Name: Class: Date: ID: A Geometry Study Guide Matching Match each vocabulary term with its definition. a. conjecture e. biconditional statement b. inductive reasoning f. hypothesis c. deductive reasoning g. counterexample d. conclusion h. conditional statement 1. an example that proves that a conjecture or statement is false 2. a statement that is believed to be true 3. the part of a conditional statement following the word then 4. the part of a conditional statement following the word if 5. the process of reasoning that a rule or statement is true because specific cases are true 6. a statement that can be written in the form if p, then q, where p is the hypothesis and q is the conclusion Match each vocabulary term with its definition. a. conclusion e. hypothesis b. converse f. truth value c. inverse g. contrapositive d. negation 7. for a statement, either true (T) or false (F) 8. operations that undo each other 9. the contradiction of a statement by using not, written as 10. the statement formed by exchanging the hypothesis and conclusion of a conditional statement 11. the statement formed by both exchanging and negating the hypothesis and conclusion Match each vocabulary term with its definition. a. logically equivalent statements f. quadrilateral b. deductive reasoning g. pentagon c. biconditional statement h. definition d. inductive reasoning i. triangle e. polygon 12. a statement that describes a mathematical object and can be written as a true biconditional statement 13. statements that have the same truth value 14. a four-sided polygon 15. a closed plane figure formed by three or more segments such that each segment intersects exactly two other segments only at their endpoints and no two segments with a common endpoint are collinear 16. the process of using logic to draw conclusions 17. a statement that can be written in the form p if and only if q 18. a three-sided polygon 1
Name: ID: A Match each vocabulary term with its definition. a. deductive reasoning e. inductive reasoning b. paragraph proof f. two-column proof c. proof g. flowchart proof d. theorem 19. a style of proof in which the statements are written in the left-hand column and the reasons are written in the right-hand column 20. a statement that has been proven 21. a style of proof in which the statements and reasons are presented in paragraph form 22. an argument that uses logic to show that a conclusion is true 23. a style of proof that uses boxes and arrows to show the structure of the proof Short Answer 24. Find the next item in the pattern 2, 3, 5, 7, 11,... 25. Complete the conjecture. The sum of two odd numbers is. 26. The table shows the population 65 years and over by age and sex according to the US Census Bureau, Census 2000 Summary file. Make a conjecture based on the data. Population 65 Years and Over by Age and Sex: 2000 (numbers in thousands) 65 to 74 years 75 to 84 years 85 years and over Women 10,088 7,482 3,013 Men 8,303 4,879 1,227 27. Show that the conjecture is false by finding a counterexample. If a > b, then a b > 0. 28. Make a table of values for the rule x 2 16x + 64 when x is an integer from 1 to 6. Make a conjecture about the type of number generated by the rule. Continue your table. What value of x generates a counterexample? 29. Identify the hypothesis and conclusion of the conditional statement. If it is raining then it is cloudy. 30. Write a conditional statement from the statement. A horse has 4 legs. 31. Determine if the conditional statement is true. If false, give a counterexample. If a figure has four sides, then it is a square. 32. Write the converse, inverse, and contrapositive of the conditional statement. If an animal is a bird, then it has two eyes. 2
Name: ID: A 33. How many true conditional statements may be written using the following statements? n is a rational number. n is an integer. n is a whole number. 34. There is a myth that a duck s quack does not echo. A group of scientists observed a duck in a special room, and they found that the quack does echo. Therefore, the myth is false. Is the conclusion a result of inductive or deductive reasoning? 35. Determine if the conjecture is valid by the Law of Detachment. Given: If Tommy makes cookies tonight, then Tommy must have an oven. Tommy has an oven. Conjecture: Tommy made cookies tonight. 36. Determine if the conjecture is valid by the Law of Syllogism. Given: If you are in California, then you are in the west coast. If you are in Los Angeles, then you are in California. Conjecture: If you are in Los Angeles, then you are in the west coast. 37. Use the Law of Syllogism to draw a conclusion from the given information. Given: If two lines are perpendicular, then they form right angles. If two lines meet at a 90 angle, then they are perpendicular. Two lines meet at a 90 angle. 38. Consider the two conditional statements. Draw a conclusion from the given conditional statements, and write the contrapositive of each conditional statement. Then, draw a conclusion from the two contrapositives. How does the first conclusion relate to the second conclusion? If you are eating a banana, then you are eating fruit. If you are eating fruit, then you are eating food. 39. Write the conditional statement and converse within the biconditional. A rectangle is a square if and only if all four sides of the rectangle have equal lengths. 40. For the conditional statement, write the converse and a biconditional statement. If a figure is a right triangle with sides a, b, and c, then a 2 + b 2 = c 2. 41. Determine if the biconditional is true. If false, give a counterexample. A figure is a square if and only if it is a rectangle. 42. Write the definition as a biconditional. An acute angle is an angle whose measure is less than 90. 43. What is the truth value of the biconditional formed from the conditional, If B is the midpoint of A and C, then AB = BC. Explain. 44. Solve the equation 4x 6 = 34. Write a justification for each step. 4x 6 = 34 Given equation +6 +6 [1] 4x = 40 Simplify. 4x = 40 4 4 [2] x = 10 Simplify. 3
Name: ID: A 45. A gardener has 26 feet of fencing for a garden. To find the width of the rectangular garden, the gardener uses the formula P = 2l + 2w, where P is the perimeter, l is the length, and w is the width of the rectangle. The gardener wants to fence a garden that is 8 feet long. How wide is the garden? Solve the equation for w, and justify each step. P = 2l + 2w Given equation 26 = 2(8) + 2w [1] 26 = 16 + 2w Simplify. 16 = 16 10 = 2w Subtraction Property of Equality Simplify. 10 2 = 2w 2 [2] 5 = w Simplify. w = 5 Symmetric Property of Equality 46. Write a justification for each step. m JKL = 100 m JKL = m JKM + m MKL [1] 100 = (6x + 8) + (2x 4) Substitution Property of Equality 100 = 8x + 4 Simplify. 96 = 8x Subtraction Property of Equality 12 = x [2] x = 12 Symmetric Property of Equality 47. Identify the property that justifies the statement. AB CD and CD EF. So AB EF. 48. Write a justification for each step, given that EG = FH. EG = FH Given information EG = EF + FG [1] FH = FG + GH Segment Addition Postulate EF + FG = FG + GH [2] EF = GH Subtraction Property of Equality 4
Name: ID: A 49. Fill in the blanks to complete the two-column proof. Given: 1 and 2 are supplementary. m 1 = 135 Prove: m 2 = 45 Proof: Statements Reasons 1. 1 and 2 are supplementary. 1. Given 2. [1] 2. Given 3. m 1 + m 2 = 180 3. [2] 4. 135 + m 2 = 180 4. Substitution Property 5. m 2 = 45 5. [3] 5
Name: ID: A 50. Use the given plan to write a two-column proof. Given: m 1 + m 2 = 90, m 3 + m 4 = 90, m 2 = m 3 Prove: m 1 = m 4 Plan: Since both pairs of angle measures add to 90, use substitution to show that the sums of both pairs are equal. Since m 2 = m 3, use substitution again to show that sums of the other pairs are equal. Use the Subtraction Property of Equality to conclude that m 1 = m 4. Complete the proof. Proof: Statements Reasons 1. m 1 + m 2 = 90 1. Given 2. [1] 2. Given 3. m 1 + m 2 = m 3 + m 4 3. Substitution Property 4. m 2 = m 3 4. Given 5. m 1 + m 2 = m 2 + m 4 5. [2] 6. m 1 = m 4 6. [3] 51. Two angles with measures (2x 2 + 3x 5) and (x 2 + 11x 7) are supplementary. Find the value of x and the measure of each angle. 6
Name: ID: A 52. Use the given flowchart proof to write a two-column proof of the statement AF FD. Flowchart proof: AB = CD; BF = FC Given AB + BF = FC + CD AB + BF = AF FC + CD = FD Segment Addition Postulate AF = FD AF FD Addition Property of Equality Complete the proof. Substitution Definition of congruent segments Two-column proof: Statements Reasons 1. AB = CD; BF = FC 1. Given 2. [1] 2. Addition Property of Equality 3. [2] 3. Segment Addition Postulate 4. AF = FD 4. Substitution 5. AF FD 5. Definition of congruent segments 7
Name: ID: A 53. Use the given two-column proof to write a flowchart proof. Given: 1 4 Prove: m 2 = m 3 Two-column proof: Statements Reasons 1. 1 4 1. Given 2. 1 and 2 are supplementary. 3 and 4 2. Definition of linear pair are supplementary. 3. 2 3 3. Congruent Supplements Theorem 4. m 2 = m 3 4. Definition of congruent segments Complete the proof. Flowchart proof: 1 4 Given [1] 2 3 m 2 = m 3 Definition of linear pair [2] Definition of congruent segments 8
Name: ID: A 54. Use the given paragraph proof to write a two-column proof. Given: BAC is a right angle. 1 3 Prove: 2 and 3 are complementary. Paragraph proof: Since BAC is a right angle, m BAC = 90 by the definition of a right angle. By the Angle Addition Postulate, m BAC = m 1 + m 2. By substitution, m 1 + m 2 = 90. Since 1 3, m 1 = m 3 by the definition of congruent angles. Using substitution, m 3 + m 2 = 90. Thus, by the definition of complementary angles, 2 and 3 are complementary. Complete the proof. Two-column proof: Statements Reasons 1. BAC is a right angle. 1 3 1. Given 2. m BAC = 90 2. Definition of a right angle 3. m BAC = m 1 + m 2 3. [1] 4. m 1 + m 2 = 90 4. Substitution 5. m 1 = m 3 5. [2] 6. m 3 + m 2 = 90 6. Substitution 7. 2 and 3 are complementary. 7. Definition of complementary angles 9
Name: ID: A 55. Use the given two-column proof to write a paragraph proof. Given: 1 and 2 are supplementary. 1 2. 2 3. Prove: 3 is a right angle. Two-column proof: Statements Reasons 1. 1 and 2 are supplementary. 1. Given 1 2. 2 3. 2. 1 and 2 are right angles. 2. Congruent supplementary angles form right angles. 3. m 2 = 90 3. Definition of a right angle 4. m 2 = m 3 4. Definition of congruent angles 5. m 3 = 90 5. Substitution 6. 3 is a right angle. 6. Definition of a right angle Complete the proof. Paragraph proof: 2 3 is a given statement. Since [1], 1 and 2 are right angles. By the definition of a right angle, m 2 = 90. By the definition of congruent angles, [2]. Then, m 3 = 90 by substitution. Therefore, 3 is a right angle by the definition of a right angle. 56. Two lines intersect to form two pairs of vertical angles. 1 with measure (20x + 7)º and 3 with measure (5x + 7y + 49)º are vertical angles. 2 with measure (3x 2y + 30)º and 4 are vertical angles. Find the values x and y and the measures of all four angles. 10
Geometry Study Guide Answer Section MATCHING 1. ANS: G PTS: 1 DIF: Basic REF: Page 75 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 2. ANS: A PTS: 1 DIF: Basic REF: Page 74 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 3. ANS: D PTS: 1 DIF: Basic REF: Page 81 TOP: 2-2 Conditional Statements 4. ANS: F PTS: 1 DIF: Basic REF: Page 81 TOP: 2-2 Conditional Statements 5. ANS: B PTS: 1 DIF: Basic REF: Page 74 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 6. ANS: H PTS: 1 DIF: Basic REF: Page 81 TOP: 2-2 Conditional Statements 7. ANS: F PTS: 1 DIF: Basic REF: Page 82 TOP: 2-2 Conditional Statements 8. ANS: C PTS: 1 DIF: Basic REF: Page 83 TOP: 2-2 Conditional Statements 9. ANS: D PTS: 1 DIF: Basic REF: Page 82 TOP: 2-2 Conditional Statements 10. ANS: B PTS: 1 DIF: Basic REF: Page 83 TOP: 2-2 Conditional Statements 11. ANS: G PTS: 1 DIF: Basic REF: Page 83 TOP: 2-2 Conditional Statements 12. ANS: H PTS: 1 DIF: Basic REF: Page 97 TOP: 2-4 Biconditional Statements and Definitions 13. ANS: A PTS: 1 DIF: Basic REF: Page 83 TOP: 2-2 Conditional Statements 14. ANS: F PTS: 1 DIF: Basic REF: Page 98 TOP: 2-4 Biconditional Statements and Definitions 15. ANS: E PTS: 1 DIF: Basic REF: Page 98 TOP: 2-4 Biconditional Statements and Definitions 16. ANS: B PTS: 1 DIF: Basic REF: Page 88 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures 17. ANS: C PTS: 1 DIF: Basic REF: Page 96 TOP: 2-4 Biconditional Statements and Definitions 18. ANS: I PTS: 1 DIF: Basic REF: Page 98 TOP: 2-4 Biconditional Statements and Definitions 19. ANS: F PTS: 1 DIF: Basic REF: Page 111 TOP: 2-6 Geometric Proof 1
20. ANS: D PTS: 1 DIF: Basic REF: Page 110 TOP: 2-6 Geometric Proof 21. ANS: B PTS: 1 DIF: Basic REF: Page 120 TOP: 2-7 Flowchart and Paragraph Proofs 22. ANS: C PTS: 1 DIF: Basic REF: Page 104 TOP: 2-5 Algebraic Proof 23. ANS: G PTS: 1 DIF: Basic REF: Page 118 TOP: 2-7 Flowchart and Paragraph Proofs SHORT ANSWER 24. ANS: 13 The prime numbers make up the pattern. The next prime is 13. PTS: 1 DIF: Basic REF: Page 74 OBJ: 2-1.1 Identifying a Pattern NAT: 12.5.1.a STA: GE1.0 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 25. ANS: even List some examples and look for a pattern. 3 + 5 = 8 3 + 7 = 10 5 + 7 = 12 5 + 9 = 14 PTS: 1 DIF: Basic REF: Page 74 OBJ: 2-1.2 Making a Conjecture NAT: 12.3.5.a STA: GE1.0 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 26. ANS: Women outnumbered men in the 65 years and over population. For every age group 65 years and over, the number of women is greater than the number of men. The data supports the conjecture that women outnumbered men in the 65 years and over population. PTS: 1 DIF: Average REF: Page 75 OBJ: 2-1.3 Application NAT: 12.3.5.a STA: GE1.0 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 2
27. ANS: a = 11, b = 3 Pick values for a and b that follow the condition a > b. Then substitute them into the second inequality to see if the conjecture holds. Values of a and b a > b Let a = 4 and b = 1. 4 > 1 Let a = 11 and b = 3. 11 > 3 Let a = 11 and b = 3. 11 > 3 a b > 0 Conclusion 4 > 0 1 The conjecture holds. 11 > 0 3 The conjecture holds. 11 3 < 0 The conjecture is false. a = 11 and b = 3 is a counterexample. The conjecture is false when a is positive and b is negative. PTS: 1 DIF: Average REF: Page 76 OBJ: 2-1.4 Finding a Counterexample NAT: 12.3.5.a STA: GE3.0 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 28. ANS: The pattern appears to be an decreasing set of perfect squares. x = 9 generates a counterexample. x values 1 2 3 4 5 6 x 2 16x + 64 49 36 25 16 9 4 The pattern appears to be a decreasing set of perfect squares. When x = 7, 7 2 16(7) + 64 = 1. This follows the pattern. When x = 8, 8 2 16(8) + 64 = 0. This follows the pattern. When x = 9, 9 2 16(9) + 64 = 1. This does not follow the pattern. Thus, x = 9 generates a counterexample. PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: GE3.0 TOP: 2-1 Using Inductive Reasoning to Make Conjectures 29. ANS: Hypothesis: It is raining. Conclusion: It is cloudy. For an if-then conditional statement, the hypothesis is the part following the word if. Hypothesis: It is raining. Conclusion: It is cloudy. PTS: 1 DIF: Basic REF: Page 81 OBJ: 2-2.1 Identifying the Parts of a Conditional Statement STA: GE3.0 TOP: 2-2 Conditional Statements NAT: 12.3.5.a 3
30. ANS: If it is a horse then it has 4 legs. Identify the hypothesis and conclusion. Hypothesis A horse If it is a horse, Conclusion has 4 legs. then it has 4 legs. PTS: 1 DIF: Average REF: Page 82 OBJ: 2-2.2 Writing a Conditional Statement NAT: 12.3.5.a STA: GE3.0 TOP: 2-2 Conditional Statements 31. ANS: False; A rectangle has four sides, and it is not a square. There are several figures with four sides that are not squares. So, the conditional statement is false. Counterexample: A rectangle has four sides, and it is not a square. PTS: 1 DIF: Basic REF: Page 82 OBJ: 2-2.3 Analyzing the Truth Value of a Conditional Statement NAT: 12.3.5.a STA: GE3.0 TOP: 2-2 Conditional Statements 32. ANS: Converse: If an animal has two eyes, then it is a bird. Inverse: If an animal is not a bird, then it does not have two eyes. Contrapositive: If an animal does not have two eyes, then it is not a bird. Conditional: p q If an animal is a bird, then it has two eyes. Converse: q p If an animal has two eyes, then it is a bird. Inverse: p q If an animal is not a bird, then it does not have two eyes. Contrapositive: q p If an animal does not have two eyes, then it is not a bird. PTS: 1 DIF: Average REF: Page 83 OBJ: 2-2.4 Application NAT: 12.3.5.a STA: GE3.0 TOP: 2-2 Conditional Statements 4
33. ANS: 3 conditional statements Create a Venn diagram that represents the set of rational numbers, integers, and whole numbers. A conditional statement will be true when the set referred to in the hypothesis is a subset of the set referred to in the conclusion. If n is a whole number, then n is an integer. If n is a whole number, then n is a rational number. If n is an integer, then n is a rational number. You can write three true conditional statements using the statements given. PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: GE3.0 TOP: 2-2 Conditional Statements 34. ANS: Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning. The scientists determined the myth was false because they heard an echo by observing the duck. Inductive reasoning is based on a pattern of observation. PTS: 1 DIF: Basic REF: Page 88 OBJ: 2-3.1 Application NAT: 12.3.5.a STA: GE3.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures 35. ANS: The conjecture is not valid, because Tommy could have an oven but he could make something besides cookies tonight. Identify the hypothesis and conclusion in the given conditional. Hypothesis: Tommy makes cookies tonight. Conclusion: Tommy must have an oven. The given statement, Tommy has an oven, matches the conclusion, but that does not mean that the hypothesis is true. Tommy could have an oven but he could use it for something besides cookies. The conjecture is not valid. PTS: 1 DIF: Average REF: Page 89 OBJ: 2-3.2 Verifying Conjectures by Using the Law of Detachment NAT: 12.3.5.a STA: GE3.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures 5
36. ANS: Yes, the conjecture is valid. Let p, q, and r represent the following. p: You are in California. q: You are in the west coast. r: You are in Los Angeles. You are given that p q and r p Since p is the conclusion of the second statement and the hypothesis of the first statement, reorder the statements like this r p and p q. By the Law of Syllogism, if r p and p q are true, then r q is true. r q is the statement, If you are in Los Angeles, then you are in the west coast. PTS: 1 DIF: Average REF: Page 89 OBJ: 2-3.3 Verifying Conjectures by Using the Law of Syllogism NAT: 12.3.5.a STA: GE3.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures 37. ANS: Conclusion: The lines form a right angle. Two lines meet at a 90 angle. It given that if two lines meet at a 90 angle, then they are perpendicular. It is also given that if two lines are perpendicular, then they form right angles. So, the conclusion is: The lines form a right angle. PTS: 1 DIF: Basic REF: Page 90 OBJ: 2-3.4 Applying the Laws of Deductive Reasoning NAT: 12.3.5.a STA: GE1.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures 38. ANS: If you are eating a banana, then you are eating food. If you are not eating fruit, then you are not eating a banana. If you are not eating food, then you are not eating fruit. If you are not eating food, then you are not eating a banana. The second conclusion is the contrapositive of the first conclusion. The conclusion from the given conditional statements is, if you are eating a banana, then you are eating food. The contrapositive of the first statement is, if you are not eating fruit, then you are not eating a banana. The contrapositive of the second statement is, if you are not eating food, then you are not eating fruit. The conclusion from the contrapositives is, if you are not eating food, then you are not eating a banana. The second conclusion is the contrapositive of the first conclusion. PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: GE3.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures 6
39. ANS: Conditional: If all four sides of the rectangle have equal lengths, then it is a square. Converse: If a rectangle is a square, then its four sides have equal lengths. Let p and q represent the following. p: A rectangle is a square. q: All four sides of the rectangle have equal lengths. The two parts of the biconditional p q are p q and q p. Conditional: If all four sides of the rectangle have equal lengths, then it is a square. Converse: If a rectangle is a square, then its four sides have equal lengths. PTS: 1 DIF: Average REF: Page 96 OBJ: 2-4.1 Identifying the Conditionals within a Biconditional Statement NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions 40. ANS: Converse: If a 2 + b 2 = c 2, then the figure is a right triangle with sides a, b, and c. Biconditional: A figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. Let p and q represent the following. p: It is a right triangle. q: a 2 + b 2 = c 2. The given conditional is p q. The converse is q p. If a 2 + b 2 = c 2, then the figure is a right triangle with sides a, b, and c. The biconditional is p q. A figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. PTS: 1 DIF: Average REF: Page 97 OBJ: 2-4.2 Writing a Biconditional Statement NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions 41. ANS: The biconditional is false. A rectangle does not necessarily have four congruent sides. Conditional: If a figure is a square, then it is a rectangle. True. Converse: If a figure is a rectangle, then it is a square. False. A rectangle does not necessarily have four congruent sides. Because the converse is false, the biconditional is false. PTS: 1 DIF: Basic REF: Page 97 OBJ: 2-4.3 Analyzing the Truth Value of a Biconditional Statement NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions 7
42. ANS: An angle is acute if and only if its measure is less than 90. Think of the definition as being reversible. Let p be an angle is acute. Let q be its measure is less than 90. Conditional: If an angle is acute, then its measure is less than 90. Converse: If an angle s measure is less than 90, then it is acute. Biconditional: An angle is acute if and only if its measure is less than 90. PTS: 1 DIF: Basic REF: Page 98 OBJ: 2-4.4 Writing Definitions as Biconditional Statements NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions 43. ANS: The conditional is true. The converse, If AB = BC then B is the midpoint of AC is false. Since the conditional is true but the converse is false, the biconditional is false. The conditional statement is the definition of a midpoint, and is a true statement. The converse is false. The picture displays a counterexample. AB = BC, but B is not on AC. Therefore, B is not the midpoint of AB. If either the conditional or the converse is false, the biconditional is false. PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions 8
44. ANS: [1] Addition Property of Equality; [2] Division Property of Equality 4x 6 = 34 Given equation +6 +6 [1] Addition Property of Equality 4x = 40 Simplify. 4x = 40 4 4 [2] Division Property of Equality x = 10 Simplify. PTS: 1 DIF: Basic REF: Page 104 OBJ: 2-5.1 Solving an Equation in Algebra NAT: 12.5.2.e STA: 1A5.0 TOP: 2-5 Algebraic Proof 45. ANS: [1] Substitution Property of Equality [2] Division Property of Equality The garden is 5 ft wide. P = 2l + 2w Given equation 26 = 2(8) + 2w [1] Substitution Property of Equality 26 = 16 + 2w Simplify. 16 = 16 10 = 2w Subtraction Property of Equality Simplify. 10 2 = 2w 2 [2] Division Property of Equality 5 = w Simplify. w = 5 Symmetric Property of Equality PTS: 1 DIF: Average REF: Page 105 OBJ: 2-5.2 Problem-Solving Application NAT: 12.5.2.e STA: 1A5.0 TOP: 2-5 Algebraic Proof 46. ANS: [1] Angle Addition Postulate [2] Division Property of Equality m JKL = m JKM + m MKL [1] Angle Addition Postulate 100 = (6x + 8) + (2x 4) Substitution Property of Equality 100 = 8x + 4 Simplify. 96 = 8x Subtraction Property of Equality 12 = x [2] Division Property of Equality x = 12 Symmetric Property of Equality PTS: 1 DIF: Average REF: Page 106 OBJ: 2-5.3 Solving an Equation in Geometry STA: GE1.0 TOP: 2-5 Algebraic Proof NAT: 12.5.2.e 9
47. ANS: Transitive Property of Congruence The Transitive Property of Congruence states that if figure A figure B and figure B figure C, then figure A figure C. PTS: 1 DIF: Basic REF: Page 106 OBJ: 2-5.4 Identifying Properties of Equality and Congruence NAT: 12.5.2.e STA: GE4.0 TOP: 2-5 Algebraic Proof 48. ANS: [1] Segment Addition Postulate [2] Substitution Property of Equality EG = FH Given information EG = EF + FG Segment Addition Postulate FH = FG + GH Segment Addition Postulate EF + FG = FG + GH Substitution Property of Equality EF = GH Subtraction Property of Equality PTS: 1 DIF: Average REF: Page 110 OBJ: 2-6.1 Writing Justifications NAT: 12.3.5.a STA: GE2.0 TOP: 2-6 Geometric Proof 49. ANS: [1] m 1 = 135 [2] Definition of supplementary angles [3] Subtraction Property of Equality Proof: Statements Reasons 1. 1 and 2 are supplementary. 1. Given 2. m 1 = 135 2. Given 3. m 1 + m 2 = 180 3. Definition of supplementary angles 4. 135 + m 2 = 180 4. Substitution Property 5. m 2 = 45 5. Subtraction Property of Equality PTS: 1 DIF: Average REF: Page 111 OBJ: 2-6.2 Completing a Two-Column Proof STA: GE2.0 TOP: 2-6 Geometric Proof NAT: 12.3.5.a 10
50. ANS: [1] m 3 + m 4 = 90 [2] Substitution Property [3] Subtraction Property of Equality Proof: Statements Reasons 1. m 1 + m 2 = 90 1. Given 2. m 3 + m 4 = 90 2. Given 3. m 1 + m 2 = m 3 + m 4 3. Substitution Property 4. m 2 = m 3 4. Given 5. m 1 + m 2 = m 2 + m 4 5. Substitution Property 6. m 1 = m 4 6. Subtraction Property of Equality PTS: 1 DIF: Average REF: Page 112 OBJ: 2-6.3 Writing a Two-Column Proof from a Plan STA: GE2.0 TOP: 2-6 Geometric Proof 51. ANS: x = 6; 85 ; 95 Step 1 Create an equation The angles are supplements and their sum equals 180. (2x 2 + 3x 5) + (x 2 + 11x 7) = 180 NAT: 12.3.5.a Step 2 Solve the equation 3x 2 + 14x 12 = 180 3x 2 + 14x 192 = 0 (3x + 32)(x 6) = 0 x = 32 3 or 6. When x = 32 3, the measurement of the second angle is x 2 + 11x 7 = 10.6. Angles cannot have negative measurements, so x = 6. Step 3 Solve for the required values The measurement of the first angle is 2x 2 + 3x 5 = 2(6) 2 + 3(6) 5 = 85. The measurement of the second angle is x 2 + 11x 7 = (6) 2 + 11(6) 7= 95. PTS: 1 DIF: Advanced NAT: 12.2.1.f STA: 6MG2.2 TOP: 2-6 Geometric Proof 52. ANS: [1] AB + BF = FC + CD [2] AB + BF = AF ;FC + CD = FD In a flowchart, reasons flow from the statement above. The statement above Reason 2 is AB + BF = FC + CD. The statement above Reason 3 is AB + BF = AF ; FC + CD = FD. PTS: 1 DIF: Average REF: Page 118 OBJ: 2-7.1 Reading a Flowchart Proof NAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 11
53. ANS: [1] 1 and 2 are supplementary; 3 and 4 are supplementary [2] Congruent Supplements Theorem In a flowchart, reasons follow statements. Using the two-column proof, the statement that leads to Reason 2 is 1 and 2 are supplementary; 3 and 4 are supplementary. The reason that follows Statement 3 is Congruent Supplements Theorem. PTS: 1 DIF: Average REF: Page 119 OBJ: 2-7.2 Writing a Flowchart Proof NAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 54. ANS: [1] Angle Addition Postulate [2] Definition of congruent angles Two-column proof: Statements Reasons 1. BAC is a right angle. 1 3 1. Given 2. m BAC = 90 2. Definition of a right angle 3. m BAC = m 1 + m 2 3. Angle Addition Postulate 4. m 1 + m 2 = 90 4. Substitution 5. m 1 = m 3 5. Definition of congruent angles 6. m 3 + m 2 = 90 6. Substitution 7. 2 and 3 are complementary. 7. Definition of complementary angles PTS: 1 DIF: Average REF: Page 120 OBJ: 2-7.3 Reading a Paragraph Proof NAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 55. ANS: [1] congruent, supplementary angles form right angles [2] m 2 = m 3 In a paragraph proof, statements and reasons appear together. The reason following the statement, 1 and 2 are right angles, is congruent, supplementary angles form right angles. The statement preceding the reason, Definition of congruent angles, is m 2 = m 3. PTS: 1 DIF: Average REF: Page 121 OBJ: 2-7.4 Writing a Paragraph Proof NAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs 12
56. ANS: x = 7; y = 9; 147 ; 147 ; 33 ; 33 Step 1 Create a system of equations. m 1 = m 3 20x + 7 = 5x + 7y + 49 15x 7y = 42 The sum of the measures of supplementary angles equals 180. m 1 + 2 = 180 20x + 7 + 3x 2y + 30 = 180 23x 2y = 143 Create a system of equations. 15x 7y = 42 23x 2y = 143 Step 2 Solve the system of equations. 15x 7y = 42 23x 2y = 143 30x + 14y = 84 Multiply the first equation by 2. 161x 14y = 1001 Multiply the second equation by 7. 131x = 917 Add the two equations together. x = 7 Divide both sides by 131. Solve for y. Substitute x = 7 into 15x 7y = 42. 15(7) 7y = 42 y = 9 The values are x = 7 and y = 9. Step 3 Solve for the four angles. Angle 1: (20(7) + 7) = 147 Angle 2: (3(7) 2(9) + 30) = 33 Angle 3: (5(7) + 7(9) + 49) = 147 Angle 4 and angle 2 are vertical and thus have equal measures. The measurement of angle 4 is 33. The measures of all four angles are 147, 147, 33, and 33. PTS: 1 DIF: Advanced NAT: 12.2.1.f TOP: 2-7 Flowchart and Paragraph Proofs 13