Presentation on theme: "2.1.4 USING LOGICAL REASONING"— Presentation transcript:
12.1.4 USING LOGICAL REASONING Logical reasoning is based on conditionals.A CONDITIONAL is an if-then statementHYPOTHESIS: The part following the “if”CONCLUSION: The part following the “then”EX:If I live in DallasHYPOTHESISIf I live in Dallasthen I must be a Mavericks fanCONCLUSIONthen I must be a Mavericks fan
2CONDITIONALS CAN BE TRUE OR FALSE TRY WRITING THE FOLLOWING STATEMENTS AS CONDITIONALS (IF-THEN STATEMENTS), and identify the hypothesis and conclusion in each.1) Vertical angles are congruent.2) October has 31 days3) Two lines parallel to a third line are parallel to each other.
3PRACTICE WITH CONCLUSIONS What can you conclude?1) If <A is a right angle thenABC2)4123
4What can you conclude?3) a. If M is the midpoint of DE thenb. If XM is the perpendicular bisector of DE thenMEDX
5COUNTEREXAMPLE – A particular example or instance of the statement that is not true CONDITIONAL: If a month has thirty days, then it is SeptemberCOUNTEREXAMPLE: AprilThe month must have thirty days but could not be September.Try this one!CONDITIONAL: If you live in a state that begins with C, then you live in a state that does not border the oceanCOUNTEREXAMPLE: California
6CONVERSE – Interchanges the hypothesis and the conclusion CONDITIONAL: If it is noon, then it is time to eat lunch.CONVERSE: If it is time to eat lunch, then it is noon.
7CONVERSES CAN BE TRUE OR FALSE Write the converses of the following conditionals and determine the truth value of each1) If two lines are both vertical, then they are parallel2) If a number is not divisible by 10, then it is not divisible by 53) If the measure of an angle is between 0o and 90o, then the angle is acute
8BICONDITIONAL – When the conditional and converse are both true, you can combine them using “if and only if” (iff)CONDITIONAL: If a polygon is a quadrilateral, then it has four sides.CONVERSE: If a polygon has four sides, then it is a quadrilateralBICONDITIONAL: A POLYGON IS A QUADRILATERAL IFF IT HAS FOUR SIDES.
9ALL GEOMETRY DEFINITIONS CAN BE WRITTEN AS BICONDITIONALS
10Inverse-Negates the hypotheses and the conclusion of a conditional statement. Conditional: If you have a funny haircut, people will notice you.Inverse: If you do not have a funny haircut , people will not notice you.
11Conditional: If Monique finds a summer job, then she will buy a car. Contrapositive-interchanges and negates both the hypothesis and the conclusion of a conditional statementConditional: If Monique finds a summer job, then she will buy a car.Find the Converse, Inverse and the Contrapositive statements.Converse: If Monique buys a car, then she will find a summer job.Inverse: If Monique does not find a summer job, then she will not buy a car.Contrapositive: If Monique does not buy a car, then she will not find a summer job.
12PRACTICE 1) If <A is a right angle then A B C Write the conditional, converse, biconditional, inverse and contrapositive for this picture.
13PRACTICE X 3) If M is the midpoint of DE then D E M Write the conditional, converse, and biconditional, inverse and contrapositive for this picture.