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2-3: Deductive Reasoning

2-3: Deductive Reasoning. Expectations: L3.1.1: Distinguish between inductive and deductive reasoning, identifying and providing examples of each.

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2-3: Deductive Reasoning

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  1. 2-3: Deductive Reasoning 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: Deductive Reasoning

  2. 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. A F B E C D 2-3: Deductive Reasoning

  3. Use a pattern to answer the question. How many diagonals does an octagon have? 2-3: Deductive Reasoning

  4. 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. 2-3: Deductive Reasoning

  5. 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: _________________. 2-3: Deductive Reasoning

  6. 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. 2-3: Deductive Reasoning

  7. Make a conclusion based on the following true statements. a. If it is raining, then it is humid. b. It is humid. 2-3: Deductive Reasoning

  8. If p => q and q are true _________________ can be made. This is referred to as affirming the consequent. 2-3: Deductive Reasoning

  9. 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. 2-3: Deductive Reasoning

  10. The ___________ is a sufficient condition for the conclusion and the conclusion is a _____________ condition of the hypothesis. 2-3: Deductive Reasoning

  11. 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. 2-3: Deductive Reasoning

  12. 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. 2-3: Deductive Reasoning

  13. Notice that, “We do not have school today” is not sufficient to guarantee that today is Sunday. 2-3: Deductive Reasoning

  14. “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. 2-3: Deductive Reasoning

  15. Notice simply saying AM ≅ MB does not guarantee that M is the midpoint of AB, so it is not a sufficient condition. 2-3: Deductive Reasoning

  16. “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. 2-3: Deductive Reasoning

  17. “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. 2-3: Deductive Reasoning

  18. “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. 2-3: Deductive Reasoning

  19. 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. 2-3: Deductive Reasoning

  20. 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. • __________________________________ • __________________________________ 2-3: Deductive Reasoning

  21. 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. 2-3: Deductive Reasoning

  22. 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. 2-3: Deductive Reasoning

  23. 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. 2-3: Deductive Reasoning

  24. Which of the following is a sufficient but NOT necessary condition for angles to be supplementary? • they are both acute angles. • they are adjacent • their measures add to 90. • they are coplanar. • they form a linear pair. 2-3: Deductive Reasoning

  25. 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. 2-3: Deductive Reasoning

  26. 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: __________________________ __________________________________. 2-3: Deductive Reasoning

  27. 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. 2-3: Deductive Reasoning

  28. Assignment • pages 89 - 91, • # 18 - 30(evens), 42, 44 and 46 2-3: Deductive Reasoning

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