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Warm Up Write a conditional statement from each of the following. 1. The intersection of two lines is a point. 2. An odd number is one more than a multiple of Write the converse of the conditional “If Pedro lives in Chicago, then he lives in Illinois.” Find its truth value. If two lines intersect, then they intersect in a point. If a number is odd, then it is one more than a multiple of 2. If Pedro lives in Illinois, then he lives in Chicago; False.

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13. H: Animal is tabbyC: it is a cat 14. H: two lines intersectC: four angles are formed 15. H: 8 oz of cereal cost $2.99C: 16 oz of cereal cost $ If a patient is ill, then you should monitor the patient’s heart rate. 17. If the batter makes 3 strikes, then the batter is out. 18. If segments are congruent, then they have equal measures. 19. T 20. F, 2 planes intersect at 1 LINE 21. T 22. Conv: In an event is unlikely to occur, then the probability of the event is 0.1. F Inv: If the probability of an event is not 0.1, then the event is likely to occur. F Contr: If an event is likely to occur, then the probability of an event is not 0.1. T 23. Conv: If the air temp is 32°F or less, then freezing rain is falling. F Inv: If freezing rain is not falling, then the air temp is greater than 32°F. F Cont: If the air temp is greater than 32°F, then freezing rain is not falling. T 24. T 25. T 26. T 27. F 28. T 29. F

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34. If an animal is a dolphin, then it is a mammal. 35. If a person is a Texan, then the person is an American. 36. If x < -4, then x < X = If a mineral is calcite, then it has a hardness of 3. T 44. If a mineral is not apatite, then it has a hardness less than 5. F 46. If a mineral has a hardness of 3, then it is not apatite. T

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When you combine a conditional statement and its converse, you create a biconditional statement. A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.” p q means p q and q p The biconditional “p if and only if q” can also be written as “p iff q” or p q. Writing Math

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Example 1a Let x and y represent the following. x: An angle is acute. y: An angle has a measure that is greater than 0 and less than 90. An angle is acute iff its measure is greater than 0° and less than 90°. Write the conditional statement and converse within the biconditional. Conditional: If an angle is acute, then its measure is greater than 0° and less than 90°. The two parts of the biconditional x y are x y and y x. Converse: If an angle’s measure is greater than 0° and less than 90°, then the angle is acute.

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Example 1b Cho is a member if and only if he has paid the $5 dues. Write the conditional statement and converse within the biconditional. Conditional: If Cho is a member, then he has paid the $5 dues. Let x and y represent the following. x: Cho is a member. y: Cho has paid his $5 dues. The two parts of the biconditional x y are x y and y x. Converse: If Cho has paid the $5 dues, then he is a member.

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Example 2a If the date is July 4th, then it is Independence Day. For the conditional, write the converse and a biconditional statement. Converse: If it is Independence Day, then the date is July 4th. Biconditional: It is July 4th if and only if it is Independence Day.

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Example 3a An angle is a right angle iff its measure is 90°. Determine if the biconditional is true. If false, give a counterexample. Conditional: If an angle is a right angle, then its measure is 90°. The conditional is true. Converse: If the measure of an angle is 90°, then it is a right angle. The converse is true. Since the conditional and its converse are true, the biconditional is true. For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false.

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Example 3b y = –5 y 2 = 25 Determine if the biconditional is true. If false, give a counterexample. Conditional: If y = –5, then y 2 = 25. The conditional is true. Converse: If y 2 = 25, then y = –5. The converse is false. The converse is false when y = 5. Thus, the biconditional is false.

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In geometry, biconditional statements are used to write definitions. A definition is a statement that describes a mathematical object and can be written as a true biconditional. In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments.

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Think of definitions as being reversible. Postulates, however are not necessarily true when reversed. Helpful Hint Example 4 4a. A quadrilateral is a four-sided polygon. 4b. The measure of a straight angle is 180°. Write each definition as a biconditional. A figure is a quadrilateral if and only if it is a 4-sided polygon. An is a straight if and only if its measure is 180°.

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