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2-4 Deductive Reasoning Ms. Andrejko Real World Vocabulary Deductive reasoning- uses facts, rules, definitions, or properties to reach logical conclusions from given statements Unlike inductive reasoning which uses patterns/observations Law of Detachment- if p q is a true conditional, and p is true, then q is is also true Law of Syllogism- if p q and q r are true statements, then p r is a true statement Note: Stating a counterexample is not a valid method of proving a conjecture. Laws Detachment: If a car is out of gas, then it will not start. Sarah’s car is out of gas. Valid conclusion: Sarah’s car will not start. Syllogism: If you get a job, then you will earn money. If you earn money, then you will buy a car. Valid conclusion: If you get a job, then you will buy a car. Examples – Inductive or deductive? 1. A dental assistant notices a patient has never been on time for an appointment, he concludes the patient will be late for her next appointment. Inductive 2. If Eduardo decides to go to a concert tonight, he will miss the football game. Tonight Eduardo went to a concert. Eduardo missed football practice. Deductive Practice– Inductive or deductive? 1. A person must be insured in order to drive a car. Andy is driving a car. Andy is insured. Deductive 2. Whenever Julie attends a tutoring session she notices that her grades have improved. Julia attends a tutoring session and she concludes that her grades will improve. Inductive Examples – Valid or Invalid? Given: If the sum of the measures of two angles is 180, then the angles are supplementary. VALID – Law of detachment m∠A + m∠B is 180. Conclusion: ∠A and ∠B are supplementary. 1. 2. Given: If the sum of the measures of two angles is 90, then the angles are complementary. m∠ABC is 45 and m∠DEF is 48. Conclusion: ∠ABC and ∠DEF are complementary. INVALID: 45 + 48 = 93 , 93 ≠ 90. Practice– Valid or Invalid? Given: If a point is the midpoint of a segment, then it divides the segment into two congruent segments. R is the midpoint of QS VALID – Law of detachment Conclusion: QR QS 1. 2. Given: If 3 points are noncollinear, they determine a plane. Points A,B, and C, lie in plane G. Conclusion: Points A, B, and C are noncollinear. INVALID: points A, B, and C, could be collinear and form a line. Examples- Law of Syllogism 1. If two angles are complementary, then the sum of their measures is 90.If the sum of the measures of two angles is 90, then both of the angles are acute. If 2 angles are complementary, then both of the angles are acute. 2. If the heat wave continues, then air conditioning will be used more frequently. If air conditioning is used more frequently, then energy costs will be higher. If the heat wave continues, then energy costs will be higher. Examples- Law of Syllogism 3. If it is Tuesday, then Marla tutors chemistry. If Marla tutors chemistry, then she arrives home at 4 P.M. If it’s a Tuesday, then Marla arrives home at 4. 4. If two lines are perpendicular, then they intersect to form right angles. Lines r and s form right angles. NO VALID CONCLUSION 5. If two angles form a linear pair, then the two angles are supplementary. If two angles are supplementary, then the sum of their measures is 180. If 2 angles form a linear pair, then the sum of their measures is 180. FOLDABLE LOGIC TAB