Given: Prove: Proof: 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. 5. SHORT RESPONSE Find x so that m n. Show your work. 1. and are corresponding angles of lines j and k. Since, j k by the Converse of Corresponding Angles Postulate. angle and angle are alternate exterior angles of lines m and n. Since m n, by the Converse of Alternate Exterior Angles Theorem. 2. and are alternate interior angles of lines j and k. Since, j k by the Converse of Alternate Interior Angles Theorem. 3. and are alternate exterior angles of lines and m. Since, m by the Converse of Alternate Exterior Angles Theorem. 4. m 6 + m 8 = 180 and are consecutive interior angles of lines and m. Since, m by the Converse of Consecutive Interior Angles Theorem. 6. PROOF Copy and complete the proof of Theorem Given: esolutions Manual - Powered by Cognero Page 1
2 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 Sample answer: Yes; since the alternate exterior angles are congruent, the backrest and footrest are parallel. Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. and are corresponding angles of lines r and s. Since, r s by the Converse of Corresponding Angles Postulate. and are alternate exterior angles of lines and m. Since, m by the Converse of Alternate Exterior Angles Theorem and are consecutive interior angles of lines r and s. Since, r s by the Converse of Consecutive Interior Angles Theorem. and are consecutive interior angles of lines r and s. Since, r s by the Converse of Consecutive Interior Angles Theorem. and are alternate interior angles of lines u and v. Since, u v by the Converse of Alternate Interior Angles Theorem. No lines can be proven parallel. and are corresponding angles of lines r and s. Since, r s by the Converse of Corresponding Angles Postulate. 10. and are alternate interior angles of lines r and s. Since, r s by the Converse of Alternate Interior Angles Theorem. esolutions Manual - Powered by Cognero Page 2
3 Find x so that m n. Identify the postulate or theorem you used. 16. By the Alternate Exterior Angles Converse, if 3x 14 = 2x + 25, then m n. 18. By the Alternate Interior Angles Converse, if x = x + 84, then m n. 17. By the Converse of Corresponding Angles Postulate, if 5x 20 = 90, then m n. 19. By the Consecutive Interior Angles Converse, if 7x x = 180, then m n. esolutions Manual - Powered by Cognero Page 3
4 20. Use the Vertical Angle Theorem followed by Consecutive Interior Angles Converse to find x. 21. By the Alternate Exterior Angles Converse, if 6x 144 = 2x, then m n. Then by Consecutive Interior Angles Converse, if 3x + 2x + 45 = 180, then m n. 22. CCSS SENSE-MAKING 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. esolutions Manual - Powered by Cognero Page 4
5 23. PROOF Copy and complete the proof of Theorem PROOF Write a two-column proof for each of the following. 25. Given: Given: 1 and 2 are supplementary. 1., (Given) 2. (Corr. postulate) 3. (Trans. Prop.) 4. (If alternate are, then lines are.) 26. Given: 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. 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 (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.) esolutions Manual - Powered by Cognero Page 5
6 27. Given: 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. 1., (Given) 2. (Def. of ) 3. (Substitution) 4. and are supplementary. (Def. of suppl. ) 5. (If consec. int. are suppl., then lines are.) 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 Given: 28. Given: 1., (Given) 2. (If alt. int. are, then lines are.) 3. (Perpendicular Transversal Theorem) 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 6
7 31. PROOF Write a two-column proof of Theorem Given: 34. r s; Sample answer: The alternate exterior angles are congruent. Since the measures of angles are equal, the lines are parallel. 1. (Given) 2. (Vertical s are ) 3. (Transitive Prop.) 4. (If corr are, then lines are.) 32. CCSS REASONING Based upon the information given in the photo of the staircase, what is the relationship between each step? Explain your answer. 35. 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 7
8 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; and are alternate interior angles for and, so if alternate interior angles are congruent, then the lines are parallel. 38. CCSS REASONING Is Theorem 5.21 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 8
9 39. CHALLENGE Use the figure to prove that two lines parallel to a third line are parallel to each other. 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. 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. OPEN ENDED 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. was copied to construct. So,. and are corresponding angles, so by the converse of corresponding angles postulate, a. We know that. Since and are linear pairs,. By substitution,. By subtracting from both sides we get., by the definition of congruent angles. Therefore, since the corresponding angles are congruent. b. We know that and. Since and are corresponding angles, they are congruent and their measures are equal. By substitution,. By dividing both sides by 2, we get. Therefore, 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 9
10 44. Which of the following facts would be sufficient to prove that line d is parallel to 46. What is the approximate surface area of the figure? A B C D If line d is parallel to the line through then with the transversals of, the alternate interior angles must be congruent. Thus. A in 2 B 108 in 2 C in 2 D 216 in 2 The formula for finding the surface area of a prism is. S = total surface area, h = height of a solid, B = area of the base, P = perimeter of the base Since the base of the prism is a rectangle, the perimeter P of the base is or 27 inches. The area of the base B is or 40.5 square inches. The height is 5 cinches. Thus B is the correct choice. 45. ALGEBRA The expression is equivalent to F 13 G H J The surface area of the prism is 216 square inches. So, the correct option is D. So, the correct option is G. esolutions Manual - Powered by Cognero Page 10
11 47. SAT/ACT If x 2 = 25 and y 2 = 9, what is the greatest possible value of (x y) 2? F 4 G 16 H 58 J 64 K 70 First solve for x and y. Next, find the greatest possible value. Substituting - 5 and 3 for x and y, respectively, leads to the greatest positive number. Another solution is to substitute 5 and -3 for x and y. The correct choice is H. esolutions Manual - Powered by Cognero Page 11
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