Geometry Honors

1. Geometry Honors Day 15 Deductive Reasoning

2. Today’s Objectives • Daily Quiz • Review Homework • Deductive Reasoning • Assignment

3. Standards: • Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. • Determine whether two propositions are logically equivalent. • Use methods of direct and indirect proof and determine whether a short proof is logically valid. • Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs.

4. Inductive vs. Deductive Reasoning • Recall that inductive reasoning bases its conclusions on observations and patterns. It is useful, but we can never prove a conjecture definitively by just providing examples. • Deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions from given statements.

5. Deductive Reasoning • Deductive reasoning – a thought process in which a conclusion is an inevitable logical consequence of its premise. • Example: 1) All men are mortal 2) Socrates is a man. • What must be the conclusion? 3) Socrates is mortal. • Deductive reasoning is powerful because, if the starting point of your reasoning is true, the conclusion has to be true.

6. Deductive Reasoning • What is the premise, and what is the conclusion of the following? • If there is lightning, then the power will go off. If the power goes off in the middle of the night, my alarm won’t go off. If my alarm does not go off in the morning, I won’t wake up on time. If I don’t wake up on time, then I can’t make my children breakfast. If my children don’t eat breakfast, they will be cranky. If my oldest daughter is cranky at school, she will hit someone. If she hits someone, she will be expelled. If she gets expelled, she won’t get into college. If she doesn’t get into college, she will be poor.

7. Deductive Reasoning • Deductive reasoning is required to prove a conjecture. There are various methods of valid deductive reasoning. • Note: Saying a conclusion is valid is different than saying it is true. Valid means that the reasoning is sound based on the given premise. • The Law of Detachment states that if is a true statement, and p is true, then q must be true. • In other words, as long as the facts are true, then the conclusion reached with deductive reasoning will have to be true.

8. Examples • Turn to p. 116 and study the examples. Then determine whether the Guided Practice problems show valid or invalid reasoning.

9. The Law of Syllogism • The Law of Syllogism states that if and are true statements, then is also a true statement. • When using the Law of Syllogism, the conclusion of the first statement must be the hypothesis of the next. • You can stack as many of these together as you wish. If they are properly set up, then the hypothesis of the first statement will lead to the conclusion of the last statement.

10. Examples • Turn to p. 118, study the examples, and complete the Guided Practice problems.

11. Postulates • Geometry is an axiomatic system. In other words, it is a system that is created by deductive reasoning, starting from a few commonly accepted premises. • Ideas that we accept without proof are called postulates (or axioms). • In a chain of deductive reasoning (cf. the Law of Syllogism), each conclusion becomes the next premise. But there must be an original premise that we start the chain with. This original premise, by its nature, will be unproven. This is where postulates come in.

12. Postulates (p. 125) • We have discussed some postulates before: • Through any two points, there is exactly one line. • Through any three noncollinear points, there is exactly one plane. • A line contains at least two points. • A plane contains at least three noncollinear points. • If two points lie in a plane, then the entire line containing those points lies in that plane. • If two lines intersect, then their intersection is exactly one point. • If two planes intersect, then their intersection is a line. • Note: The book numbers these postulates (2.1, 2.2, etc.), but you don’t have to worry about the numbering.

13. Postulates and Reasoning • Postulates can be used to support our reasoning process. • Look at Example 2 on p. 126 and answer the Guided Practice problems. • When we solve problems, we should always be prepared to justify our reasoning with the postulates or theorems we are applying.

14. Introducing Proofs • A proof is a detailed description of the logical process used to deduce a fact (called a theorem) from previously known information. • A theorem is a conjecture that has been proven true by deductive reasoning. • Proofs allow us to make universal statements or rules that apply to all situations of a given description. • For example, how do we know that the Pythagorean Theorem can be used on all right triangles? • We don’t. Yet. • A proof is a logical argument in which every statement is supported by a reason that we have accepted as true. • The reasons might be based off of definitions, mathematical properties, common knowledge we accept without proof (i.e., postulates), or previously proven theorems.

15. Informal Proofs • The proof process is as follows (p. 127): • List the given information, and draw a diagram to illustrate the set-up. • State the theorem or conjecture to be proven. • Create a deductive argument by forming a logical chain of statements linking the given to what you are trying to prove. • Justify each statement with a reason. (Can include given information, definitions, algebraic properties, postulates, and theorems.) • State what you have proven. • An informal proof (or paragraph proof) involves writing in paragraph form an explanation as to why a conjecture is true. • Let’s go over Example 3 and Guided Practice 3 on p. 127. • Now that we’ve proven this idea, we can call it a theorem. (Your book calls it the Midpoint Theorem, but I am less concerned with names than with understanding the concept behind them.)

16. Can you…? • Distinguish between inductive and deductive reasoning? • Use the Laws of Detachment and Syllogism to determine if deductive reasoning is valid? • Support deductive reasoning using postulates, theorems, definitions, arithmetic properties, etc? • Follow and understand a simple informal proof?

17. Homework 7 • Workbook, pp. 21, 24 • Book, p.122, #35-40, 42, 43, 45