5-6 Proving Lines Parallel
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1 Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 1. j k; converse of corresponding angles postulate 3. alternate exterior angles converse 5. SHORT RESPONSE Find x so that m n. Show your work RECREATION Is it possible to prove that the backrest and footrest of the lounging beach chair are parallel? If so, explain how. If not, explain why not. Sample answer: Yes; since the alternate exterior angles are congruent, the backrest and footrest are parallel. esolutions Manual - Powered by Cognero Page 1
2 Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 9. u v; Alternate Exterior Angles Converse r s; Consecutive Interior Angles Converse u v; Alternate Interior Angles Converse r s; Corresponding Angles Converse Find x so that m n. Identify the postulate or theorem you used ; Conv. Corr. s Post ; Consec. Int. s Conv. esolutions Manual - Powered by Cognero Page 2
3 21. 36; Alt. Ext. s Conv. 23. PROOF Copy and complete the proof of Theorem 3.6. Given: 1 and 2 are supplementary. Prove: esolutions Manual - Powered by Cognero Page 3
4 PROOF Write a two-column proof for each of the following. 25. Given: Prove: Proof: Statements (Reasons) , (Given) (Corr. s postulate) (Trans. Prop.) 4. (If alternate s are, then lines are.) 27. Given: Prove: Proof: Statements (Reasons) 1. ABC ADC, m A + m ABC = 180 (Given) 2. m ABC = m ADC (Def. of s) 3. m A + m ADC = 180 (Substitution) 4. A and ADC are supplementary. (Def. of suppl. s) 5. (If consec. int. s are suppl., then lines are.) 29. MAILBOXES Mail slots are used to make the organization and distribution of mail easier. In the mail slots shown, each slot is perpendicular to each of the sides. Explain why you can conclude that the slots are parallel. Refer to Page 212. The Converse of the Perpendicular Transversal Theorem states that two coplanar lines perpendicular to the same line are parallel. Since the slots are perpendicular to each of the sides, the slots are parallel. Since any pair of slots is perpendicular the sides, they are also parallel. esolutions Manual - Powered by Cognero Page 4
5 31. PROOF Write a two-column proof of Theorem 3.7. Given: 1 2 Prove: Proof: Statements (Reasons) (Given) (Vertical s are ) (Transitive Prop.) 4. (If corr s are, then lines are.) Determine whether lines r and s are parallel. Justify your answer. 33. r s; Sample answer: The corresponding angles are congruent. Since the measures of angles are equal, the lines are parallel. 35. r s; Sample answer: The alternate exterior angles are congruent. Since the measures of angles are equal, the lines are parallel. esolutions Manual - Powered by Cognero Page 5
6 37. ERROR ANALYSIS Sumi and Daniela are determining which lines are parallel in the figure at the right. Sumi says that since 1 2, Daniela disagrees and says that since 1 2, Is either of them correct? Explain. Daniela; 1 and 2 are alternate interior angles for and, so if alternate interior angles are congruent, then the lines are parallel. 39. CHALLENGE Use the figure to prove that two lines parallel to a third line are parallel to each other. Sample Answer: Given: a b and b c Prove: a c Proof: Statements (Reasons) 1. a b and b c (Given) (Alternate Interior 's Theorem) (Vertical. 's are ) (Alternate Interior. 's Theorem) (Trans. Prop.) 6. a c (Alternate Interior. 's Converse Theorem) esolutions Manual - Powered by Cognero Page 6
7 41. CHALLENGE Refer to the figure at the right. a. If m 1 + m 2 = 180, prove that a c. b. Given that a c, if m 1 + m 3 = 180, prove that t c. a. We know that m 1 + m 2 = 180. Since 2 and 3 are linear pairs, m 2 + m 3 = 180. By substitution, m 1 + m 2 = m 2 + m 3. By subtracting m 2 from both sides we get m 1 = m , by the definition of congruent angles. Therefore, since the corresponding angles are congruent. b. We know that and m 1 + m 3 = 180. Since 1 and 3 are corresponding angles, they are congruent and their measures are equal. By substitution, m 3 + m 3 = 180 or 2 m 3 = 180. By dividing both sides by 2, we get m 3 = 90. Therefore, t c since they form a right angle. 43. WRITING IN MATH Can a pair of angles be supplementary and congruent? Explain your reasoning. Yes; sample answer: A pair of angles can be both supplementary and congruent if the measure of both angles is 90, since the sum of the angle measures would be ALGEBRA The expression is equivalent to F 13 G H J G 47. SAT/ACT If x 2 = 25 and y 2 = 9, what is the greatest possible value of (x y) 2? F 4 G 58 H 64 J 70 H esolutions Manual - Powered by Cognero Page 7
8 49. Write an equation in slope-intercept form of the line having the given slope and y-intercept. 51. ROAD TRIP Anne is driving 400 miles to visit Niagara Falls. She manages to travel the first 100 miles of her trip in two hours. If she continues at this rate, how long will it take her to drive the remaining distance? 6 hours Find a counterexample to show that each conjecture is false. 53. Given: points W, X, Y, and Z Conjecture: W, X, Y, and Z are noncollinear. Find the perimeter or circumference and area of each figure. Round to the nearest tenth m; 3.5 m Find x and y so that and are perpendicular. 10, 8.3 esolutions Manual - Powered by Cognero Page 8
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