2-2 Statements, Conditionals, and Biconditionals

Use the following statements to write a 3. compound statement for each conjunction or SOLUTION: disjunction. Then find its truth value. Explain your reasoning. is a disjunction. A disjunction is true if at least p: A week has seven days. one of the statements is true. q is false, since there q: There are 20 hours in a day. are 24 hours in a day, not 20 hours. r is true since r: There are 60 minutes in an hour. there are 60 minutes in an hour. Thus, is true, 1. p and r because r is true.

SOLUTION: ANSWER: p and r is a conjunction. A conjunction is true only There are 20 hours in a day, or there are 60 minutes when both statements that form it are true. p is true in an hour. since a week has seven day. r is true since there are is true, because r is true. 60 minutes in an hour. Then p and r is true, 4. because both p and r are true. SOLUTION: ANSWER: ~p is a negation of statement p, or the opposite of A week has seven days, and there are 60 minutes in statement p. The or in ~p or q indicates a an hour. p and r is true, because p is true and r is disjunction. A disjunction is true if at least one of the true. statements is true. ~p would be : A week does not have seven days, 2. which is false. q is false since there are 24 hours in SOLUTION: a day, not 20 hours in a day. Then ~p or q is false, is a conjunction. A conjunction is true only because both ~p and q are false. when both statements that form it are true. p is true ANSWER: since a week has seven days. q is false since there A week does not have seven days, or there are 20 are 24 hours in a day, not 20 hours in a day. Thus, hours in a day. ~p or q is false, because ~p is false is false, because q is false. and q is false.

ANSWER: 5. A week has seven days, and there are 20 hours in a day. is false, because q is false. SOLUTION: is a disjunction. A disjunction is true if at least one of the statements is true. p is true since a week has seven days. r is true, since there are 60 minutes in an hour.Thus, is true, because p is true and r is true.

ANSWER: A week has seven days, or there are 60 minutes in an hour. is true, because p is true and r is true.

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6. 9. The measure of an acute angle is between 0 and 90.

SOLUTION: SOLUTION: ~p and ~r is the conjunction of the negations of p To write these statements in if-then form, identify and r. A conjunction is true if both statements are the hypothesis and conclusion. The word if is not true. part of the hypothesis. The word then is not part of ~p is : A week does not have seven days, which is the conclusion. false. ~r is : There are not 60 minutes in an hour, If the angle is acute, then its measure is between 0 which is false. Then is false, because and 90. both ~p and ~r are false. ANSWER: ANSWER: If the angle is acute, then its measure is between 0 A week does not have seven days, and there are not and 90. 60 minutes in an hour. is false, because 10. Equilateral triangles are equiangular. ~p is false and ~r is false. SOLUTION: Write each statement in if-then form. To write these statements in if-then form, identify 7. Sixteen-year-olds are eligible to drive. the hypothesis and conclusion. The word if is not SOLUTION: part of the hypothesis. The word then is not part of To write these statements in if-then form, identify the conclusion. the hypothesis and conclusion. The word if is not If a triangle is equilateral, then it is equiangular. part of the hypothesis. ANSWER: The word then is not part of the conclusion. If a triangle is equilateral, then it is equiangular. If you are sixteen years old, then you are eligible to Determine the truth value of each conditional drive. statement. If true, explain your reasoning. If false, give a counterexample. ANSWER: If you are sixteen years old, then you are eligible to 11. If then x = 4. drive. SOLUTION: If (–4)2 = 16. The hypothesis of the 8. Cheese contains calcium. conditional is true, but the conclusion is false. This SOLUTION: counterexample shows that the conditional To write these statements in if-then form, identify statement is false. the hypothesis and conclusion. The word if is not ANSWER: part of the hypothesis. The word then is not part of False; if x = –4, (–4)2 =16. The hypothesis of the the conclusion. conditional is true, but the conclusion is false. This If it is cheese, then it contains calcium. counterexample shows that the conditional ANSWER: statement is false If it is cheese, then it contains calcium.

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12. If you live in Atlanta, then you live in Georgia. 15. If the measure of a right angle is 95, then bees are lizards. SOLUTION: The conditional is false. You could live in Atlanta, SOLUTION: Georgia or Atlanta, Kansas. The conditional statement "If the measure of a right angle is 95, then bees are lizards." is true. The ANSWER: hypothesis is false, since the measure of a right False; Atlanta, Kansas; The hypothesis of the angle is 90. A conditional with a false hypothesis is conditional is true, but the conclusion is false. This always true, so this conditional statement is true. counterexample shows that the conditional statement is false. ANSWER: True; the hypothesis is false, since the measure of a 13. If tomorrow is Friday, then today is Thursday. right angle is 90. A conditional with a false SOLUTION: hypothesis is always true, so this conditional The conditional statement "If tomorrow is Friday, statement is true. then today is Thursday." is true. When this 16. If pigs can fly, then 2 + 5 = 7. hypothesis is true, the conclusion is also true, since Friday is the day that follows Thursday. So, the SOLUTION: conditional statement is true. The conditional statement "If pigs can fly, then 2 + 5 = 7" is true. The hypothesis is false, since pigs ANSWER: cannot fly. A conditional with a false hypothesis is True; when this hypothesis is true, the conclusion is always true, so this conditional statement is true. also true, since Friday is the day that follows Thursday. So, the conditional statement is true. ANSWER: True; the hypothesis is false, since pigs cannot fly. A 14. If an animal is spotted, then it is a Dalmatian. conditional with a false hypothesis is always true, so SOLUTION: this conditional statement is true. False 17. If Jacqueline turned 14 years old last year, then she The animal could be a leopard. The hypothesis of will turn 15 this year. the conditional is true, but the conclusion is false. This counterexample shows that the conditional SOLUTION: statement is false. Hypothesis: Jacqueline turned 14 years old last year Conclusion: She will turn 15 this year ANSWER: False; the animal could be a leopard. The hypothesis 14 + 1 = 15, so if the hypothesis is true, then the conclusion is true, so the conditional is true. of the conditional is true, but the conclusion is false. This counterexample shows that the conditional ANSWER: statement is false. True; the number 15 is one more than 14.

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18. If a number is between 10 and 12, then it is 11. 20. All whole numbers are integers.

SOLUTION: SOLUTION: Consider all numbers between 10 and 12, 11 is If a number is a whole number, then it is an integer. between them, but so are 10.5 and 11.5 and many others. The converse is formed by exchanging the ANSWER: hypothesis and conclusion of the conditional. False; 11.5 is between 10 and 12. Converse: If a number is an integer, then it is a whole number. False; Sample answer: –3. JUSTIFY ARGUMENTS Write the converse, inverse, and contrapositive of each true The inverse is formed by negating both the conditional statement. hypothesis and conclusion of the conditional. Determine whether each related conditional is Inverse: If a number is not a whole number, then it true or false. If a statement is false, find a is not an integer. False: Sample answer: –3. counterexample. 19. If a number is divisible by 4, then it is divisible by 2. The contrapositive is formed by negating both the SOLUTION: hypothesis and the conclusion of the converse of the conditional. The converse is formed by exchanging the Contrapositive: If a number is not an integer, then it hypothesis and conclusion of the conditional. is not a whole number; true. Converse: If a number is divisible by 2, then it is divisible by 4; false. Sample answer: 6 is divisible by ANSWER: 2 but is not divisible by 4. If a number is a whole number, then it is an integer. Converse: If a number is an integer, then it is a The inverse is formed by negating both the whole number. False; Sample answer: –3. Inverse: hypothesis and conclusion of the conditional. If a number is not a whole number, then it is not an Inverse: If a number is not divisible by 4, then it is integer. False: Sample answer: –3. Contrapositive: If not divisible by 2; false. Sample answer: 6 is not a number is not an integer, then it is not a whole divisible by 4 but is divisible by 2. number; true.

The contrapositive is formed by negating both the Rewrite each statement as a biconditional hypothesis and the conclusion of the converse of the statement. Then determine whether the conditional. biconditional is true or false. Contrapositive: If a number is not divisible by 2, then 21. There is no school on Saturday. it is not divisible by 4; true. SOLUTION: ANSWER: Write a biconditional with "if and only if".

Converse: If a number is divisible by 2, then it is There is no school if and only if it is Saturday. divisible by 4; false. Sample answer: 6 is divisible by 2 but is not divisible by 4. Inverse: If a number is not This statement is false, because there is no school divisible by 4, then it is not divisible by 2; false. on Sundays and holidays. Sample answer: 6 is not divisible by 4 but is divisible by 2. Contrapositive: If a number is not divisible by ANSWER: 2, then it is not divisible by 4; true. There is no school if and only if it is Saturday; false.

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22. An integer is a rational number. 25. A dodecagon is a polygon with 12 sides.

SOLUTION: SOLUTION: Write the statement as a biconditional using if and Write a biconditional using if and only if. only if. A polygon is a dodecagon if and only if it has 12 A number is an integer if and only if it is a rational sides; true this is the definition of a dodecagon. number; false because 0.5 is not an integer, but it is a rational number. ANSWER: A polygon is a dodecagon if and only if it has 12 ANSWER: sides; true. A number is an integer if and only if it is a rational number; false. 26. Two angles whose measure add to are complementary. 23. An angle with a measure between and is acute. SOLUTION: The parts of the statement are "two angles whose SOLUTION: measures add to " and "two angles are Make a biconditional with and "if and only if" complementary" statement A biconditional is of the form if and only if. An angle is acute if and only if it has a measure between 0 and 90 degrees; true, because the Two angles have measures that add to if and definition of acute is an angle between 0 and 90 only if the angles are complementary. degrees. This is true, because it is the definition of ANSWER: complementary angles. An angle is acute if and only if it has a measure between and ; true. ANSWER: 24. An obtuse triangle has one obtuse angle. Two angles have measures that add to if and SOLUTION: only if the angles are complementary; true. Write a biconditional using if and only if. 27. A number x such that is positive.

A triangle is obtuse if and only if it has one obtuse SOLUTION: angle; true as this is the definition of an obtuse A number x is greater than -0.5 if and only if x is triangle. positive; false, because x could be 0.

ANSWER: ANSWER: A triangle is obtuse if and only if it has one obtuse A number x is greater than -0.5 if and only if x is angle; true. positive; false.

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Use the following statements and figure to 30. write a compound statement for each SOLUTION: conjunction or disjunction. Then find its truth value. Explain your reasoning. Negate p, then find the disjunction . A disjunction is true if at least one of the statements is p: is the angle bisector of . true. q: Points C, D, and B are collinear. r is true since . The negation of p is is not the angle bisector of , which is false. Thus, r or ~p is true because r is true.

ANSWER: is not the angle bisector of . r or ~p is true because r is true. 28. p and r 31. r and q SOLUTION: p and r is a conjunction. A conjunction is true only SOLUTION: when both statements that form it are true. p is r and q is a conjunction. A conjunction is true only is the angle bisector of , which is true. r is when both statements that form it are true. r is true , which is true. Thus, p and r is true since . q is false, since points C, D, and because p is true and r is true. B are not collinear. Thus r and q is false because q is false. ANSWER: ANSWER: is the angle bisector of and . p and r is true because p is true and r is true. and Points C, D, and B are collinear. r and q is false because q is false. 29. q or p 32. SOLUTION: SOLUTION: q or p is a disjunction. A disjunction is true if at least Negate both p and r and find the disjunction. A one of the statements is true. q is false since points disjunction is true if at least one of the statements is C, D, and B are not collinear. p is true since is true. the angle bisector of . Thus, q or p is true because p is true. ~p is is not the angle bisector of , which ANSWER: is false. ~r is is not congruent to , which is Points C, D, and B are collinear, or is the angle false. Thus, ~p or ~r is false because ~p is false bisector of . q or p is true because p is true. and ~r is false. ANSWER: is not the angle bisector of , or . ~p or ~r is false because ~p is false and ~r is false.

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33. REASONING Use the following statements to write a compound statement for each SOLUTION: conjunction or disjunction. Then find its truth Negate both p and r and find the conjunction. A value. Explain your reasoning. conjunction is true only when both statements that p: Springfield is the capital of Texas. form it are true. q: Illinois borders the Atlantic Ocean. r: Illinois shares a border with Kentucky. ~p is is not the angle bisector of , s: Illinois is to the west of Missouri. which is false. ~r is is not congruent to , which is false. Thus, ~p and ~r is false because ~p is false and ~r is false.

ANSWER: is not the angle bisector of , and . ~p and ~r is false because ~p is false and ~r is false 34.

SOLUTION: is a conjunction. A conjunction is true only when both statements that form it are true. p is Springfield is the capital of Illinois, which is true. r is Illinois shares a border with Kentucky, which is true. Then is true, because p is true and r is true.

ANSWER: Springfield is the capital of Illinois, and Illinois shares a border with Kentucky. is true because p is true and r is true.

35.

SOLUTION: is a conjunction. A conjunction is true only when both statements that form it are true. p is Springfield is the capital of Illinois, which is true. q is Illinois borders the Atlantic Ocean, which is false. Then. is false because q is false.

ANSWER: Springfield is the capital of Illinois, but Illinois does not border the Atlantic Ocean. is false because q is false.

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36. 38.

SOLUTION: SOLUTION: Negate r and find the disjunction . A Negate both p and r and find the conjunction disjunction is true if at least one of the statements is . A conjunction is true only when both true. statements that form it are true. ~r is Illinois does not share a border with ~p is Springfield is not the capital of Illinois, which Kentucky, which is false. s is Illinois is west of is false. ~r is Illinois does not share a border with Missouri, which is false. Then is false, Kentucky, which is false. Then is false because ~r is false and s is false. because ~p is false and ~r is false.

ANSWER: Illinois does not share a border with Kentucky, or ANSWER: Illinois is west of Missouri is false because Springfield is not the capital of Illinois, and Illinois ~r is false and s is false. does not share a border with Kentucky. is false because ~p is false and ~r is false.

37. 39.

SOLUTION: SOLUTION: is a disjunction. A disjunction is true if at least Negate both s and p and find the disjunction one of the statements is true. . A disjunction is true if at least one of the r is Illinois shares a border with Kentucky, which is statements is true. true. q is Illinois borders the Atlantic Ocean, which ~s is Illinois is not west of Missouri, which is true. is false. Then is true because r is true. ~p is Springfield is not the capital of Illinois, which is false. Then is true because ~s is true. ANSWER:

Illinois shares a border with Kentucky, or Illinois

borders the Atlantic Ocean. is true because r is true. ANSWER: Illinois is not west of Missouri, or Springfield is not the capital of Illinois. is true because ~s is true.

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Write each statement in if-then form. 43. The area of a circle is 40. Get a free milkshake with any combo purchase. SOLUTION: SOLUTION: To write these statements in if-then form, identify To write these statements in if-then form, identify the hypothesis and conclusion. The word if is not the hypothesis and conclusion. The word if is not part of the hypothesis. The word then is not part of part of the hypothesis. The word then is not part of the conclusion. the conclusion. If a figure is a circle, then the area is . If you buy a combo, then you get a free milkshake. ANSWER: ANSWER: If a figure is a circle, then the area is . If you buy a combo, then you get a free milkshake. 44. Collinear points lie on the same line.

41. Everybody at the party received a gift. SOLUTION: To write these statements in if-then form, identify SOLUTION: the hypothesis and conclusion. The word if is not To write these statements in if-then form, identify part of the hypothesis. The word then is not part of the hypothesis and conclusion. The word if is not the conclusion. part of the hypothesis. The word then is not part of If points are collinear, then they lie on the same the conclusion. line.

If you were at the party, then you received a gift. ANSWER: If points are collinear, then they lie on the same line. ANSWER: If you were at the party, then you received a gift. 45. A right angle measures 90 degrees.

42. The intersection of two planes is a line. SOLUTION: To write these statements in if-then form, identify SOLUTION: the hypothesis and conclusion. The word if is not To write these statements in if-then form, identify part of the hypothesis. The word then is not part of the hypothesis and conclusion. The word if is not the conclusion. part of the hypothesis. The word then is not part of

the conclusion. If an angle is right, then the angle measures 90

degrees. If two planes intersect, then the intersection is a line. ANSWER: If an angle is right, then the angle measures 90 ANSWER: degrees. If two planes intersect, then the intersection is a line.

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CONSTRUCT ARGUMENTS Determine the 49. If an angle is acute, then it has a measure of 45. truth value of each conditional statement. If true, explain your reasoning. If false, give a SOLUTION: counterexample. 46. If a banana is blue, then an apple is a vegetable.

SOLUTION: False; the angle drawn is an acute angle whose The conditional statement "If a banana is blue, then measure is not 45. The hypothesis of the conditional an apple is a vegetable." is true. The hypothesis is is true, but the conclusion is false. This false, since a banana is never blue. A conditional counterexample shows that the conditional with a false hypothesis is always true, so this statement is false. conditional statement is true. ANSWER: ANSWER: True; the hypothesis is false, since a banana is never blue. A conditional with a false hypothesis is always False; the angle drawn is an acute angle whose true, so this conditional statement is true. measure is not 45. The hypothesis of the conditional is true, but the conclusion is false. This 47. If a number is odd, then it is divisible by 5. counterexample shows that the conditional statement is false. SOLUTION: False; 9 is an odd number, but not divisible by 5. The 50. If a polygon has six sides, then it is a regular hypothesis of the conditional is true, but the polygon. conclusion is false. This counterexample shows that SOLUTION: the conditional statement is false.

ANSWER: False; 9 is an odd number, but not divisible by 5. The False; this polygon has six sides, but is not regular. hypothesis of the conditional is true, but the The hypothesis of the conditional is true, but the conclusion is false. This counterexample shows that conclusion is false. This counterexample shows that the conditional statement is false. the conditional statement is false.

48. If a dog is an amphibian, then the season is summer. ANSWER:

SOLUTION: The conditional statement "If a dog is an amphibian, False; this polygon has six sides, but is not regular. then the season is summer." is true. The hypothesis The hypothesis of the conditional is true, but the is false, since a dog is not an amphibian. A conclusion is false. This counterexample shows that conditional with a false hypothesis is always true, so the conditional statement is false. this conditional statement is true.

ANSWER: True; the hypothesis is false, since a dog is not an amphibian. A conditional with a false hypothesis is always true, so this conditional statement is true.

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CONSTRUCT ARGUMENTS Determine the 53. If red paint and blue paint mixed together make truth value of each conditional statement. If white paint, then 3 – 2 = 0. true, explain your reasoning. If false, give a SOLUTION: counterexample. 51. If an angle’s measure is 25, then the measure of the The conditional statement "If red paint and blue angle’s complement is 65. paint mixed together make white paint, then 3 – 2 = 0" is true. The hypothesis is false, since red and blue SOLUTION: paint make green paint. A conditional with a false The conditional statement "If an angle’s measure is hypothesis is always true, so this conditional 25, then the measure of the angle’s complement is statement is true. 65." is true. When this hypothesis is true, the conclusion is also true, since an angle and its ANSWER: complement’s sum is 90. So, the conditional True; the hypothesis is false, since red and blue statement is true. paint make green paint. A conditional with a false hypothesis is always true, so this conditional ANSWER: statement is true. True; when this hypothesis is true, the conclusion is also true, since an angle and its complement’s sum 54. If two angles are congruent, then they are vertical is 90. So, the conditional statement is true. angles. SOLUTION: 52. If North Carolina is south of Florida, then the capital of Ohio is Columbus.

SOLUTION: The conditional statement "If North Carolina is south False; the angles are congruent, but they are not of Florida, then the capital of Ohio is Columbus." is vertical angles. The hypothesis of the conditional is true. The hypothesis is false, since North Carolina is true, but the conclusion is false. not south of Florida. A conditional with a false This counterexample shows that the conditional hypothesis is always true, so this conditional statement is false. statement is true. ANSWER: ANSWER: True; the hypothesis is false, since North Carolina is not south of Florida. A conditional with a false hypothesis is always true, so this conditional statement is true. False; the angles are congruent, but they are not vertical angles. The hypothesis of the conditional is true, but the conclusion is false. This counterexample shows that the conditional statement is false.

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55. If an animal is a bird, then it is an eagle. 57. If two lines intersect, then they form right angles.

SOLUTION: SOLUTION: The statement "If an animal is a bird, then it is an eagle." is false. The animal could be a falcon. The hypothesis of the conditional is true, but the conclusion is false. This counterexample shows that False; These lines intersect, but do not form right the conditional statement is false. angles. The hypothesis of the conditional is true, but ANSWER: the conclusion is false. This counterexample shows False; the animal could be a falcon. The hypothesis that the conditional statement is false. of the conditional is true, but the conclusion is false. ANSWER: This counterexample shows that the conditional statement is false.

56. If two angles are acute, then they are supplementary. False; these lines intersect, but do not form right SOLUTION: angles. The hypothesis of the conditional is true, but the conclusion is false. This counterexample shows that the conditional statement is false.

False; and are acute, but their sum is 90°. The hypothesis of the conditional is true, but the conclusion is false. This counterexample shows that the conditional statement is false.

ANSWER:

False; and are acute, but their sum is 90°. The hypothesis of the conditional is true, but the conclusion is false. This counterexample shows that the conditional statement is false.

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Write the converse, inverse, and 59. If you live in Chicago, you live in Illinois. contrapositive of each true conditional SOLUTION: statement. Determine whether each related conditional is The converse is formed by exchanging the true or false. If a statement is false, find a hypothesis and conclusion of the conditional. counterexample. Converse: If you live in Illinois, then you live in 58. A right triangle has an angle measure of 90. Chicago. The converse is false. Counterexample: You can live in Urbana. SOLUTION: If a triangle is right, then it has an angle measure of The inverse is formed by negating both the 90. hypothesis and conclusion of the conditional. Inverse: If you do not live in Chicago, then you do The converse is formed by exchanging the not live in Illinois. The inverse is false. hypothesis and conclusion of the conditional. Counterexample: You can live in Urbana. Converse: If a triangle has an angle measure of 90, then it is a right triangle. The converse is true. The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the The inverse is formed by negating both the conditional. hypothesis and conclusion of the conditional. Contrapositive: If you do not live in Illinois, then you Inverse: If a triangle is not right, then it does not do not live in Chicago. The contrapositive is true. have an angle measure of 90. The inverse is true. ANSWER: The contrapositive is formed by negating both the Converse: If you live in Illinois, then you live in hypothesis and the conclusion of the converse of the Chicago. False: You can live in Galveston. Inverse: conditional. If you do not live in Chicago, then you do not live in Contrapositive: If a triangle does not have an angle Illinois. False: You can live in Galveston. measure of 90, then it is not a right triangle. The Contrapositive: If you do not live in Illinois, then you contrapositive is true. do not live in Chicago; true.

ANSWER: If a triangle is right, then it has an angle measure of 90. Converse: If a triangle has an angle measure of 90, then it is a right triangle; true. Inverse: If a triangle is not right, then it does not have an angle measure of 90; true. Contrapositive: If a triangle does not have an angle measure of 90, then it is not a right triangle; true.

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60. If a bird is an ostrich, then it cannot fly. 61. If two angles have the same measure, then the angles are congruent. SOLUTION: The converse is formed by exchanging the SOLUTION: hypothesis and conclusion of the conditional. The converse is formed by exchanging the Converse: If a bird cannot fly, then it is an ostrich. hypothesis and conclusion of the conditional. The converse is false. Converse: If two angles are congruent, then they Counterexample: The bird could be a penguin. have the same measure. The converse is true.

The inverse is formed by negating both the The inverse is formed by negating both the hypothesis and conclusion of the conditional. hypothesis and conclusion of the conditional. Inverse: If a bird is not an ostrich, then it can fly. Inverse: If two angles do not have the same The inverse is false. measure, then the angles are not congruent. The Counterexample: The bird could be a penguin. inverse is true.

The contrapositive is formed by negating both the The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the hypothesis and the conclusion of the converse of the conditional. conditional. Contrapositive: If a bird can fly, then the bird is not Contrapositive: If two angles are not congruent, then an ostrich; The contrapositive is true. they do not have the same measure. The contrapositive is true. ANSWER: Converse: If a bird cannot fly, then it is an ostrich. ANSWER: False; The bird could be a penguin. Inverse: If a bird Converse: If two angles are congruent, then they is not an ostrich, then it can fly. False; The bird have the same measure; true. Inverse: If two angles could be a penguin. Contrapositive: If a bird can fly, do not have the same measure, then the angles are then the bird is not an ostrich; true. not congruent; true. Contrapositive: If two angles are not congruent, then they do not have the same measure; true.

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62. All congruent segments have the same length. 63. All squares are rectangles.

SOLUTION: SOLUTION: If segments are congruent, then they have the same If a figure is a square, then it is a rectangle. length. The converse is formed by exchanging the The converse is formed by exchanging the hypothesis and conclusion of the conditional. hypothesis and conclusion of the conditional. Converse: If a figure is a rectangle, then it is a Converse: If segments have the same length, then square. The converse is false. they are congruent. The converse is true. Counterexample: A rectangle does not have to have all sides congruent. The inverse is formed by negating both the hypothesis and conclusion of the conditional. The inverse is formed by negating both the Inverse: If segments are not congruent, then they do hypothesis and conclusion of the conditional. not have the same length. The inverse is true. Inverse: If a figure is not a square, then it is not a rectangle. The inverse is false. The contrapositive is formed by negating both the Counterexample: The figure could be a rectangle, hypothesis and the conclusion of the converse of the even though it is not a square. conditional. Contrapositive: If segments do not have the same The contrapositive is formed by negating both the length, then they are not congruent. The hypothesis and the conclusion of the converse of the contrapositive is true. conditional. Contrapositive: If a figure is not a rectangle, then it ANSWER: is not a square. The contrapositive is true. If segments are congruent, then they have the same length. Converse: If segments have the same length, ANSWER: then they are congruent; true. Inverse: If segments If a figure is a square, then it is a rectangle. are not congruent, then they do not have the same Converse: If a figure is a rectangle, then it is a length; true. Contrapositive: If segments do not have square. False. A rectangle does not have to have all the same length, then they are not congruent; true. sides congruent. Inverse: If a figure is not a square, then it is not a rectangle. False. The figure could be a rectangle, even though it is not a square. Contrapositive: If a figure is not a rectangle, then it is not a square; true.

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Rewrite each statement as a biconditional 67. The midpoint of a segment bisects the segment. statement. Then determine whether the SOLUTION: biconditional is true or false. 64. Lines that do not intersect are horizontal. Write a biconditional using if and only if.

SOLUTION: A point is the midpoint of a segment if and only if it Write a biconditional with if and only if. is on a line that bisects the segment; true a bisector is a line through the midpoint. Lines do not intersect if and only if they are horizontal; false they do not intersect if they are ANSWER: parallel or skew, they could in any orientation. A point is the midpoint of a segment if and only if it is on a line that bisects the segment; true. ANSWER: 68. Real numbers are irrational numbers. Lines do not intersect if and only if they are horizontal; false. SOLUTION: Write a biconditional using if and only if. 65. Points that lie in the same plane are coplanar. A number is a real number if and only if it is an SOLUTION: irrational number; false all irrational numbers are Write a biconditional using if and only if. real numbers, but all rational numbers are not irrational, but they are real. Points are coplanar if and only if they lie in the same plane; true this is the definition of coplanar. ANSWER: A number is a real number if and only if it is an ANSWER: irrational number; false. Points are coplanar if and only if they lie in the same plane; true. 69. Perpendicular lines meet at right angles.

66. Right angles measure 90 degrees. SOLUTION: Write as a biconditional using if and only if. SOLUTION: Write a biconditional using if and only if. Lines are perpendicular if and only if they meet at right angles; true as this is the definition of An angle is right if and only if it measures 90 perpendicular. degrees; true this is the definition of a right angle. ANSWER: ANSWER: Lines are perpendicular if and only if they meet at An angle is right if and only if it measures 90 right angles; true. degrees; true.

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JUSTIFY ARGUMENTS Write the statement 73. contrapositive indicated, and use the information at the left to SOLUTION: determine the truth value of each statement. If a statement is false, give a counterexample. The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the Animals with stripes are zebras. conditional. 70. conditional If an animal is not a zebra, then it does not have stripes. The contrapositive is false. Counterexample: SOLUTION: A tiger has stripes. To write these statements in conditional form, identify the hypothesis and conclusion. The word if ANSWER: is not part of the hypothesis. The word then is not If an animal is not a zebra, then it does not have part of the conclusion. stripes; false: a tiger has stripes. If an animal has stripes, then it is a zebra. The Write the conditional and converse for each conditional is false. Counterexample: A tiger has statement. Determine the truth values for the stripes. conditionals and converses. If false, write a counterexample. Write a biconditional if ANSWER: possible. If an animal has stripes, then it is a zebra; false: a 74. Regular quadrilaterals are squares. tiger has stripes. SOLUTION: 71. converse Write as if then. Conditional: If a quadrilateral is regular then it is a SOLUTION: square; true all sides and angles are congruent in a The converse is formed by exchanging the square. hypothesis and conclusion of the conditional. Reverse the order of the conditional. If an animal is a zebra, then it has stripes. The Converse: If a quadrilateral is a square then it is converse is true. regular; true all sides and angles are congruent in a square. ANSWER:

If an animal is a zebra, then it has stripes; true. Write using if and only if. Biconditional: A quadrilateral is regular if and only if 72. inverse it is a square; true all sides and angles are congruent SOLUTION: in a square. The inverse is formed by negating both the ANSWER: hypothesis and conclusion of the conditional. Conditional: If a quadrilateral is regular then it is a If an animal does not have stripes, then it is not a square; true. zebra. The inverse is true. Converse: If a quadrilateral is a square then it is regular; true. ANSWER: Biconditional: A quadrilateral is regular if and only if If an animal does not have stripes, then it is not a it is a square; true. zebra; true.

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75. Equilateral triangles have all sides the same length. 77. Integers are rational numbers.

SOLUTION: SOLUTION: Write as if then. Write the statement using if then. Conditional: If a triangle is equilateral then all sides Conditional: If a number is an integer then it is a have the same length; true this is definition of rational number; true. equilateral. All integers can be written as fraction with one as denominator. Reverse the parts of the if then Converse: If a triangle has all sides the same length, Reverse the order of the conditional and test the then it is equilateral; true this is definition of new statement. equilateral. Converse: If a number is a rational number then it is an integer; false 0.5 is a rational number that is not Write with if and only if. an integer. Biconditional: A triangle is equilateral if and only if all sides have the same length; true this is definition ANSWER: of equilateral. Conditional: If a number is an integer then it is a rational number; true. ANSWER: Converse: If a number is a rational number then it is Conditional: If a triangle is equilateral then all sides an integer; false 0.5 is a rational number that is not have the same length; true. an integer. Converse: If a triangle has all sides the same length, then it is equilateral; true. Biconditional: A triangle is equilateral if and only if all sides have the same length; true.

76. Right angles have measures greater than acute angles.

SOLUTION: Right angles have measures greater than acute angles.

Write as if then and test statement. Conditional: If an angle is right, then it measures greater than an acute angle; true.

Reverse the statement. Converse: If an angle measures greater than an acute angle, then it is right; false an obtuse angle is also greater than an acute angle.

ANSWER: Conditional: If an angle is right, then it measures greater than an acute angle; true. Converse: If an angle measures greater than an acute angle, then it is right; false an obtuse angle is also greater than an acute angle.

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78. VEHICLES Different vehicles are characterized 79. ART Write the following statement in if-then form: by different structural features. Write each At the Andy Warhol Museum in Pittsburgh, statement in if-then form. Pennsylvania, most of the collection is Andy • A convertible describes any vehicle with a fully Warhol's artwork. retractable top. SOLUTION: • A coupe is any car with only two full-size passenger doors. To write these statements in if-then form, identify • A pickup is any vehicle with an open cargo bed in the hypothesis and conclusion. The word if is not the rear. part of the hypothesis. The word then is not part of the conclusion. SOLUTION: If the museum is the Andy Warhol Museum, then To write these statements in if-then form, identify most of the collection is Andy Warhol's artwork. the hypothesis and conclusion. The word if is not part of the hypothesis. The word then is not part of ANSWER: the conclusion. If the museum is the Andy Warhol Museum, then most of the collection is Andy Warhol's artwork. If you drive a convertible, then you have a vehicle with a fully retractable top. If you drive a coupe, then you have a car with only two full-size passenger doors. If you drive a pickup, then you drive a vehicle with an open cargo bed in the rear.

ANSWER: If you drive a convertible, then you have a vehicle with a fully retractable top. If you drive a coupe, then you have a car with only two full-size passenger doors. If you drive a pickup, then you drive a vehicle with an open cargo bed in the rear.

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80. SCIENCE The water on Earth is constantly 81. SPORTS In football, touchdowns are worth 6 changing through a process called the water cycle. points, extra point conversions are worth 2 points, Write the three conditionals below in if-then form. and safeties are worth 2 points. a. Write three conditional statements in if-then form for scoring in football. b. Write the converse of the three true conditional statements. State whether each is true or false. If a statement is false, find a counterexample.

a. As runoff, water flows into bodies of water. SOLUTION: b. Plants return water to the air through a. To write these statements in if-then form, identify transpiration. the hypothesis and conclusion. The word if is not c. Water bodies return water to the air through part of the hypothesis. The word then is not part of evaporation. the conclusion. SOLUTION: If a football team makes a touchdown, they get 6 To write these statements in if-then form, identify points; If a football team makes a two-point the hypothesis and conclusion. The word if is not conversion, they get 2 points; If a football team part of the hypothesis. The word then is not part of makes a safety, they get 2 points. the conclusion. b. The converse is formed by exchanging the a. If water runs off, it flows into bodies of water. hypothesis and conclusion of the conditional. b. If plants return water to the air, they transpire. If a football team gets 6 points, they made a c. If water bodies return water to the air, it is touchdown. The converse is true. through evaporation. If a football team gets 2 points, they made a two- point conversion. The converse is false. ANSWER: Counterexample: They could have gotten a safety; a. If water runs off, it flows into bodies of water. If a football team gets 2 points, they made a safety. b. If plants return water to the air, they transpire. the converse is false. Counterexample, They could c. If water bodies return water to the air, it is have gotten a two-point conversion. through evaporation. ANSWER: a. Sample answer: If a football team makes a touchdown, they get 6 points; If a football team makes a two-point conversion, they get 2 points; If a football team makes a safety, they get 2 points. b. Sample answer: If a football team gets 6 points, they made a touchdown. True; If a football team gets 2 points, they made a two-point conversion. False; they could have gotten a safety; If a football team gets 2 points, they made a safety. False; they could have gotten a two-point conversion.

82. SCIENCE Chemical compounds are grouped and described by the elements that they contain. Acids contain hydrogen (H). Bases contain hydroxide (OH). Hydrocarbons contain only hydrogen (H) and eSolutions Manual - Powered by Cognero Page 20 2-2 Statements, Conditionals, and Biconditionals

carbon (C). 83. Can the conditional "If , then x = 6," be combined with its converse to form a true biconditional? Explain.

SOLUTION: No, the biconditional cannot be formed, because the conditional is not true. The value of x could also be a. Write three conditional statements in if-then form -6. for classifying chemical compounds. b. Write the converse of the three true conditional ANSWER: statements. No, the biconditional cannot be formed, because the conditional is not true. The value of x could also be SOLUTION: -6. a. To write these statements in if-then form, identify the hypothesis and conclusion. The word if is not 84. If the contrapositive of a conditional is true, can you part of the hypothesis. The word then is not part of rewrite the conditional as a true biconditional? Explain. the conclusion. If a compound is an acid, it contains hydrogen. If a SOLUTION: compound is a base, it contains hydroxide. If a No, the contrapositive has the same truth as the compound is a hydrocarbon, it contains only conditional, which means the conditional is true, but hydrogen and carbon. the converse must also be true, but that information b. The converse is formed by exchanging the is not given. hypothesis and conclusion of the conditional. ANSWER: If a compound contains hydrogen, it is an acid. The No, the contrapositive has the same truth as the converse is false. Counterexample: A hydrocarbon conditional, which means the conditional is true, but contains hydrogen. the converse must also be true, but that information If a compound contains hydroxide, it is a base. The is not given. converse is true. If a compound contains only hydrogen and carbon, it 85. Compare the mathematical meanings of the symbols and in and . is a hydrocarbon. The converse is true. SOLUTION: ANSWER: The expression with the right arrow means the a. Sample answer: If a compound is an acid, it conditional "if p is true then q is also true." contains hydrogen. The expression with the double arrow means there If a compound is a base, it contains hydroxide. If a is a conditional "p is true if and only if q is true." compound is a hydrocarbon, it contains only The arrow tells which way(s) the conditional goes. hydrogen and carbon. b. Sample answer: If a compound contains ANSWER: hydrogen, it is an acid. The expression with the right arrow means the conditional "if p is true then q is also true." False; a hydrocarbon contains hydrogen. If a The expression with the double arrow means there compound contains hydroxide, it is a base; true. If a is a conditional "p is true if and only if q is true." compound contains only hydrogen and carbon, it is a The arrow tells which way(s) the conditional goes. hydrocarbon; true.

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86. MULTIPLE REPRESENTATIONS In this LOGIC To negate a statement containing the problem, you will investigate a law of logic by using words all or for every, you can use the phrase at conditionals. least one or there exists. To negate a statement a. LOGICAL Write three true conditional containing the phase there exists, you can use the statements, using each consecutive conclusion as phrase for all or for every. the hypothesis for the next statement. p: All polygons are convex. b. LOGICAL Write a conditional using the ~p: At least one polygon is not convex. hypothesis of your first conditional and the q: There exists a problem that has no solution. ~q: conclusion of your third conditional. Is the For every problem, there is a solution. conditional true if the hypothesis is true? c. VERBAL Given two conditionals If a, then b Sometimes these phrases may be implied. For and If b, then c, make a conjecture about the truth example, The square of a real number is value of c when a is true. Explain your reasoning. nonnegative implies the following conditional and its negation. SOLUTION: p: For every real number x, x2 ≥ 0. a.With consecutive conclusion, the conclusion is the 2 hypothesis of the next statement. If a, then b. If b, ~p: There exists a real number x such that x < 0 . then c, If c, then d. If you live in New York City, then you live in New Use the information given to write the negation York State; If you live in New York State, then you of each statement. live in the United States; If you live in the United 87. There exists a segment that has no midpoint. States, then you live in North America. SOLUTION: b. The hypothesis of the first statement is "If you To negate a statement containing the phrase "there live in New York City". The conclusion of the third exists", use the word "every". statement is " you live in North America". Then the Every segment has a midpoint. new conditional is "If you live in New York City, then you live in North America" . The new ANSWER: conditional statement is true. Every segment has a midpoint. c. If a is true, then c is true. If we know that a is true, then we know that b is true, and if we know 88. Every student at Hammond High School has a that b is true, then we know that c is true. locker. Therefore, when a is true, c is true. SOLUTION: ANSWER: To negate a statement containing the word "every", a. Sample answer: If you live in New York City, use the phase "at least one". then you live in New York State; If you live in New There exists at least one student at Hammond High York State, then you live in the United States; If you School who does not have a locker. live in the United States, then you live in North America. ANSWER: b. If you live in New York City, then you live in There exists at least one student at Hammond High North America; yes. School who does not have a locker. c. Sample answer: If a is true, then c is true. If we know that a is true, then we know that b is true, and if we know that b is true, then we know that c is true. Therefore, when a is true, c is true. eSolutions Manual - Powered by Cognero Page 22 2-2 Statements, Conditionals, and Biconditionals

89. All squares are rectangles. 93. ERROR ANALYSIS Nicole and Kiri are evaluating the conditional If 15 is a prime number, SOLUTION: then 20 is divisible by 4. Both think that the To negate a statement containing the word "all", use conditional is true, but their reasoning differs. Is the phase "at least one". either of them correct? Explain. There exists at least one square that is not a rectangle.

ANSWER: There exists at least one square that is not a rectangle. SOLUTION: Kiri is correct. For the conditional statement "If 15 90. There exists a real number x such that is a prime number, then 20 is divisible by 4." p is 15 SOLUTION: is "a prime number", which is false and q is "20 is To negate a statement containing the phrase "there divisible by 4" is true. When the hypothesis of a exists", use the word "every". conditional is false, the conditional is always true. For every real number x, x2 . Nicole, did not consider the truth value of the hypothesis. ANSWER: ANSWER: For every real number x, x2 . Sample answer: Kiri; when the hypothesis of a 91. There exists an even number x such that conditional is false, the conditional is always true. .

SOLUTION: To negate a statement containing the phrase "there exists", use the phrase "for every". For every even number x, .

ANSWER: For every even number x, .

92. Every real number has a real square root.

SOLUTION: To negate a statement containing the word "every", use the phrase "there exists" . There exists a real number that does not have a real square root.

ANSWER: There exists a real number that does not have a real square root.

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94. CHALLENGE You have learned that statements 95. REASONING You are evaluating a conditional with the same truth value are logically equivalent. statement in which the hypothesis is true, but the Use logical equivalence to summarize the conclusion is false. Is the inverse of the statement conditional, converse, inverse, and contrapositive for true or false? Explain your reasoning. the statements p and q. SOLUTION: SOLUTION: The inverse of a conditional statement in which the The conditional statement is (p → q). The converse hypothesis is true is true. Since the conclusion is is formed by exchanging the hypothesis and false, the conclusion of the conditional (q → p). The inverse is converse of the statement must be true. The formed by negating both the hypothesis and converse and inverse are logically equivalent, so the conclusion of the conditional (~p → ~q). The inverse is also true. contrapositive is formed by negating both the Consider the following truth table. hypothesis and the conclusion of the converse of the conditional (~q → ~p). Conditional Converse Inverse p q p→q q→p ~p→~q If the conditional is true, then the contrapositive is T F F T T also true. These are logically equivalent.

Similarly, if the converse is true, then the inverse is ANSWER: also true. These are also logically equivalent. True; since the conclusion is false, the converse of the statement must be true. The converse and inverse are logically equivalent, so the inverse is also ANSWER: true. If the conditional is true, then the contrapositive is also true. These are logically equivalent.

Similarly, if the converse is true, then the inverse is also true. These are also logically equivalent.

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96. OPEN ENDED Write a conditional statement in 97. CHALLENGE The inverse of conditional A is which the converse, inverse, and contrapositive are given below. Write conditional A, its converse, and all true. Explain your reasoning. its contrapositve. Explain your reasoning. If I received a detention, then I did not arrive at SOLUTION: school on time. If four is divisible by two, then birds have feathers. In order for the converse, inverse, and SOLUTION: contrapositive to be true, the hypothesis and the The inverse is formed by negating both the conclusion must both be either true or false. hypothesis and conclusion of the conditional. The hypothesis q of the inverse statement is I p = " four is divisible by two" and q = "birds have received a detention. The conclusion p of the feathers" inverse statement is I did not arrive at school on time.

So the conditional A is : If I did not arrive at school on time, then I received a detention.

ANSWER: The converse is formed by exchanging the Sample answer: If four is divisible by two, then birds hypothesis and conclusion of the conditional. have feathers. In order for the converse, inverse, So the converse of statement A is : If I and contrapositive to be true, the hypothesis and the arrived at school on time, then I did not receive a conclusion must both be either true or false. detention.

The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the conditional. The contrapositive of Statement A is : If I did not receive a detention, then I arrived at school on time.

ANSWER: The hypothesis q of the inverse statement is I received a detention. The conclusion p of the inverse statement is I did not arrive at school on time. So the conditional A is : If I did not arrive at school on time, then I received a detention. So the converse of statement A is : If I arrived at school on time, then I did not receive a detention. The contrapositive of Statement A is : If I did not receive a detention, then I arrived at school on time.

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98. WRITING IN MATH Describe the relationship 100. REASONING Because a conditional statement between a conditional, its converse, its inverse, and and its converse must both be true for a its contrapositive. biconditional to be true, the order of the hypothesis and the conclusion do not matter. How does this SOLUTION: affect the truth values of the statements? Explain. The converse is formed by exchanging the SOLUTION: hypothesis and conclusion of the conditional. The Since the conditional and its converse are both true, inverse is formed by negating both the hypothesis interchanging the order of the hypothesis and and conclusion of the conditional.The contrapositive conclusion does not affect the truth values of the is formed by negating both the hypothesis and the statements. conclusion of the converse of the conditional. Since they are logically equivalent, a conditional and ANSWER: its contrapositive always have the same truth value. Since the conditional and its converse are both true, The inverse and converse of a conditional are also interchanging the order of the hypothesis and conclusion does not affect the truth values of the logically equivalent and have the same truth value. statements. The conditional and its contrapositive can have the same truth value as its inverse and converse, or it 101. MULTI-STEP Use conditional statements I can have the opposite truth value of its inverse and through IV to answer the following questions. converse. I. Two lines are perpendicular, and the lines ANSWER: intersect. Sample answer: Since they are logically equivalent, II. All triangles have an acute angle. a conditional and its contrapositive always have the III. The sum of the measures of two same truth value. The inverse and converse of a supplementary angles is 180 degrees. conditional are also logically equivalent and have the IV. An angle is a right angle, and it has a same truth value. The conditional and its measurement of 90 degrees.

contrapositive can have the same truth value as its a. Rewrite each of the conditional statements in if- inverse and converse, or it can have the opposite then form. Then, write the converse statement. truth value of its inverse and converse. 99. WRITING IN MATH If a biconditional is true, b. Which conditional statements have a true what do you know about the conditional and the converse? converse? If the biconditional is false, what do you A I know about the conditional and converse? B II C III SOLUTION: D IV If a biconditional is true, then the conditional and the converse are both true. If a biconditional is false, c. Which statements, paired with their converses then the conditional, the converse, or both are false. can be written as a biconditional? Select all of the true statements. ANSWER: A I If a biconditional is true, then the conditional and the B II converse are both true. If a biconditional is false, C III then the conditional, the converse, or both are false. D IV

SOLUTION:

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a. 102. Which of the following can be used to prove that a Write each first as an if then, then reverse the conditional statement is false? order. I. If two lines are perpendicular then the lines A counterexample intersect. Converse: If two lines intersect, then they are perpendicular. B converse II. If a figure is a triangle then it has an acute angle. Converse: If a figure has an acute angle, then it is a C conclusion triangle. III. If the sum of the measures of two angles is 180 D contrapositive degrees, then the angles are supplementary. Converse: If two angles are supplementary, then the SOLUTION: sum of their measures is 180 degrees. A counterexample can be used to prove a IV. If an angle is a right angle, then the angle conditional statement is false by finding an example measures 90 degrees. Converse: If an angle where the hypothesis is true and the conclusion is measures 90 degrees, then the angle is a right false. angle. b. C, D these are the definitions, but A is not true, ANSWER: because lines can intersect and not be perpendicular, A and B is not true, because quadrilaterals and other polygons can also have acute angles. c. C, D see explanation in part b.

ANSWER: a. I. If two lines are perpendicular then the lines intersect. Converse: If two lines intersect, then they are perpendicular. II. If a figure is a triangle then it has an acute angle. Converse: If a figure has an acute angle, then it is a triangle. III. If the sum of the measures of two angles is 180 degrees, then the angles are supplementary. Converse: If two angles are supplementary, then the sum of their measures is 180 degrees. IV. If an angle is a right angle, then the angle measures 90 degrees. Converse: If an angle measures 90 degrees, then the angle is a right angle. b. C, D c. C, D

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103. Consider the conditional statements below. 104. Which of the following statements is logically equivalent to the statement below. Select all logically I. Every eagle is a bird. equivalent statements. II. A butterfly is an insect. III. If an animal is a tiger, then it lives underwater. All octagons are polygons.

Which of the conditional statements has a true A If a figure is not a polygon, then it is not an contrapositive? octagon. A I only B All polygons are octagons. B II only C If a figure is not an octagon, then it is not C III only a polygon. D I and II only D Every polygon is also an octagon. E I, II, and III E A figure is an octagon if and only if it is a polygon. SOLUTION: A conditional and its contrapositive are logically SOLUTION: equivalent. Logically equivalent statements have the Compare the statements. same truth value. To identify the conditional statements that have a true contrapositive, identify All octagons are polygons. the true conditional statements. A If a figure is not a polygon, then it is not an I. Every eagle is a bird. octagon. = Contrapositive The conditional statement is If a creature is an B All polygons are octagons. = Converse eagle, then it is a bird. This conditional statement C If a figure is not an octagon, then it is not is true, so the contrapositive is also true. a polygon. = Inverse D Every polygon is also an octagon. = Converse II. A butterfly is an insect. E A figure is an octagon if and only if it is a polygon The conditional statement is If a creature is a = Biconditional butterfly, then it is an insect. This conditional statement is true, so the contrapositive is also true. The converse is false, because octagons are only polygons with 8 sides, so only A is true. III. If an animal is a tiger, then it lives underwater. This conditional statement is false, so the ANSWER: contrapositive is also false. A

Statements I and II have a true contrapositive.

ANSWER: D

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105. Use the following statements to write a compound statement. Then find the truth value.

p: A triangle has two congruent sides. q: A triangle has no congruent sides. r: A triangle is equilateral.

a. b.

SOLUTION: p: A triangle has two congruent sides. q: A triangle has no congruent sides. r: A triangle is equilateral.

a. = A triangle has two congruent sides, no congruent sides, or is equilateral. True, because there are three sides in a triangle. b. = A triangle has two congruent sides and no congruent sides, or is equilateral. False, a triangle cannot have both 0 and 2 sides congruent.

ANSWER: a. A triangle has two congruent sides, no congruent sides, or is equilateral; true. b. A triangle has two congruent sides and no congruent sides, or is equilateral; false.