Study Guide and Review

Transcription

1 State whether each sentence is or false. If false, replace the underlined term to make a sentence. 1. A postulate is a statement that requires proof. A postulate is a statement that does not require a proof. So, the sentence is false. A theorem is a statement that requires a proof. false; theorem 2. The first part of an if-then statement is the conjecture. The first part of an if-then statement is the hypothesis. So, the sentence is false. false; hypothesis 3. Deductive reasoning uses the laws of mathematics to reach logical conclusions from given statements. True 4. The contrapositive is formed by negating the hypothesis and conclusion of a conditional. The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the conditional. So, the sentence is false. The inverse is formed by negating the hypothesis and conclusion of a conditional. false; inverse 5. A conjunction is formed by joining two or more statements with the word and. True 6. A theorem is a statement that is accepted as without proof. A theorem is a statement that requires a proof. So, the sentence is false. A postulate is a statement that is accepted as without proof. false; postulate 7. The converse is formed by exchanging the hypothesis and conclusion of a conditional. True 8. To show that a conjecture is false, you would provide a disjunction. To show that a conjecture is false, you would provide a counter example. So, the sentence is false. false; counterexample 9. The inverse of a statement p would be written in the form not p. The inverse is formed by negating both the hypothesis and conclusion of the conditional. So, the sentence is false. The negation of a statement p would be written in the form not p. false; negation 10. In a two-column proof, the properties that justify each step are called reasons. True esolutions Manual - Powered by Cognero Page 1

2 Determine whether each conjecture is or false. If false, give a counterexample. 11. If are supplementary angles, then form a linear pair. Two supplementary angles form a linear pair only if they share a common side. So, the sentence is false. Counter example: Two nonadjacent supplementary angles. false; two nonadjacent supplementary angles 12. If W( 3, 2), X( 3, 7), Y(6, 7), Z(6, 2), then quadrilateral WXYZ is a rectangle. The sides are horizontal lines and are vertical lines. So, the quadrilateral has four right angles. Therefore, it is a rectangle, by definition. 13. PARKS Jacinto enjoys hiking with his dog in the forest at his local park. While on vacation in Big Bend National Park in Texas, he was disappointed that dogs were not allowed on most hiking trails. Make a conjecture about why his local park and the national park have differing rules with regard to pets. Sample answer: The national park may be home to wildlife species not found in the local park. Dogs or other pets may threaten or chase these unfamiliar animals or insects. Sample answer: Dogs or other pets may threaten or chase wildlife that might not be present in his local park. Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. Explain. p : A plane contains at least three noncollinear points. q: A square yard is equivalent to three square feet. r: The sum of the measures of two complementary angles is Negate q finding the opposite truth values. Then find the disjunction. A disjunction is if at least one of the statements is. A square yard is not equivalent to three square feet or the sum of the measures of two complementary angles is 180º. Here, ~q is a statement. Since one of the statements is, the disjunction is also. A square yard is not equivalent to three square feet or the sum of the measures of two complementary angles is 180º;. esolutions Manual - Powered by Cognero Page 2

3 Negate r finding the opposite truth values. Then find the conjunction. A conjunction is only when both statements that form it are. A plane contains at least three noncollinear points is True. The sum of the measures of two complementary angles is not 180 is. Since both the statements are, the conjunction is also. A plane contains at least three noncollinear points and the sum of the measures of two complementary angles is not 180;. Negate p finding the opposite truth values. Then find the disjunction. A disjunction is if at least one of the statements is. The statement "A plane does not contain at least three noncollinear points" is false. The statement "a square yard is equivalent to three square feet" is false. Since both the statements are false, the disjunction is also false. A plane does not contain at least three noncollinear points or a square yard is equivalent to three square feet; false. 17. PETS The Venn diagram shows the results of a pet store survey to determine the pets customers owned. a. How many customers had only fish? b. How many had only cats and dogs? c. How many had dogs as well as fish? a. Find (fish ~dogs ~cats). A total of 18 customers had only fish. b. Find (dogs cats ~fish). A total of 14 had only cats and dogs. c. (dogs fish ~cats). A total of = 22 had dogs as well as fish. a. 18 b. 14 c. 22 Determine the truth value of each conditional statement. If, explain your reasoning. If false, give a counterexample. 18. If you square an integer, then the result is a positive integer. The conditional statement "If you square an integer, then the result is a positive integer." is. When this hypothesis is "you square an integer", the conclusion "the result is a positive integer" is also, since the product of an integer multiplied by that same integer is always positive. So, the conditional statement is. esolutions Manual - Powered by Cognero Page 3

4 19. If a hexagon has eight sides, then all of its angles will be obtuse. A conditional with a false hypothesis is always. Here, the hypothesis a hexagon has eight sides is false as a hexagon has six sides. Therefore, the conditional statement is. 20. Write the converse, inverse, and contrapositive of the following conditional. Then, determine whether each related conditional is or false. If a statement is false, find a counterexample. If two angles are congruent, then they have the same degree measure. Converse: The converse is formed by exchanging the hypothesis and conclusion of the conditional. If two angles have the same degree measure, then they are congruent. The statement is by the definition of congruence. Inverse: The inverse is formed by negating both the hypothesis and conclusion of the conditional. If two angles are not congruent, then they do not have the same degree measure. The statement is by the definition of congruence. Contrapositive: The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the conditional. If two angles do not have the same degree measure, then they are not congruent. The statement is by the definition of congruence. Converse: If two angles have the same degree measure, then they are congruent. True. Inverse: If two angles are not congruent, then they do not have the same degree measure. True. Contrapositive: If two angles do not have the same degree measure, then they are not congruent. True. Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no valid conclusion and explain your reasoning. 21. Given: If a quadrilateral has diagonals that bisect each other, then it is a parallelogram. The diagonals of quadrilateral PQRS bisect each other. If p q is a statement then if p is, then q. Here, the statement if a quadrilateral has diagonals that bisect each other, then it is a parallelogram is a statement and the diagonals of quadrilateral PQRS bisect each other. So, PQRS is a parallelogram by the Law of Detachment. PQRS is a parallelogram; Law of Detachment. 22. Given: If Liana struggles in science class, then she will receive tutoring. If Liana stays after school on Thursday, then she will receive tutoring. By the Law of Syllogism, if p q and q r are statements, then p r is a statement. The Law of Syllogism does not apply since the conclusion of the first statement is not the hypothesis of the second statement. The conclusion is invalid. Invalid; the Law of Syllogism does not apply since the conclusion of the first statement is not the hypothesis of the second statement. esolutions Manual - Powered by Cognero Page 4

5 23. EARTHQUAKES Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain. Given: If an earthquake measures a 7.0 or higher on the Richter scale, then it is considered a major earthquake that could cause serious damage. The 1906 San Francisco earthquake measured 8.0 on the Richter scale. Conclusion: The 1906 San Francisco earthquake was a major earthquake that caused serious damage. If p q is a statement then if p is, then q. Here, the statement if an earthquake measures a 7.0 or higher on the Richter scale, then it is considered a major earthquake that could cause serious damage is a statement and the 1906 San Francisco earthquake measured 8.0 on the Richter scale. So, the 1906 San Francisco earthquake was a major earthquake that caused serious damage is a valid conclusion by the Law of Detachment. Valid; Law of Detachment Determine whether each statement is always, sometimes, or never. Explain. 24. Two planes intersect at a point. If two planes intersect, they form a line. So, the statement "Two planes intersect at a point." is never. 25. Three points are contained in more than one plane. If the three points are non-collinear, then there exists a plane containing all the three points. But if the points are collinear, we can find three different planes with each plane containing one of the three points. So, the statement is sometimes. Sometimes; if the three points are collinear, they will be contained in multiple planes, but if they are noncollinear, they will be contained in only one plane. 26. If line m lies in plane X and line m contains a point Q, then point Q lies in plane X. If a plane contains a line, then every point of that line lies in the plane. Therefore, the statement " If line m lies in plane X and line m contains a point Q, then point Q lies in plane X." is always. Never; if two planes intersect, they form a line. Always; if a plane contains a line, then every point of that line lies in the plane. esolutions Manual - Powered by Cognero Page 5

6 27. If two angles are complementary, then they form a right angle. 1 and 2 are complementary. Two complementary angles form a right angle only if they are adjacent. Otherwise they do not form a right angle. So, the statement is sometimes. Sometimes; if the angles are adjacent, they will form a right angle, but if they are not adjacent, they will not. 28. NETWORKING Six people are introduced at a business convention. If each person shakes hands with each of the others, how many handshakes will be exchanged? Include a model to support your reasoning. The first person will shake hands with the other 5 people and the second person with the other 4 people as the hand shake between the first and the second persons has already been counted. Similarly, the third person will shake hands with 3 other people and so on. So, the total number of hand shakes will be = 15. So, a total of 15 handshakes will be exchanged at the convention. 15 handshakes; State the property that justifies each statement. 29. If 7(x 3) = 35, then 35 = 7(x 3). The Symmetric Property of Equality is used to transform the equation 7(x 3) = 35to 35 = 7(x 3). Symm. Prop. esolutions Manual - Powered by Cognero Page 6

7 30. If 2x + 19 = 27, then 2x = 8. Use the Subtraction Property of Equality to subtract 19 from each side of 2x + 19 = 27 to obtain 2x = Copy and complete the following proof. Given: 6(x 4) = 42 Prove: x = 3 Subt. Prop (3x + 1) = 15x + 5 The Distributive Property to simplify 5(3x + 1) to 15x + 5. Distr. Prop. The 1st contains the given information. The 2nd row uses the Distributive property to remove the parenthesis. The 3rd row uses addition to add 24 to each side.. The 4th row uses division to divide each side by x 2 = 7x 2 The Reflexive Property of Equality describes 7x 2 = 7x 2. Reflex. Prop. 33. If 12 = 2x + 8 and 2x + 8 = 3y, then 12 = 3y. Use the Transitive Property of Equality to combine 12 = 2x + 8 and 2x + 8 = 3y to 12 = 3y. Trans. Prop. esolutions Manual - Powered by Cognero Page 7

8 35. Write a two-column proof to show that if PQ = RS, PQ = 5x + 9, and RS = x 31, then x = 10. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two segments of equal length and expressions for each segment. Once you prove the values are equal, you will need to find the variable in the expression. Use the properties that you have learned about congruent segments and equivalent expressions in algebra to walk through the proof. Given: PQ = RS, PQ = 5x + 9, and RS = x 31 Prove: x = 10 Proof: 1. PQ = RS, PQ = 5x + 9,RS = x 31 (Given) 2. 5x + 9 = x 31 (Substitution Property) 3. 4x + 9 = 31 (Subtraction. Property) 4. 4x = 40 (Subtraction Property) 5. x = 10 (Division. Property) 1. PQ = RS, PQ = 5x + 9,RS = x 31 (Given) 2. 5x + 9 = x 31 (Subs. Prop.) 3. 4x + 9 = 31 (Subt. Prop.) 4. 4x = 40 (Subt. Prop.) 5. x = 10 (Div. Prop.) 36. GRADES Jerome received the same quarter grade as Paula. Paula received the same quarter grade as Heath. Which property would show that Jerome and Heath received the same grade? Use the Transitive Property of Equality. Write a two-column proof. 37. Given: X is the midpoint of Prove: VW = ZY You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given the midpoint of two segments Use the properties that you have learned about congruent segments, midpoints, and equivalent expressions in algebra to walk through the proof. Given: X is the midpoint of Prove: VW = ZY Proof: 1. X is the midpoint of (Given) 2. (Definition. of midpoint) 3. WX = YX, VX = ZX (Definition of congruence) 4. VX = VW + WX, ZX = ZY + YX (Segment. Addition Postulate.) 5. VW + WX = ZY + YX (Substitution) 6. VW = ZY (Subtraction Prop.) 1. X is the midpoint of (Given) 2. (Def. of midpoint) 3. WX = YX, VX = ZX (Def. of ) 4. VX = VW + WX, ZX = ZY + YX (Seg. Add. Post.) 5. VW + WX = ZY + YX(Subs.) 6. VW = ZY (Subt. Prop.) Trans. Prop. esolutions Manual - Powered by Cognero Page 8

9 38. Given: AB = DC Prove: AC = DB You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two segments of equal length. Use the properties that you have learned about congruent segments and equivalent expressions in algebra to walk through the proof. 39. GEOGRAPHY Leandro is planning to drive from Kansas City to Minneapolis along Interstate 35. The map he is using gives the distance from Kansas City to Des Moines as 194 miles and from Des Moines to Minneapolis as 243 miles. What allows him to conclude that the distance he will be driving is 437 miles from Kansas City to Minneapolis? Assume that Interstate 35 forms a straight line. Interstate 35 forms a straight line and Des Moines is between Kansas City and Minneapolis. So, by Segment Addition Postulate the total distance from Kansas City to Minneapolis is the sum of the distances from Kansas City to Des Moines and Des Moines to Minneapolis, that is 437 miles. Given: AB = DC Prove: AC = DB Proof: 1. AB = DC (Given) 2. BC = BC (Reflexive Property) 3. AB + BC = DC + BC (Addition Property) 4. AB + BC = AC, DC + BC = DB (Segment Addition Property) 5. AC = DB (Substitution) 1. AB = DC (Given) 2. BC = BC (Refl. Prop.) 3. AB + BC = DC + BC (Add. Prop.) 4. AB + BC = AC, DC + BC = DB (Seg. Add. Post.) 5. AC = DB (Subs.) Seg. Add. Post. Find the measure of each angle. 40. Since the measure of the linear pair of 5 is 90, 90 esolutions Manual - Powered by Cognero Page 9

10 41. Angle 6 and the angle with measure 53 form a linear pair. So, 43. PROOF Write a two-column proof. Given: Prove: The 7 and the angle with measure 53 are vertical angles. So, they are congruent. 53 You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two sets of congruent angles. Use the properties that you have learned about congruent angles and equivalent expressions in algebra to walk through the proof. Given: Prove: Proof: 1. (Given) 2. (Definition of congruence) 3. (Addition Property) 4. (Angle Addition Postulate) 5. (Substitution) 6. (Definition. of congruence) 1. (Given) 2. (Def. of ) 3. (Add. Prop.) 4. Post.) ( Add. 5. (Subs.) 6. (Def. of ) esolutions Manual - Powered by Cognero Page 10