Lesson 2.2 Analyze Conditional Statements

1. Lesson 2.2 Analyze Conditional Statements Objective: The learner will write definitions as conditional statements.

2. Warm Up Determine if each statement is true or false. 1.The measure of an obtuse angle is less than 90°. 2. All perfect-square numbers are positive. 3. Every prime number is odd. 4. Any three points are coplanar. F T F T

3. Vocabulary • Conditional Statement: a logical statement that has a hypothesis and a conclusion. • If-then form: format for a conditional statement. • Hypothesis: the “if” part • Conclusion: the “then” part • Hypothesis must always be true.

6. Example 1: p q • If it is raining, then there are clouds in the sky. • If the car is a Mustang, then it is a Ford. q p

7. Writing Conditional Statements • All birds have feathers. • Two angles are supplementary if they are a linear pair. If an animal is a bird, then it has feathers. If two angles are a linear pair, then they are supplementary.

8. More Examples: Rewrite as conditional statements • All 90° angles are right angles. • When n = 9, n² = 81. • Tourists at the Alamo are in Texas. If an angle is 90, then it is a right angle If n = 9, then n^2 = 81. If tourists are at the Alamo, then they are in Texas.

9. Writing a Conditional Statement Write a conditional statement from the following. If an animal is a blue jay, then it is a bird. The inner oval represents the hypothesis, and the outer oval represents the conclusion.

10. Negation: the opposite of the original statement. • The ball is red. • Negation: • The cat is not black. • Negation: p The ball is not red. ~p p ~p The cat is black. p  ~p

12. Converse: flip-flop the hypothesis and conclusion. • If it is raining, then I will carry an umbrella. • Converse: • If I am in Nutley, then I’m in New Jersey. • Converse: pq qp If I carry an umbrella, then it is raining. pq qp If I’m in New Jersey, then I am in Nutley. ** The converse is not always true **

14. Inverse: Negate both the hypothesis and conclusion • If it is a Corvette, then it is a Chevy. • Inverse: • If you are a soccer player, then you are an athlete. • Inverse: pq ~p~ q If it is not a Corvette, then it is not a Chevy. pq If you are not a soccer player, then you are not an athlete. ~p~q ** The inverse is not always true **

16. Contrapositive: negate and flip-flop the hypothesis and conclusion. T Converse: Inverse: Contrapositive: F Angle A could be ANY degree b/w 90-180 F If angle A is not 99 and it’s 100, then it IS obstuse T Are these statements true?

17. Write the conditional, converse, inverse, and contrapositive for the conditional statement: All whales are mammals • Conditional: If an animal is a whale, then it is a mammal. • Converse: If it is a mammal, then it is a whale. • Inverse: If it is not a whale, then it is not a mammal. • Contrapositive: If it is not a mammal, then it is not a whale.

18. Verifying Statements • You must show the conclusion is true every time the hypothesis is true. • It only takes one counterexample to show it’s false.

19. Use “Guitar players are musicians.” to write the following. • “If-then” • Converse • Inverse • Contrapositive Which statements are true? Give a counterexample if it is false.

20. Equivalent Statements: when two statements are both true or false. • Conditional Statements and the contrapositive are either both true or false. • Converse and inverse are either both true or false.

21. Biconditional Statement: When a “If-Then” and its converse are true you can write them as a single statement. • Example: • Perpendicular lines: If the angle measure is 90◦, then then it is a right angle. • Converse: If the angle is a right angle, then the its measure is 90◦. • Biconditional: An angle is a right angle if and only if the its measure is 90◦

22. If Mary is in theater class, she will be in the fall play. If Mary is in the fall play, she must be taking theater class. • Biconditional: Mary is in the fall play, if and only if she is taking a theater class.

23. HW • # 7-15, 19-21, 25, 33, 35