2. CAHSEE W. UP

3. GEOMTRY GAME PLAN Date9/24/13 Tuesday Section / TopicNotes #19: 2.2 Definitions & Biconditional Statements Lesson GoalSTUDENTS WILL BE ABLE TO RECOGNIZE, ANALYZE AND USE BICONDITIONAL STATEMENTS. Geometry California Standard 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Homework #19 P.76 #18-28 all P. 82 #13-37 EOO

4. Lesson 2.2 Definitions and Biconditional Statements “The better part of one’s life consists of his friendships.” –Abraham Lincoln

5. REVIEW: Converse Switch the hypothesis & conclusion parts of a conditional statement. Ex: Write the converse of: “If you are a brunette, then you have brown hair.” If you have brown hair, then you are a brunette.

6. Negatize the hypothesis & conclusion parts of a conditional statement. Ex: Write the inverse of: “If you are a brunette, then you have brown hair.” If you are not a brunette, then you don’t have brown hair. Inverse

7. Contrapositive Switch the hypothesis & conclusion parts of a conditional statement and make it negative Ex: Write the contrpositive of: “If you are a brunette, then you have brown hair.” If you don’t have brown hair, then you are not a brunette.

8. With a partner!

10. Perpendicular lines Two lines that intersect to form a right angle (90 degrees). n m n m means “is perpendicular to”

11. Line perpendicular to a plane A line that intersects a plane in a point & is perpendicular to every line in the plane that intersects the line. Q l

12. Definitions A definition uses known words to describe new words. ◦ Like conditional statements, definitions can be written in “if/then” format ◦ But for definitions, the converse is always true. ◦ Ex. Perpendicular lines: ◦ (1) If two lines are perpendicular, then they intersect to form a right angle AND ◦ (2) If two lines intersect to form a right angle, then they are perpendicular.

13. Ex: Use definitions to justify your True or False answers. D E B ACAC

14. Biconditional statements Statement that are equivalent to writing a conditional statement AND its converse ◦ containing the phrase “if and only if” ◦ Conditional: If Rebecca sleeps all morning, then she is sick. ◦ Converse: If Rebecca is sick, she sleeps all morning. ◦ Biconditional: Rebecca sleeps all morning if and only if she is sick.

15. Biconditional statements are true sometimes and false sometimes In order for it to be true, the conditional statement and its converse must both be true ◦ Then we say the statement is true “forwards” and “backwards.” Hints

16. Rewrite the biconditional statement as a conditional statement and its converse. ◦ Two angles are congruent if and only if they have the same measure. ◦ Conditional: If two angles are congruent, then they have the same measure. ◦ Converse: If two angles have the same measure, then they are congruent. I Do!

17. Rewrite the biconditional statement as a conditional statement and its converse. ◦ A ray bisects an angle if and only if it divides the angle into two congruent angles. ◦ Conditional: If a ray bisects an angle, then it divides the angle into two congruent angles. ◦ Converse: If a ray divides an angle into two congruent angles, then the ray bisects the angle. We Do!

18. Rewrite the biconditional statement as a conditional statement and its converse.  Two lines intersect if and only if their intersection is exactly one point. ◦ Conditional: If two lines intersect, then they intersect in exactly one point. ◦ Converse: If two lines contain exactly one point, then the two lines intersect. You Do!

19. Consider the following statement: x 2 < 49 if and only if x<7. Step 1: Is this a biconditional statement?  Yes, it contains the phrase “if and only if”  Step 2: Are the conditional and converse true?  Conditional: If x 2 < 49, then x<7. True.  Converse: If x<7, then x 2 < 49. False. If x = –8, then (-8) 2 = 64 which is not less than 49. THE BICONDITIONAL IS FALSE I Do!

20. We Do!

21. Consider the following statement: y = -3 if and only if y 2 = 9. Conditional: If y = -3, then y 2 = 9. True. If y = -3, then (-3) 2 = 9. Converse: If y 2 = 9, then y = -3. False. 3 2 = 9, so y can also be positive 3, THE BICONDITIONAL IS FALSE You Do!

22. TOD ! #1 Is the biconditional statement true or false. y = -3 if and only if y 2 = 9. #2Is the biconditional statement true or false. An angle measures 94º if and only if it is obtuse.

23. TOD! #1 y = -3 if and only if y 2 = 9. Conditional: If y = -3, then y 2 = 9. True. If y = -3, then (-3) 2 = 9. Converse: If y 2 = 9, then y = -3. False. 3 2 = 9, so y can also be positive 3, BICONDITIONAL STATEMENT FALSE #2An angle measures 94º if and only if it is obtuse. Conditional: If an angle measures 94º, then it is obtuse. TRUE. Converse: If an angle is obtuse, then it measures 94º. FALSE. If an angle is obtuse, then it can measure any degree between 90º and 180º. BICONDITIONAL STATEMENT FALSE

24. Give a counterexample that demonstrates that the converse of the statement is false. If an angle measures 94°, then it is obtuse. If two angles measure 42° and 48°, then they are complementary. More examples…

25. Rewrite each of the following statements in “If- then” form as the conditional, converse, inverse, contrapositive, and biconditional. 1) Celebrities have many fans. Conditional: If you are a celebrity, then you have many fans. Converse: If you have many fans, then you are a celebrity.

26. Inverse: If you are not a celebrity, then you do not have many fans. Contrapositive: If you do not have many fans, then you are not a celebrity. Biconditional: You are a celebrity if and only if you have many fans.

27. 2) Penguins are birds that cannot fly. Conditional: If a bird is a penguin, then it cannot fly. Converse: If a bird cannot fly, then it is a penguin. Inverse: If a bird is not a penguin then it can fly. Contrapositive: If a bird can fly, then it is not a penguin. Bionditional: A bird is a penguin if and only if it cannot fly.

28. 3) Angles that form a linear pair are supplementary. Conditional: If two angles form a linear pair, then they are supplementary. Converse: If two angles are supplementary, then they form a linear pair. Inverse: If two angles do not form a linear pair, then they are not supplementary. Contrapositive: If two angles are not supplementary, then they do not form a linear pair. Biconditional: Two angles form a linear pair if and only if they are supplementary.

29. 4) Complementary angles are acute. Conditional: If two angles are complementary, then they are acute. Converse: If two angles are acute, then they are complementary. Inverse: If two angles are not complementary, then they are not acute. Contrapositive: If two angles are not acute, then they are not complementary. Biconditional: Two angles are complementary if and only if they are acute.

30. 5) I will go to school on Monday. Conditional: If I go to school, then it’s Monday. Converse: If it’s Monday, then I will go to school. Inverse: If I don’t go to school, then it is not Monday. Contrapositive: If it is not Monday, then I will not go to school. Biconditional: I will go to school if and only if it is Monday.