Given: Prove: Proof: 3-5 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. 6. PROOF Copy and complete the proof of Theorem Given: j k; converse of corresponding angles postulate 2. j k; alternate interior angles converse 3. alternate exterior angles converse 4. m 6 + m 8 = 180 consecutive interior angles converse 5. SHORT RESPONSE Find x so that m n. Show your work. 7. 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. 20 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. Find x so that m n. Identify the postulate or theorem you used ; Alt. Ext. s Conv. 8. r s; Converse of Corresponding Angles Postulate u v; Alternate Exterior Angles Converse 22; Conv. Corr. s Post r s; Alternate Interior Angles Converse r s; Consecutive Interior Angles Converse ; Alt. Int. s Conv. 12. u v; Consecutive Interior Angles Converse 13. u v; Alternate Interior Angles Converse ; Consec. Int. s Conv. No lines can be proven. 15. r s; Corresponding Angles Converse ; Vert. s Thm and Consec. Int. s Conv. esolutions Manual - Powered by Cognero Page 2
3 24. CRAFTS Jacqui is making a stained glass piece. She cuts the top and bottom pieces at a 30 angle. If the corners are right angles, explain how Jacqui knows that each pair of opposite sides are parallel ; Alt. Ext. s Conv. 22. ORGANIZE IDEAS Wooden picture frames are often constructed using a miter box or miter saw. These tools allow you to cut at an angle of a given size. If each of the four pieces of framing material is cut at a 45 angle, will the sides of the frame be parallel? Explain your reasoning. Yes; when two pieces are put together, they form a 90 angle. Two lines that are perpendicular to the same line are parallel. Since the corners are right angles, each pair of opposite sides is perpendicular to the same line. Therefore, each pair of opposite sides is parallel. PROOF Write a two-column proof for each of the following. 25. Given: 23. PROOF Copy and complete the proof of Theorem 3.6. Given: 1 and 2 are supplementary , (Given) (Corr. s postulate) (Trans. Prop.) 4. (If alternate s are, then lines are.) esolutions Manual - Powered by Cognero Page 3
4 26. Given: 28. Given: (Given) 2. 2 and 4 are supplementary. (Cons. Int. s) 3. m 2 + m 4 = 180 (Def. of suppl. s) 4. m 3 + m 4 = 180 (Substitution) 5. 3 and 4 are supplementary. (Def. of suppl. s) 6. (If cons. int. s are suppl., then lines are.) 27. Given: 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.) , (Given) 2. (If alt. int. s are, then lines are.) 3. (Perpendicular Transversal Theorem) 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. 30. PROOF Write a paragraph proof of Theorem 3.8. Given: Since and, the measures of angle 1 and angle 2 are 90. Since and have the same measure, they are congruent. By the converse of Corresponding Angles Postulate,. esolutions Manual - Powered by Cognero Page 4
5 31. PROOF Write a two-column proof of Theorem 3.7. Given: r s; Sample answer: The alternate exterior angles are congruent. Since the measures of angles are equal, the lines are parallel (Given) (Vertical s are ) (Transitive Prop.) 4. (If corr s are, then lines are.) 32. ANALYZE RELATIONSHIPS Based upon the information given in the photo of the staircase, what is the relationship between each step? Explain your answer. Refer to Page r s; Sample answer: The alternate exterior angles are congruent. Since the measures of angles are equal, the lines are parallel. Each step is parallel to each other because the corresponding angles are congruent. 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. esolutions Manual - Powered by Cognero Page 5
6 36. MULTIPLE REPRESENTATIONS In this problem, you will explore the shortest distance between two parallel lines. a. GEOMETRIC Draw three sets of parallel lines k and, s and t, and x and y. For each set, draw the shortest segment shown below. and label points A and D as 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. b. TABULAR Copy the table below, measure ABC and BCD, and complete the table. c. VERBAL Make a conjecture about the angle the shortest segment forms with both parallel lines. a. Daniela; 1 and 2 are alternate interior angles for and, so if alternate interior angles are congruent, then the lines are parallel. 38. JUSTIFY ARGUMENTS Is Theorem 3.8 still true if the two lines are not coplanar? Draw a figure to justify your answer.. No; sample answer: In the figure shown, and but b. c. Sample answer: The angle that the segment forms with the parallel lines will always measure 90. esolutions Manual - Powered by Cognero Page 6
7 39. ORGANIZE IDEAS Use the figure to prove that two lines parallel to a third line are parallel to each other. 41. JUSTIFY ARGUMENTS Refer to the figure. Sample Answer: Given: a b and b c a c 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) 40. ORGANIZE IDEAS Draw a triangle ABC. a. Construct the line parallel to through point A. b. Use measurement to justify that the line you constructed is parallel to. c. Use mathematics to justify this construction. a. b. Sample answer: Using a straightedge, the lines are equidistant. So they are parallel. c. Sample answer: is a transversal for and. ABC was copied to construct EAD. So, ABC EAD. ABC and EAD are corresponding angles, so by the converse of corresponding angles postulate, 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. 42. WRITING IN MATH Summarize the five methods used in this lesson to prove that two lines are parallel. Sample answer: Use a pair of alternate exterior angles that are congruent and cut by transversal; show that a pair of consecutive interior angles are supplementary; show that alternate interior angles are congruent; show two coplanar lines are perpendicular to same line; show corresponding angles are congruent. 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 180. esolutions Manual - Powered by Cognero Page 7
8 44. In this diagram, 3 4. What is the appropriate reason for concluding q r? 47. A student started the diagram below by drawing and. In order to construct so it is parallel to, which construction should the student perform? A Alternate Interior Angles Theorem B Converse of Alternate Interior Angles Theorem C Converse of Corresponding Angles Postulate D Converse of Consecutive Interior Angles Theorem B 45. ACT/SAT Two lines a and b are cut by a transversal. The measures of two consecutive interior angles are (4x + 1) and (5x - 10). What value of x will make a parallel to b? F 11 G 21 H 85 J 95 K 159 G F Construct 2 congruent to 1. G Construct 3 congruent to 5. H Construct 3 congruent to 1. J Construct 4 congruent to 5. J 46. Engineers designed streets in a city so that 1 2. Which statement is true? A 1 B 1 C 2 D D esolutions Manual - Powered by Cognero Page 8
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