2.1 conditionals, 2.2 Biconditionals, 5.4 inverse and contrapositive

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Presentation transcript:

2.1 conditionals, 2.2 Biconditionals, 5.4 inverse and contrapositive Reasoning and Proofs 2.1 conditionals, 2.2 Biconditionals, 5.4 inverse and contrapositive

Agenda Warm-up Homework check 2.1 conditionals, 2.2 biconditionals, Practice Homework

Warm-Up 2/7/2013 1. 2. 3. 4. 5. 6.

2-1, 2-2 Reasoning and Proof Negate – to add or remove the word “Not” Ex. Negate: “<A is a right <.” Becomes: <A is not a right <. Ex. Negate: “I am not cold.” Becomes: I am cold.

Conditional Statement: an if-then statement. (p  q) All conditional statements have 2 parts. A hypothesis A conclusion.

Hypothesis: (p) Follows “IF” Conclusion: (q) Follows “THEN” Ex: If today is Valentines Day, then the month is February. **When stating these do NOT include the words “if” or “then” Hypothesis: Today is Valentines Day. Conclusion: The month is February.

EX: Three points are collinear, if they are all in a line EX: Three points are collinear, if they are all in a line. Hypothesis: Three points are all in a line. Conclusion: The three points are collinear.

Write the statement as a conditional: Trees have roots. NC State fans are awesome.

Converse: SWITCHES the hypothesis and the conclusion EX: Conditional: If two lines intersect to form right angles, then they are perpendicular Write the converse: If two lines are perpendicular then the two lines intersect to form right angles.

Biconditional: p if and only if q (p iff Biconditional: p if and only if q (p iff. q)  occurs when a statement and its converse are true.

Counterexample: an example that shows the hypothesis can be true while the conclusion is false, thereby proving the statement is false. Ex. Conditional: If two angles are adjacent then they are a linear pair. Counterexample: <ABC and <CBD are Adjacent but not a linear Pair.

Examples Ex1. An angle that measures 90 ̊ is a right <. Write As a conditional: If an angle measures 90 ̊, then it is a right <. Write the Converse: If an angle is a right <, then it measures 90 ̊

Ex. If possible, write the biconditional of each statement Ex. If possible, write the biconditional of each statement. If not explain. 2. If an angle is a straight angle, then it measures 180 ̊ Converse: If an angle measures 180 ̊, then it is a straight angle. (True) Biconditional: An angle is a straight < if and only if it measures 180 ̊.

3. If 2 angles are a linear pair, then their sum is 180 ̊ 3. If 2 angles are a linear pair, then their sum is 180 ̊. Converse: If 2 angles’ sum is 180, then they are a linear pair. –False Counterexample: No biconditional because the converse is false!

Most good definitions can be written as biconditionals Most good definitions can be written as biconditionals. **You can determine a definition is not good by finding a counterexample.

Classwork Complete the converse and biconditional worksheet.

Homework Worksheet Scrapbook Project Rough Draft due Friday (tomorrow!) Scrapbook Project Due NEXT Friday Quiz Monday 2/11 Test Thursday 2/14