OTCQ True or False 1.An Equiangular triangle is always equilateral. 2.An Equiangular triangle is never an isoceles triangle. 3.A scalene triangle can be.

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OTCQ True or False 1.An Equiangular triangle is always equilateral. 2.An Equiangular triangle is never an isoceles triangle. 3.A scalene triangle can be a right triangle.

OTCQ True or False 1.An Equiangular triangle is always equilateral. TRUE 2.An Equiangular triangle is never an isoceles triangle. FALSE Isosceles means at least 2 sides/angles congruent. 1.A scalene triangle can be a right triangle. True. Scalene means no sides angles congruent. A 30 ◦ - 60 ◦ – 90 ◦ right triangle would be scalene.

Aims 2-1 and 2-2 How do we define the rules of logic? (part 1) NY GG 24 & GG 25 Homework: Write 2 “if then” statements and label/form the converse, inverse and contrapositive of each.

Objectives: 1. SWBAT think/reason with logic. 2. SWBAT define terms in logic and apply logic rules in their arguments.

Definitions: Inductive reasoning: the method of thinking/reasoning from a series of observed examples to a conclusion/conjecture. Example: Observation: Ali hits 10 free throws in a row. Conclusion/conjecture: Ali never misses a free throw.

Definitions continued: Counterexample: an observation that proves that an inductive conclusion/conjecture is false. Example: Observation: Ali misses next shot Conclusion/conjecture: Ali never misses a free throw. PROVED FALSE!

Definitions: Deductive reasoning: the method of thinking/reasoning from assumptions, definitions and general statements accepted by all as true to a conditional statement. Example: Assumption#1:Barney is a dog. Assumption #2All dogs bark. Conditional Statement:If Barney is a dog, then Barney barks.

Conditional Statement: A statement that matches the form “If (definition/assumption/premise), then (conjecture/conclusion).” Abbreviated: p q Or saidp then q Ex: If Barney is a dog, then Barney barks. p q

Converse of a Conditional Statement: A conditional statement may have its p and q switch places. Original p q Converse: q p Or saidq then p original: If Barney is a dog, then Barney barks. Converse:If Barney barks, then Barney is a dog. q p

Inverse of a Conditional Statement: A conditional statement may have its p and q negated with the word “not”, or the words “it is not the case that.” Original p q Inverse: ~ p ~ q Or saidnot p then not q Original : If Barney is a dog, then Barney barks. Inverse 2 forms: It is not the case that if Barney is a dog, then Barney barks. If Barney is not a dog, then Barney does not bark. ~ p ~ q

Contrapositive of a Conditional Statement: A conditional statement may have its p and q negated and switch places. Original pq Contrapositive: ~ q ~ p Or said ~ q then ~ p Original: If Barney is a dog, then Barney barks. Contrapositive: If Barney does not bark, then Barney is not a dog. ~ q ~ p

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 q Converse:q p Inverse: ~ p ~q Contrapositive: ~ q ~ p

Consider Inductive Example: Observation:It rained all week. Conclusion:It always rains. Can you disprove with a counterexample?

Sometimes you must rewrite a conditional statement to put it in “if- then” form Ex: “All Zebras have stripes.” May be rewritten as: “If an animal is a Zebra then it has stripes.”

You try: rewrite the below conditional statement to put it in “if-then” form “Vertical angles are congruent”

You try: rewrite a conditional statement to put it in “if-then” form “Vertical angles are congruent” Is rewritten as “If two angles are vertical then they are congruent.”

Regents Prep

Regents

Regents: The original statement and the contrapositive are logically equivalent.

Regents. Time permitting start homework. Hwk read 2-1 and 2-2. Problems on page 47. Do not rewrite questions.