1. 3/15/20162-3: Deductive Reasoning1 Expectations: L3.1.1: Distinguish between inductive and deductive reasoning, identifying and providing examples of each. L3.1.3: Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics. Identify and give examples of each. L3.3.3: Explain the difference between a necessary and a sufficient condition within the statement of a theorem. Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied.

2. 3/15/20162-3: Deductive Reasoning2 Diagonals A segment is a diagonal of a polygon iff its endpoints are 2 non-consecutive vertices of a polygon. ex: AC, BE and DF are diagonals for polygon ABCDEF. B C D E F A

3. 3/15/20162-3: Deductive Reasoning3 Use a pattern to answer the question. How many diagonals does an octagon have?

4. 3/15/20162-3: Deductive Reasoning4 Patterns are not proof – they are conjecture. Remember, this is inductive reasoning which is not valid for making a proof. The following slides give us some properties of deductive reasoning which is valid for proving statements true.

5. 3/15/20162-3: Deductive Reasoning5 Law of Detachment If p => q is true and p is a true statement, then ___ must be true. ex: 1. If today is Monday, then tomorrow is Tuesday. 2. Today is Monday. Conclude: _________________.

6. 3/15/20162-3: Deductive Reasoning6 Make a conclusion based on the following true statements. a. If the air conditioner is on, then it is hot outside. b. The air conditioner is on.

7. 3/15/20162-3: Deductive Reasoning7 Make a conclusion based on the following true statements. a. If it is raining, then it is humid. b. It is humid.

8. 3/15/20162-3: Deductive Reasoning8 If p => q and q are true _________________ can be made. This is referred to as affirming the consequent.

9. 3/15/20162-3: Deductive Reasoning9 Necessary and Sufficient Conditions In the statement of a theorem in “if- then” form, we can talk about sufficient conditions for the truth of the statement and necessary conditions of the truth of the statement. This is really just another way of looking at the Law of Detachment and Affirming the Consequent.

10. 3/15/20162-3: Deductive Reasoning10 The ___________ is a sufficient condition for the conclusion and the conclusion is a _____________ condition of the hypothesis.

11. 3/15/20162-3: Deductive Reasoning11 Necessary Consider the statement p => q. We say q is a necessary condition for (or of) p. Ex: “If if is Sunday, then we do not have school.” A necessary condition of it being Sunday is that we do not have school.

12. 3/15/20162-3: Deductive Reasoning12 Sufficient Condition A sufficient condition is a condition that all by itself guarantees another statement must be true. Ex: If you legally drive a car, then you are at least 15 years old.” Driving legally guarantees that a person must be at least 15 years old.

13. 3/15/20162-3: Deductive Reasoning13 Notice that, “We do not have school today” is not sufficient to guarantee that today is Sunday.

14. 3/15/20162-3: Deductive Reasoning14 “If M is the midpoint of segment AB, then AM ≅ MB.” Given that M is the midpoint, it is necessary (true) that AM ≅ MB. This means that M being the midpoint is a ____________ condition for AM  MB.

15. 3/15/20162-3: Deductive Reasoning15 Notice simply saying AM ≅ MB does not guarantee that M is the midpoint of AB, so it is not a sufficient condition.

16. 3/15/20162-3: Deductive Reasoning16 “If a triangle is equilateral, then it is isosceles.” A triangle having 3 congruent sides (equilateral) guarantees that at least 2 sides are congruent, so a triangle being equilateral is sufficient to say it is isosceles.

17. 3/15/20162-3: Deductive Reasoning17 “If a person teaches mathematics, then they are good at algebra.” Because Trevor is a math teacher, can we conclude he is good at algebra. Justify your answer.

18. 3/15/20162-3: Deductive Reasoning18 “If a person teaches mathematics, then they are good at algebra.” Betty is 32 and is very good at algebra. Can we correctly conclude that she is a math teacher? Justify.

19. 3/15/20162-3: Deductive Reasoning19 Bi-Conditional Statements If a statement and its converse are both true it is called a bi-conditional statement and can be written in ________________ form.

20. 3/15/20162-3: Deductive Reasoning20 Ex: “If an angle is a right angle, then its measure is exactly 90°” and “If the measure of an angle is exactly 90°, then it is a right angle” are true converses of each other so they can be combined into a single statement. ____________________________________________________________________

21. 3/15/20162-3: Deductive Reasoning21 Necessary and Sufficient If a statement is a bi-conditional statement then either part is a necessary and sufficient condition for the entire statement. Remember all definitions are bi-conditional statements.

22. 3/15/20162-3: Deductive Reasoning22 A triangle is a right triangle iff it has a right angle. Being a right triangle is necessary and sufficient for a triangle to have a right angle and possessing a right angle is necessary and sufficient for a triangle to be a right triangle.

23. 3/15/20162-3: Deductive Reasoning23 Necessary, Sufficient, Both or Neither Given the true statement: “If a quadrilateral is a rhombus, then its diagonals are perpendicular.” Is the following statement necessary, sufficient, both or neither? The diagonals of ABCD are perpendicular.

24. Which of the following is a sufficient but NOT necessary condition for angles to be supplementary? A. they are both acute angles. B. they are adjacent C. their measures add to 90. D. they are coplanar. E. they form a linear pair. 3/15/20162-3: Deductive Reasoning24

25. 3/15/20162-3: Deductive Reasoning25 Necessary, Sufficient, Both or Neither Given the true statement: “A quadrilateral is a rhombus if and only if its 4 sides are congruent.” Is the following statement necessary, sufficient, both or neither? The sides of ABCD are all congruent.

26. 3/15/20162-3: Deductive Reasoning26 Law of Syllogism: Transitive Law of Logic (A Form of Logical Argument) If p => q and q =>r, then __________. ex: 1. If a polygon is a square, then it is a rhombus. 2. If a polygon is a rhombus, then it is a parallelogram. Conclude: __________________________ __________________________________.

27. 3/15/20162-3: Deductive Reasoning27 Make a conclusion based on the following. a. If a quadrilateral is a square, then it has 4 right angles. b. If a quadrilateral has 4 right angles, then it is a rectangle.

28. 3/15/20162-3: Deductive Reasoning28 Assignment pages 89 - 91, # 18 - 30(evens), 42, 44 and 46