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LESSON 3.6 Answers for the lesson “Prove Theorems about Perpendicular Lines” Skill Practice 1. } AB; it’s > to the parallel lines. 2. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 3. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. 4. If two lines are perpendicular, then they intersect to form four right angles. 5. 258 6. 908 7. 528 8. Lines Perpendicular to a Transversal Theorem Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 9. Since the two angles labeled x8 form a linear pair of congruent angles, t >n; since the two lines are perpendicular to the same line, they are parallel to each other. 10. Alternate Exterior Angles Converse Theorem 11. Sample answer: Draw a line. Construct a second line perpendicular to the first line. Construct a third line perpendicular to the second line. 12. Fold the paper into thirds lengthwise and then in half across its width. 13. There is no information to indicate that y i z or y >x. ‹]› 14. } AC is not > to AB . 15. 13 16. 95 17. 33 18. Lines n and p; they are perpendicular to line k. 19. Lines f and g; they are perpendicular to line d. 20. Lines z and y; they are perpendicular to line w. Lines v, w, and x; lines v and w are perpendicular to line y, and lines w and x are perpendicular to line z. 21. A 22. The distance between nonparallel lines is not constant. 23. 4.1 24. 2.8 } } 25. Ï 13 < 3.6; Ï 26 < 5.1 26. Sample answer: m1 5 908 since it is a right angle; m2 1 308 5 908, therefore m2 5 608; m5 5 908 since it is vertical to 1; m3 5 308 since it is vertical to the angle measuring 308; m4 5 208 since m3 1 m4 1 408 5 908. 27. 2.5 Geometry Answer Transparencies for Checking Homework 85 28. Construct a perpendicular line to the plane passing through the point. Find the length of the perpendicular segment from the point to the point where the segment intersects the plane; if and only if the line is parallel to the plane. Problem Solving 29. Point C; the shortest distance is the length of the perpendicular segment. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 30. Check students sketches; perpendicular segments; shortest distance between 2 parallel lines is a perpendicular line, so the perpendicular line would require less paint than the diagonal ones. 31. 2. Linear Pair Postulate 3. m1 1 m2 5 1808 5. Definition of congruent angles 8. Division Property of Equality 9. 1 is a right angle. 10. Definition of perpendicular lines 32. Given: a > b Prove: 1, 2, 3, and 4 are right angles. Statements (Reasons) 1. a > b (Given) 2. 1 is a right angle. (Definition of perpendicular lines) 3. m1 5 908 (Definition of right angle) 4. 1 and 4 are vertical angles. (Definition of vertical angles) 5. m4 5 908 (Vertical Angles Congruence Theorem) 6. 4 is a right angle. (Definition of right angle) 7. 1 and 2 form a linear pair. (Definition of a linear pair) 8. m1 1 m2 5 1808 (Linear Pair Postulate) 9. 908 1 m2 5 1808 (Substitution) 10. m2 5 908 (Subtraction Property of Equality) 11. 2 is a right angle. (Definition of right angle) 12. 2 and 3 are vertical angles. (Definition of vertical angles) 13. m3 5 908 (Definition of vertical angles) 14. 3 is a right angle. (Definition of right angle) Geometry Answer Transparencies for Checking Homework 86 33. Given: h i k, j > h 34. Given: m > p, n > p Prove: m i n Prove: j > k j m n 1 1 2 p h 2 k Statements (Reason) 1. m > p, n > p Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. Statements (Reasons) 1. h i k, j > h (Given) 2. 1 > 2 (Corresponding Angles Postulate) 3. m1 5 m2 (Definition of Congruent Angles) 2. and 2 are right angles. (Perpendicular lines intersect to form four right angles.) 3. 1 > 2 (Right Angle Congruence Theorem) 4. m i n (Corresponding Angles Converse Postulate) 4. 1 is a right angle. (Perpendicular lines intersect 35. no to form four right angles.) 37. yes 5. m1 5 908 (Definition of right angle) 6. m2 5 908 (Given) 36. no 38. yes (Substitution) 7. 2 is a right angle. (Definition of right angle) 8. j > k (Definition of perpendicular lines) Geometry Answer Transparencies for Checking Homework 87 Mixed Review of Problem Solving 2. 15; 1 5 1. a. y 5 35x, y 5 20x; Cost (dollars) y 150 skating rink 100 50 0 bowling alley 0 1 2 3 4 5 x Number of hours b. No; they don’t have the same slope. c. cost per hour of renting d. y 5 20x 1 25; y 150 Cost (dollars) Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. each facility 0 Sample answer: 2y 2 3x 5 22 4. 96.6. Sample answer: It’s the perpendicular distance to the line. 5. a. 340 ft 17 b. } 25 100 50 3. Sample answer: 4x 1 6y 5 8; c. Monongahela Incline. bowling alley Sample answer: 0.7 . 0.68 0 1 2 3 4 5 x Number of hours Yes; y 5 20x and y 5 20x 1 25 have the same slope. Geometry Answer Transparencies for Checking Homework 88