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OTCQ True or False 1. An Isosceles triangle has 2 congruent sides and no congruent angles. 2. An Equiangular triangle has three angles ◦ that measure 70 . 3. A right triangle may be shaped with ◦ ◦ ◦ angles of 1 , 89 and 90 . OTCQ 092409 True or False 1. An Isosceles triangle has 2 congruent sides and no congruent angles. False. An isosceles has 2 or 3 congruent sides and 2 or 3 congruent angles. 2. An Equiangular triangle has three angles that ◦ ◦ measure 70 . False. 3 angles of 70 would add 3. ◦ up to 210 . In every triangle, the 3 angles add up ◦ to exactly 180 . No more no less. ◦ A right triangle may be shaped with angles of 1 , ◦ ◦ ◦ ◦ ◦ ◦ 89 and 90 . True 1 + 89 + 90 = 180 Aims 2-1 and 2-2 How do we define the rules of logic? (part 2) Homework : construct 2 valid argument/proofs of your own to present to class. NY GG 24 & GG 25 Objectives: 1. SWBAT think/reason with logic. 2. SWBAT define terms in logic and apply logic rules in their arguments. Summary of part 1. Inductive Reasoning: from observed examples to conjectures. Counterexamples: one example that disproves an inductive conjecture. Deductive reasoning: from assumed truths to conjectures. Conditional statements: Basic: If (premise), then (conclusion): p Converse: q Inverse: ~p Contrapositive: ~q q p ~q ~p Consider Inductive Example: Observation: The sun was out all week. Conclusion: It is always sunny. Can you disprove with a counterexample? Constructing Logical Arguments. Law of Syllogism: A valid form of deductive reasoning is the Law of Syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Law of Syllogism If p q and q r are true statements, then p r is a true statement. Law of Syllogism If p q and q r are true statements, then p r is a true statement. Example: If I go to Miami, then I will be in Florida. p q If I am in Florida, then I will be south of NY. q r If I go to Miami, then I will be south of NY. p r Consider Deductive Example: Assumptions/Givens: 1. A right triangle may have 2 congruent sides/angles (45-45-90 right triangle). 2. An isosceles triangle has 2 or 3 congruent sides/angles. Can you restate these in the form of a syllogism? Steps to create a valid 2 hop proof argument with the law of Syllogism. 1. Order and translate all given/assumptions into p q, q r and p r form. 2. Write proof as follows: Line 1: p q Line 2: q r Conclusion: p r Step 1. Try to order from general to specific. Comes with practice. Try to keep translations short: 1. If a right triangle has angles of (45-45-90), then it will have 2 congruent sides. 2. If a triangle has 2 congruent sides, then it is an isosceles triangle. Conclusion: If a right triangle has angles of (45-45-90), then it is an isosceles triangle. Law of Syllogism If p q and q r are true statements, then p r is a true statement. In theory you can have an infinite hop proof as long as you do not skip a hop: Line 1: p q hop1 Line 2: q r hop2 Line 3: r s hop3 Line 4: s t hop4 Conclusion: p t One way to check your proof is to draw the red lines. If you cannot not connect to the bottom line before your conclusion, your proof is incomplete Line 1: p q hop1 Line 2: q r hop2 Line 3: r s hop3 Line 4: s t hop4 Conclusion: p t Using the Law of Syllogism. If a valid conclusion is possible, write it. 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. 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. Use the Law of Syllogism. If a valid conclusion is possible, write it. 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 two angles are complimentary, 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 … Explain which is valid. Homework : construct 2 valid argument/proofs of your own to present to class. Ex #1 (1) If it is Tuesday, then Marla tutors chemistry. (2) If Marla tutors chemistry, then she arrives home at 4 P.M. (3) If Marla arrives at home at 4 P.M., then it is Tuesday. Explain which is valid. Homework : construct 2 valid argument/proofs of your own to present to class. ex #2 (1) If a marine animal is a starfish, then it lives in the intertidal ocean zone. (2) If an animal is in the intertidal zone, then it lives in the least stable ocean zone. (3) If a marine animal is a starfish, then it lives in the least stable ocean zone.