Points, Planes, & Lines, Study notes of Geometry

Download Points, Planes, & Lines and more Study notes Geometry in PDF only on Docsity! Geometry CC RHS Unit 1 Points, Planes, & Lines 1 GEOMETRY Unit 1 Points, Planes, & Lines Prepared by: G. Alvarez-Garcia, S. George, & R. Mercadante Geometry CC RHS Unit 1 Points, Planes, & Lines 2 1.1 Points, Lines, and Planes Definition The “easy” definition What does it look like? How is it written? A point has no ________________. A line extends in _______ dimension. A plane extends in _______ dimensions. A line segment AB consists of ____________ A and B, and all the points on line AB that are ________________ A and B. A ray AB consists of the initial point ____ and all the points on the line AB that lie on _____________ __________ of A as point B. THINK ABOUT IT! Is there a difference between lines ̅̅ ̅̅ and ̅̅ ̅̅ ? Is there difference between ⃗⃗⃗⃗ ⃗ and ⃗⃗⃗⃗ ⃗? Example 1: Give all the names of the figures shown below. 1. 2. 3. 4. 5. 6. E FC D E F G I J J O P P R E F G I J O P Q M N J O P O Geometry CC RHS Unit 1 Points, Planes, & Lines 5 Regents Test Basic Concepts and Definitions Questions 1 through 9 refer to the following: 1) PR and TS determine a plane. TRUE FALSE 2) Points T, Q, and S are collinear. TRUE FALSE 3) S is between P and R. TRUE FALSE 4) Points P, Q, and R are collinear. TRUE FALSE 5) PR and TS intersect at Q. TRUE FALSE 6) Points T, Q, and S are non-collinear. TRUE FALSE 7) Points P, Q, and T are non-collinear. TRUE FALSE 8) Q is between P and R. TRUE FALSE 9) Points S, Q, and R are collinear. TRUE FALSE Questions 10 through 16 refer to the following: In the diagram below, points P, N, and Q are collinear. Indicate whether the given statement is True or False. 10) Points M, N, K and Q are coplanar. TRUE FALSE 11) Points P, K, and Q determine a plane. TRUE FALSE 12) Only one plane contains points P, N, and Q. TRUE FALSE 13) PQ and NK intersect at N. TRUE FALSE Geometry CC RHS Unit 1 Points, Planes, & Lines 6 14) Points N, K, and M determine a plane. TRUE FALSE 15) KN and M determine a plane. TRUE FALSE Geometry CC RHS Unit 1 Points, Planes, & Lines 7 16) Points P, K, N, and Q are coplanar. TRUE FALSE 17) If two planes intersect, then their intersection is a line. TRUE FALSE 18) PQ has no endpoints. TRUE FALSE 19) PQ has only one endpoint. TRUE FALSE 20) A line segment has exactly one midpoint. TRUE FALSE 21) Tell whether a point, a line, or a plane is illustrated by . 22) Tell whether a point, a line, or a plane is illustrated by . 23) PQ has no endpoints. TRUE FALSE 24) FG has only one endpoint. TRUE FALSE 25) PQ has two endpoints. TRUE FALSE 26) Tell whether a point, a line, or a plane is illustrated by the top of a desk. 1900 - 1 - Page 2 Geometry CC RHS Unit 1 Points, Planes, & Lines 10 13) Define a postulate. 14) A theorem is a statement accepted without proof. TRUE FALSE 15) A postulate is a statement that must be proven. TRUE FALSE 16) A postulate can be used in the proof of a theorem. TRUE FALSE 17) A postulate can not be used in the proof of a theorem. TRUE FALSE 18) A postulate is a true statement accepted without proof. TRUE FALSE 19) Explain the difference between a postulate and a theorem. 20) All statements in Geometry are definitions or postulates. TRUE FALSE Geometry CC RHS Unit 1 Points, Planes, & Lines 11 Regents Test Points, Planes, & Lines: Concepts involving Definitions Questions 1 through 5 refer to the following: Arrange the following terms in the order in which their definitions should be stated. 1) ray, angle, endpoint of a ray 2) isosceles triangle, triangle, base of an isosceles triangle 3) congruent angles, angle bisector, angle 4) angle, acute triangle, acute angle 5) midpoint of a segment, segment, congruent segments 6) Point is a defined term. TRUE FALSE 7) Point is an undefined term. TRUE FALSE Questions 8 through 11 refer to the following: Explain why the following is not a good definition. 8) A line segment is a geometric figure. 9) Coplanar points are points in the same plane. 10) An isosceles triangle is when a triangle has two sides the same length. 11) A square is something not round. 12) Plane is a defined term. TRUE FALSE 13) Line is a defined term. TRUE FALSE 14) Define AB written out with words. Geometry CC RHS Unit 1 Points, Planes, & Lines 12 15) Define AB written out with words. 16) Plane is an undefined term. TRUE FALSE 17) Define AB written out with words. 18) Ray is not an undefined term. TRUE FALSE 19) Line is an undefined term. TRUE FALSE 20) Define AB written out with words. Geometry CC RHS Unit 1 Points, Planes, & Lines 15 5. M is between N and L. If , , and , find NM and explain. 6. Let ̅̅ ̅̅ bisect ̅̅̅̅ at point O, where and – , find JO and explain. ooh! I <3 PROOFS! Given: is the midpoint of ̅̅ ̅̅ ; Prove: Statements Reasons 1. is the midpoint of ̅̅ ̅̅ ; 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. 6. Geometry CC RHS Unit 1 Points, Planes, & Lines 16 Regents Test Points, Planes, & Lines: Lines and Line Segments Questions 1 through 17 refer to the following: Use the figure below to name a segment, ray, or point that best completes the given statement. 1) the length of NK is G 2) G is the midpoint of MT 3) NK = NO + G 4) EA = ER - G 5) TW = TK - G 6) EA + AR = OK + G 7) WK C G 8) the ray opposite TK is G 9) G is the midpoint of NK Geometry CC RHS Unit 1 Points, Planes, & Lines 17 10) T is the midpoint of G 11) ER C G 12) AR = AW + G 13) the length of MW is G 14) the ray opposite AM is G 15) another name for ET is G 16) R is the midpoint of G 17) another name for AR is G 18) Find PR, given that Q is the midpoint of PR and QR = 8. 19) Find PQ, given that Q is the midpoint of PR and PR = 20. 20) Find the value of x if B is the midpoint of AC, AB = x - 3, and AC = 6x - 38. Geometry CC RHS Unit 1 Points, Planes, & Lines 20 11) Find the distance between A and R. 12) Find the distance between E and M. 13) Find the distance between S and A. 14) Find the distance between S and R. 15) Find the distance between M and N. 16) Find the distance between R and A. 17) Find the distance between H and A. 18) Find the distance between N and F. 19) Find the distance between E and F. 20) Find the distance between E and N. Questions 21 through 28 refer to the following: Find the required distance if A, B and C are collinear points and point B is between points A and C. 21) AB = 2, AC = 18, BC = G Geometry CC RHS Unit 1 Points, Planes, & Lines 21 22) AC = 10, BC = 7, AB = G 23) AC = 19, BC = 8, AB = G 24) AB = 12, AC = 25, BC = G 25) AB = 4, BC = 9, AC = G Geometry CC RHS Unit 1 Points, Planes, & Lines 22 26) BC = 11, AB = 5, AC = G 27) AB = 6, BC = 9, AC = G 28) BC = 20, AB = 7, AC = G Geometry CC RHS Unit 1 Points, Planes, & Lines 25 35. Find the distance between point O and Point N 36. Find the distance between point L and point N Congruence, Distance & Length: HOMEWORK 37. Find the distance between point N and point K 38. Find the distance between point Q and point L 39. Find the distance between point L and point P 40. Find the distance between point K and point Q Geometry CC RHS Unit 1 Points, Planes, & Lines 26 Congruence, Distance & Length: CLASSWORK Figure for #41-52 41. Find the distance between point G and point L 42. Find the distance between point K and point M 43. Find the distance between point A and Point D 44. Find the distance between point B and point F 45. Find the distance between point C and point H 46. Find the distance between point E and point N Congruence, Distance & Length: HOMEWORK 47. Find the distance between point L and point A 48. Find the distance between point D and point L 49. Find the distance between point N and point K 50. Find the distance between point K and point H 51. Find the distance between point C and point I Geometry CC RHS Unit 1 Points, Planes, & Lines 27 52. Find the distance between point B and point M Congruence, Distance & Length: CLASSWORK 53. Given: RV = 24 SU = 8 RS = ST = TU RS = ? 54. Given: RV = 24 SU = 8 RS = ST = TU UV = ? 55. Given: RV = 37 RS = TU UV = 9 ST + UV = 15 ST = ? RS = ? SU = ? 56. Given: RV = 25 RT = 6 RS = ST = TU UV = 3x +1 x = ? 57. Points F, X, and D are collinear, with F between X and D. Draw a diagram and solve for X, if FX = 2x+13, FD = 7x-6, and XD = 2x+70. 58. P, A, and Z are collinear. Z is between P and A. Draw a diagram and solve for x, given: ZP = 6x-13, ZA = 9x+27, and AP = 3x +74. #53-59 Geometry CC RHS Unit 1 Points, Planes, & Lines 30 Geometry CC RHS Unit 1 Points, Planes, & Lines 31 PARCC type question: 71. The figure shows line j, point C on line j, and . Part A: Consider the partial construction of a line segment congruent to on line j. What would be the final step in the construction? a) Draw a point D between points C and A b) Draw a point D between points C and B c) Draw a point D on line j between point C and the given arc d) Draw a point D on line j at the given arc. Part B: Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction? a) Because the distance between A and D match the distance between B and C, AD = BC, making them congruent. b) Because the distance between A and C match the distance between B and D, AC = BD, making them congruent. c) Because the distance between A and B match the distance between C and D, AB = CD, making them congruent. d) Because the distance between A and j match the distance between B and D, Aj = BD, making them congruent. AB j A B C j A B C AB Geometry CC RHS Unit 1 Points, Planes, & Lines 32 Points, Lines and Planes Review Multiple Choice 72. Intersecting lines are _________________ non-coplanar a. always b. sometimes c. never 73. Two non-parallel lines _________________ intersect at one point. a.always b.sometimes c.never 74. Two points are _________________ collinear. a.always b.sometimes c.never 75. Three points are _________________ collinear. a.always b.sometimes c.never 76. Three points are _________________ coplanar. a.always b.sometimes c.never 77. Four points are _________________ coplanar. a.always b.sometimes c.never 78. When two planes intersect, they do so: a.at a point b.along a line c.along a plane 79. All points along a line are _________________ collinear. a.always b.sometimes c.never 80. Opposite rays (circle all that apply): a.are collinear b.have the same endpoint Geometry CC RHS Unit 1 Points, Planes, & Lines 35 Points, Lines and Planes Review Constructed Response 91. Points J, K and L are collinear with J between L and K. KJ = 2x + 3, LK = 9x – 7 and LJ = 4x + 8. Draw a diagram, solve for x, and find the measures for KJ, LJ & LK. 92. Points B, D, F, H and J are collinear and are in the following order  D is between J and B  F is between D and J  B is between F and H  JH = 47, FB = 12, FD = DB = BH and JF = 3x - 7 Draw a diagram of the points, solve for x and solve for JF. Geometry CC RHS Unit 1 Points, Planes, & Lines 36 Answer Key 1. KLI & HLJ 2. HLK, HLI, KLJ, ILJ 3. LK, LI, LJ, LH 4. 3 5. Infinite 6. VRS 7. QVW, TUR, QUS, VRS 8. RW, RU 9. TURW, j 10. Infinite 11. Q 12. Q 13. R 14. P 15. PSOU or R 16. O 17. U 18. O 19. U 20. POSQ or V 21. F 22. J 23. FHJI 24. ACJI 25. CH 26. A 27. J 28. F 29. EDJI 30. CHEJ 31. JI 32. D 33. 5 34. 12 35. 4 36. 3 37. 6 38. 14 39. 12 40. 17 41. 2.0 42. 0.6 43. 1.5 44. 2.3 45. 2.4 46. 3.7 47. 5.0 48. 3.5 49. 1.2 50. 1.3 51. 2.9 52. 5.0 53. 4 54. 12 55. ST=6 RS=11 SU=17 56. 5 57. 9 58. 5 59. 2 60. 6 61. 18 62. ST = 9 RS = 7 RT = 16 63. RS = 18 RU = 24 64. 9 65. 9 66. 8 67. Check Constructions 68. Check Constructions 69. Check Constructions 70. Check Constructions 71. Part A: choice d Part B: choice c Geometry CC UNIT 1 Points, Planes, & Lines 37 Review Answer Key Multiple Choice 72. C 73. A 74. A 75. B 76. A 77. B 78. B 79. A 80. A, B, D 81. A 82. B 83. A 84. B 85. D 86. C 87. B 88. C 89. A, B, C, D 90. Part A: A Part B: B Geometry CC UNIT 1 Points, Planes, & Lines 40 1 2 Geometry CC UNIT 1 Points, Planes, & Lines 41 3 4 5 Geometry CC UNIT 1 Points, Planes, & Lines 42 6 7 Geometry CC UNIT 1 Points, Planes, & Lines 45 12 13 Geometry CC UNIT 1 Points, Planes, & Lines 46 14 15 Geometry CC UNIT 1 Points, Planes, & Lines 47 16 17 Geometry CC UNIT 1 Points, Planes, & Lines 50 24 25 Geometry CC UNIT 1 Points, Planes, & Lines 51 26 27 28 Geometry CC UNIT 1 Points, Planes, & Lines 52 29 30 31 Geometry CC UNIT 1 Points, Planes, & Lines 55 37 38 39 Geometry CC UNIT 1 Points, Planes, & Lines 56 40 41 Geometry CC UNIT 1 Points, Planes, & Lines 57 42 43 Geometry CC UNIT 1 Points, Planes, & Lines 60 48 49 50 Geometry CC UNIT 1 Points, Planes, & Lines 61 51 52 53 Geometry CC UNIT 1 Points, Planes, & Lines 62 54 55