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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 syllogism style 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 Syllogism tester A valid syllogism follows this structure. p q L1 __________________________. q r L2 ____________________________. p r Conclusion:._______________________ Is this a valid syllogism? L1 If W is late, then G will eat the cake. L2 If G eats the cake, then X will cry. Conclusion: If w is late, then x will cry. Is this a valid syllogism? yes p q L1 If W is late, then G will eat the cake. q r L2 If G eats the cake, then X will cry. p r Conclusion: If w is late, then x will cry. o o o 1. 45 – 45 - 90 right triangles have 2 congruent sides/angles. 2. Triangles with 2 or 3 congruent sides/angles are isosceles triangles. Can you restate these statements in the form of “if then” statements and put together a syllogism argument? Line 1: p Line 2: q Conclusion: p q r r 1. If a right triangle has angles of (45-45-90), then it will be a triangle with 2 congruent sides/angles. Syllogism Structure: Line 1: p Line 2: q Conclusion: p q r r 2. If a triangle has 2 or 3 congruent sides/angles, 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 infinitely long syllogism as long as follow the structure: Line 1: p q Line 2: q r Line 3: r s Line 4: s t Conclusion: p t Using the Law of Syllogism. If a valid conclusion is possible, write it. 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. Use the Law of Syllogism. If a valid conclusion is possible, write it. 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. Using the Law of Syllogism. If a valid conclusion is possible, write it. 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. 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 … Homework : construct 2 valid argument/proofs of your own to present to class. Ex #1 : Is this valid? (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. Homework : construct 2 valid argument/proofs of your own to present to class. ex #2 Is this valid? (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.