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Points, Lines, and Planes

Points, Lines, and Planes

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Points, Lines, and Planes

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  1. Points, Lines, and Planes GEOMETRY LESSON 1-2 (For help, go to the Skills Handbook, page 722.) 1.y = x + 5 2.y = 2x – 4  3.y = 2x y = –x + 7 y = 4x – 10 y = –x + 15 4. Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points. Solve each system of equations. 1-2

  2. Solutions 1. By substitution, x + 5 = –x + 7; adding x – 5 to both sides results in 2x = 2; dividing both sides by 2 results in x = 1. Since x = 1, y = (1) + 5 = 6. (x, y) = (1, 6) 2. By substitution, 2x – 4 = 4x – 10; adding –4x + 4 to both sides results in –2x = –6; dividing both sides by –2 results in x = 3. Since x = 3, y = 2(3) – 4 = 6 – 4 = 2. (x, y) = (3, 2) 3. By substitution, 2x = –x + 15; adding x to both sides results in 3x = 15; dividing both sides by 3 results in x = 5. Since x = 5, y = 2(5) = 10. (x, y) = (5, 10) 4. The 6 different lines are AB, AC, AD, BC, BD, and CD. Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  3. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Point -A single location in space -Indicated by an “infinitely small” dot -Named by a capital letter Example: Good to Know: -A geometric figure is a set of points -Space is defined as the set of all points 1-1

  4. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Line -A series of points that extend infinitely in two opposite directions -Drawn with an arrow at each end, indicating that it extends indefintely Example: 1-1

  5. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Line A line can be named either one of two ways: 1. A line can be named by placing the line symbol ( ) over any two points that fall on the line. Example: 1-1

  6. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Line A line can be named either one of two ways: 2. A line can be named with a single, lowercase letter Example: 1-1

  7. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Collinear points -All points that lie on the same line are said to be collinear Example: • Points C, A and B are collinear • Points C, A and D are NOT collinear • Are points D, B and E collinear? • YES! 1-1

  8. In the figure below, name three points that are collinear and three points that are not collinear. Points Y, Z, and W lie on a line, so they are collinear. Points, Lines, and Planes GEOMETRY LESSON 1-2 Any other set of three points do not lie on a line, so no other set of three points is collinear. For example, X, Y, and Z and X, W, and Z form triangles and are not collinear. 1-2

  9. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Planes -A flat surface with no thickness -Comprised of an infinite number of lines -Extends without end in the direction of all of its lines Example: 1-1

  10. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Planes A plane can be named in either of two ways: 1. A single capital letter Example: 1-1

  11. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Planes A plane can be named in either of two ways: 2. the combination of three of its non-collinear points Example: 1-1

  12. Name the plane shown in two different ways. Points, Lines, and Planes GEOMETRY LESSON 1-2 You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following: plane RST plane RSU plane RTU plane STU plane RSTU 1-2

  13. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Definitions Coplanar Points and lines on the same plane are said to be coplanar -line b, line c, and point K are coplanar -line a and line b are NOT coplanar -line a and point K are coplanar, but not on the plane shown! 1-1

  14. Use the diagram below. What is the intersection of plane HGC and plane AED? The back and left faces of the cube intersect at HD. Planes HGC and AED intersect vertically at HD. Points, Lines, and Planes GEOMETRY LESSON 1-2 As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED. 1-2

  15. Shade the plane that contains X, Y, and Z. Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z. Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  16. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Postulates Postulate 1-1: Through any two points, there is exactly ONE line Postulate 1-2: If two lines intersect, then they intersect in exactly ONE point. Postulate 1-3: If two planes intersect, then they intersect in exactly ONE line. Postulate 1-4 Through any three noncollinear points, there is exactly one plane. 1-1

  17. 9. Answers may vary. Sample: AE, EC, GA 10. Answers may vary. Sample: BF, CD, DF 11.ABCD 12.EFHG 13.ABHF 14. EDCG 15.EFAD 16.BCGH 17.RS 18.VW 19.UV 20.XT 21. planes QUX and QUV 22. planes XTS and QTS 23. planes UXT and WXT 24.UVW and RVW Points, Lines, and Planes GEOMETRY LESSON 1-2 Pages 13–16 Exercises 1. no 2. yes; line n 3. yes; line n 4. yes; line m 5. yes; line n 6. no 7. no 8. yes; line m 1-2

  18. 25. 26. 27. 28. 29. 30. S 31.X 32.R 33.Q 34. X Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  19. 35. no 36. yes 37. no 38. coplanar 39. coplanar 40. noncoplanar 41. coplanar 42. noncoplanar 43. noncoplanar 44. Answers may vary. Sample: The plane of the ceiling and the plane of a wall intersect in a line. 45. Through any three noncollinear points there is exactly one plane. The ends of the legs of the tripod represent three noncollinear points, so they rest in one plane. Therefore, the tripod won’t wobble. 46. Postulate 1-1: Through any two points there is exactly one line. 47. Answer may vary. Sample: 48. 49. not possible Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  20. 56. no 57. no 58. yes 50. 51. not possible 52. yes 53. yes 54. no 55. yes Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  21. 59. yes 60. always 61. never 62. always 63. always 64. sometimes 65. never 66. a. 1 b. 1 c. 1 d. 1 e. A line and a point not on the line are always coplanar. 67. Post. 1-4: Through three noncollinear points there is exactly one plane. 68. Answers may vary. Sample: Post. 1-3: If two planes intersect, then they intersect in exactly one line. 69. A, B, and D 70. Post. 1-1: Through any two points there is exactly one line. Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  22. 71. Post. 1-3: If two planes intersect, then they intersect in exactly one line. 72. The end of one leg might not be coplanar with the ends of the other three legs. (Post. 1-4) 73. yes 74. yes 75. no 76. no 77. yes Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  23. 78. no 79. Infinitely many; explanations may vary. Sample: Infinitely many planes can intersect in one line. 80. By Post. 1-1, points D and B determine a line and points A and D determine a line. The distress signal is on both lines and, by Post. 1-2, there can be only one distress signal. 81.a. Since the plane is flat, the line would have to curve so as to contain the 2 points and not lie in the plane; but lines are straight. b. One plane; Points A, B, and C are noncollinear. By Post. 1-4, they are coplanar. Then, by part (a), AB and BC are coplanar. 82. 1 Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  24. 1 4 83. 84. 1 85.A 86.I 87. B 88.H 89.[2] a.ABD, ABC, ACD, BCD b.AD, BD, CD [1] one part correct 90. The pattern 3, 9, 7, 1 repeats 11 times for n = 1 to 44. For n = 45, the last digit is 3. 91. I, K 92. 42, 56 93. 1024, 4096 94. 25, –5 95. 34 96. 44 Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2

  25. Points, Lines, and Planes GEOMETRY LESSON 1-2 Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. 1-2

  26. HE Points, Lines, and Planes GEOMETRY LESSON 1-2 Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. D, J, and H planes BCGF and CGHD 1 Sample: Planes AEHD and BFGC never intersect. 1-2