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CAHSEE W. UP

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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

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

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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.

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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

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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.

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With a partner!

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

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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

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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.

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

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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.

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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

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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!

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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!

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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!

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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!

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We Do!

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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!

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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.

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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

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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…

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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.

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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.

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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.

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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.

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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.

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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.

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Notes on Logic Continued

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