Geometry Chapter 2 © 2004 Charlean Mullikin All rights reserved (mull243@bellsouth.net)

Conditionals A conditional is a statement that can be written in the If – Then form. If the team wins the semi-final, then they will play in the championship.

Conditionals The “If” part is called the Hypothesis. If the team wins the semi-final, then they will play in the championship. The “Then” part is called the Conclusion.

Hypothesis Conclusion Conditionals Hypothesis Conclusion p q When If Then All verbs Then Subject adjectives are... If

Practice Write each statement in the if-then form: 1. The lights go out when lightning strikes the power lines. 1. If lightning strikes the power lines, then the lights will go out . 2. All squirrels are mammals. 2. If an animal is a squirrel, then it is a mammal. 3.Cheerleaders can do stunts. 3. If you are a cheerleader, then you can do stunts. 4. Complementary angles have a sum of 90. 4. If two angles are complementary angles, then they have a sum of 90. 5. The product of two odd integers is odd. 5. If two integers are odd, then their product is odd.

If Hypothesis Then Conclusion Conditionals Conditional If Hypothesis Then Conclusion p q Converse If Then Hypothesis Conclusion Conclusion Hypothesis q p

Conditionals p q If the team wins the semi-final, then they will play in the championship. q p Converse If the team plays in the championship, then they won the semi-final. If they will play in the championship, then the team wins the semi-final.

Conditionals p q q p ~ p ~ q ~ q ~ p Conditional Converse Inverse If the team wins the semi-final, then they will play in the championship. p q Converse If the team plays in the championship, then they won the semi-final. q p Inverse If the team does not win the semi-final, then they will not play in the championship. ~ p ~ q Contrapositive If the team does not play in the championship, then they did not win the semi-final. ~ q ~ p

True False it could be a lion False it could be a lion True All tigers are cats. Conditional If an animal is a tiger, then it is a cat. True p q False Converse If an animal is a cat, then it is a tiger. it could be a lion q p False Inverse If an animal is not a tiger, then it is not a cat. it could be a lion ~ p ~ q Contrapositive If an animal is not a cat, then it is not a tiger. True ~ q ~ p

Biconditionals True True p q q p p  q Conditional If an angle is acute, then it has a measure less than 90. p q Converse If an angle has a measure less than 90, then it is an acute angle. True q p If both the conditional and its Converse are true, then it can Be written as a biconditional. Biconditional p  q “if and only if” An angle is acute if and only if It has a measure less than 90.

Practice http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-02-834817-6&chapter=1&lesson=4

Notice how the original Conditional has been Law of Detachment (1) p q Notice how the original Conditional has been Broken apart into two pieces. (Detached) (2) p (3) q

Law of Detachment p q If you pass the driving test, then you will get your license. p Brian passed his driving test. q Brian got his license.

(1) p q (2) q r (3) p r Law of Syllogism Notice how all three statements are conditionals with three basic ideas. The repeating part cancels out to give the conclusion. Law of Syllogism (1) p q (2) q r (3) p r

Law of Syllogism p q q r p r If you pass the driving test, then you will get your license. q r If you get your license, then you can drive to school. p r If you pass the driving test, then you can drive to school.

If Marita gets a speeding ticket, then she will have to pay a fine. Law of Syllogism p q If Marita oveys the speed limit, then she will not get a speeding ticket. Not q r If Marita gets a speeding ticket, then she will have to pay a fine. No Conclusion

If a quadrilateral is a rectangle, then the diagonals are congruent. Law of Syllogism p q If a quadrilateral is a rectangle, then the measure of each angle is 90. q r If the measure of each angle of a quadrilateral is 90, then the diagonals are congruent. p r If a quadrilateral is a rectangle, then the diagonals are congruent.

Law of Syllogism p q p r No Conclusion If I lose my textbook, then I will fail my test. p r If I lose my textbook, then my grades will go down. No Conclusion

If it is not a conditional and not a p statement, Then there is Working the Laws If it is not a conditional and not a p statement, Then there is NO CONCLUSION! Identify the p and q If no, is it a p? Then check for Detachment. If yes, check for syllogism. Is 2nd statement another conditional ?

p q p q r q p r q Working the Laws No conclusion Law of Detachment Law of Syllogism