Do Now Write the converse of each true statement. If true, combine the statements to write a true biconditional. If the converse is false, give a counterexample. a) If an angle measures 30 o, then it is acute. b) If two angles are congruent, then they have the same measure. Do Now Write the converse of each true statement. If true, combine the statements to write a true biconditional. If the converse is false, give a counterexample. a) If an angle measures 30 o, then it is acute. Converse: If an angle is acute, then its measure is 30. Counterexample: 15 angle is acute b) If two angles are congruent, then they have the same measure. Converse: If two angles have the same measure, then they are congruent. BICONDITIONAL: Two angles are congruent if and only if they have the same measure.
2.2 p82 #13-21, 24-35, 59-62 2.3 Deductive Reasoning Mastery Objective You will be able to: Distinguish between inductive and deductive reasoning Identify an argument as valid or invalid Recognize two types of valid argument: THE LAW OF DETACHMENT THE LAW OF SYLLOGISM Language Objective You will be able to: Read and discuss the validity of an argument
Deductive Reasoning vs Inductive Reasoning Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument. This differs from inductive reasoning, in which previous examples and patterns are used to form a conjecture. Example: Decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning. a) For the past three Wednesdays the cafeteria has served macaroni and cheese for lunch. Dana concludes that the cafeteria will serve macaroni and cheese for lunch this Wednesday. Inductive or Deductive? Why? b) If you live in Nevada and are between the ages of 16 and 18, then you must take driver s education to get your license. Marcus lives in Nevada, is 16 years old, and has his driver s license. Therefore, Marcus took driver s education. Inductive or Deductive? Why?
Arguments can be either VALID or INVALID. A valid argument is one in which the truth of the premises GUARANTEES the truth of the conclusion. In an invalid argument, the conclusion can be false EVEN THOUGH all of the premises are true. Two classic forms of VALID argument are: 1. THE LAW OF DETACHMENT 2. THE LAW OF SYLLOGISM
p Law of Detachment q is true, and p occurs or is true, then q is true. premise 1. premise 2. Conclusion: p q p Therefore, q Law of Syllogism If p q and q r, therefore p r premise 1. premise 2. Conclusion: p q q r Therefore, p r
Examples: State whether the argument is valid. a. Jamal knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Jamal misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday. b. If two angles form a linear pair, then they are supplementary; A and B are supplementary. So, A and B form a linear pair. Examples: Over the summer, Mike visited Alabama. Given the following true statements,can you conclude that Mike visited the Civil Rights Memorial? (1) If Mike visits Alabama, then he will spend a day in Montgomery. (2) If Mike spends a day in Montgomery, then he will visit the Civil Rights Memorial.
Homework: 2.3 p91 #8-20, 23-25, 30-35, 45-49
Concept check! Is this the same as the Law of Detachment? p q q Therefore, p. Examples: Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. (1) If you are a student, then you have lots of homework. (2) If you have lots of homework, then you have no social life. (3) If you are a student, then you have no social life. (1) If the lines are perpendicular, then they intersect to form a right angle (2) Line l is perpendicular to line m. (3) Lines l and m intersect to form a right angle.
Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. (1) If the quadrilateral is a square, then it has four right angles. (2) Quadrilateral ABCD has four right angles. (3) Quadrilateral ABCD is a square. (1) Vertical angles are congruent. ~ (2) A = B (3) A and B are vertical angles. Examples: Write a conclusion using the true statements. If no conclusion is possible, write no conclusion. If you get good grades, your parents will be happy. If your parents are happy, they will let you go snowboarding over Christmas break. You got good grades. If you sleep in, you will miss the bus. You missed the bus.
Examples: Write a conclusion using the true statements. If no conclusion is possible, write no conclusion. If you get good grades, your parents will be happy. If your parents are happy, they will let you go snowboarding over Christmas break. You got good grades. If you sleep in, you will miss the bus. You missed the bus. Examples: Write the premises in an order that will make a valid argument. Then make a conclusion from the argument. If I am tired, I won't do my homework. 3. 2. If I get home late, then I will be tired. If I don't do my homework, I will get a bad grade in Geometry. 4. If I go to the concert, then I will get home late. 1. I went to the concert! 5. Therefore, I will get a bad grade in Geometry. :(