitions and properties. Once we have proven a Theorem, it can be used to prove other Theorems, and so on.

19. Chapter 2-5 Proving Statements about Segments Pg. 102 DITI: Let’s look at Examples #1, 2, and 3 pp. 102 – 103

20. NOTE! NAME of SEGMENT: Has endpoints A and B MEASURE or DISTANCE between A and B You can NOT +, -, x, NAMES of segments, only MEASURES Use ‘Definition of Congruent Segments’ to switch

21. HOMEWORK 2 - 5 Practice Worksheet 2-4 Side B Pg 105: #3 – 5 (use paragraph proof if you want) #7, 9, 11, 17

22. Chapter 2-6 Proving Statements about Angles Pg. 109 DITI: Use two column proofs, postulates, definitions and properties and known Theorems to prove Theorems. Obviously, you can only use a Theorem from previous sections in the book to prove Theorems in subsequent sections. Let’s read thru the examples on pp 109 – 112. NOW…let’s do an in-class handoutin-class handout

23. Chapter 2-6 Proving Statements about Angles Pg. 109 Chapter 2-5/2-6 further discussion

24. HOMEWORK 2 - 6 2-5 Day 2 Homework Worksheet Handout Pg 112: #5-9 ODD, 12-17, 23-26 (there will be angle questions like Pg 113 #12-17) Pg 118 Chapter Review: #1 – 4, 11 – 21 Plus, Study for TEST on CH 1 and 2