1. Conditional Statements Mrs. Spitz Modifyied by Mrs. Ortiz-Smith Geometry

2. Objective Recognize and analyze a conditional statement

3. Conditional Statement A logical statement has 2 parts: hypothesis & conclusion Can be written in “if-then” form

4. Conditional Statement Hypothesis is the part after the word “If” Conclusion is the part after the word “then”

5. Ex: Underline the hypothesis & circle the conclusion. If you are a brunette, then you have brown hair. hypothesisconclusion

6. Ex: Rewrite the statement in “if-then” form Vertical angles are congruent. If there are 2 vertical angles, then they are congruent. If 2 angles are vertical, then they are congruent.

7. Ex: Rewrite the statement in “if-then” form An object weighs one ton if it weighs 2000 lbs. If an object weighs 2000 lbs., then it weighs one ton.

8. Counterexample Is used to show that a conditional statement is false.

9. Ex: Provide a counterexample. If x 2 =81, then x must equal 9. counterexample: x could be -9 because (-9) 2 =81, but x≠9.

10. Negation Writing the opposite of a statement. Ex: negate x=3 x≠3 Ex: negate t>5 t 5

11. R U A GEN!US? To Thales the primary question was not what do we know, but how do we know it. ~ Aristotle

12. Lesson 2-1 Conditional Statements 12 Symbolic Logic Symbols can be used to connect statements. Hypothesis is represented by “p” Conclusion is represented by “q” if p, then q

13. Lesson 2-1 Conditional Statements 13 if p, then q represents p  q Example: p: a number is prime q: a number has exactly two divisors If a number is prime, then it has exactly two divisors. p  q: Symbolic Logic

14. Lesson 2-1 Conditional Statements 14 represents the word not ~ Example: p: the angle is obtuse The angle is not obtuse ~p means that the angle could be acute, right, or straight. ~p: Symbolic Logic

15. Example: p: I am not happy ~p: I am happy ~p took the “not” out- it would have been a double negative (not not) Symbolic Logic

16. Converse Switch the hypothesis & conclusion. Ex: “If you are a brunette, then you have brown hair.” If you have brown hair, then you are a brunette. Forms of Conditional Statements

17. Lesson 2-1 Conditional Statements 17 Forms of Conditional Statements Converse: Switch the hypothesis and conclusion (q  p) p  q If two angles are vertical, then they are congruent. q  p If two angles are congruent, then they are vertical.

18. Inverse Negate the hypothesis & conclusion. Ex: “If you are a brunette, then you have brown hair.” If you are not a brunette, then you do not have brown hair. Forms of Conditional Statements

19. Lesson 2-1 Conditional Statements 19 Forms of Conditional Statements Inverse: State the opposite of both the hypothesis and conclusion. (~p  ~q) p  q : If two angles are vertical, then they are congruent. ~p  ~q: If two angles are not vertical, then they are not congruent.

20. Contrapositive Negate, then switch the hypothesis & conclusion. Ex: “If you are a brunette, then you have brown hair.” If you do not have brown hair, then you are not a brunette. Forms of Conditional Statements

21. Lesson 2-1 Conditional Statements 21 Forms of Conditional Statements Contrapositive: Switch the hypothesis and conclusion and state their opposites. (~q  ~p) p  q : If two angles are vertical, then they are congruent. ~q  ~p: If two angles are not congruent, then they are not vertical.

22. The original conditional statement & its contrapositive will always have the same meaning. The converse & inverse of a conditional statement will always have the same meaning.

24. Conditional Statements Biconditional When a conditional statement and its converse are both true, they may be combined using the phrase if and only if (abbreviated: iff)

25. Biconditional Example: Statement: If an angle is right, then it has a measure of 90 . Converse: If an angle measures 90 , then it is a right angle. Biconditional: An angle is right if and only if it measures 90 .