In Exercises 47–49, sketch a figure for the given information. Then state the postulate that your figure illustrates.

47. A B ↔ modified eh b with left right arrow above  and E F ↔ modified e f with left right arrow above  intersect in point C.
48. The noncollinear points A, B, and C are all contained in plane N.
49. Planes LNP and MVK intersect in N M ↔ . modified n m with left right arrow above , .

50. Telecommunications A cell phone tower at point A receives a cell phone signal from the southeast. A cell phone tower at point B receives a signal from the same cell phone from due west. Trace the diagram below and find the location of the cell phone. Describe how Postulates 1-1 and 1-2 help you locate the phone.

    

51. Estimation You can represent the hands on a clock at 6:00 as opposite rays. Estimate the other 11 times on a clock that you can represent as opposite rays.
52. Open-Ended What are some basic words in English that are difficult to define?

Coordinate Geometry Graph the points and state whether they are collinear.

53. ( 1 , 1 ) , ( 4 , 4 ) , ( − 3 , − 3 ) open 1 comma 1 close comma open 4 comma 4 close comma open negative 3 comma negative 3 close
54. (2, 4), (4, 6), (0, 2)
55. ( 0 , 0 ) , ( − 5 , 1 ) , ( 6 , − 2 ) open 0 comma 0 close comma open negative 5 comma 1 close comma open 6 comma negative 2 close
56. (0, 0), (8, 10), (4, 6)
57. ( 0 , 0 ) , ( 0 , 3 ) , ( 0 , − 10 ) open 0 comma 0 close comma open 0 comma 3 close comma open 0 comma negative 10 close
58. ( − 2 , − 6 ) , ( 1 , − 2 ) , ( 4 , 1 ) open negative 2 comma negative 6 close comma open 1 comma negative 2 close comma open 4 comma 1 close

C Challenge

59. How many planes contain the same three collinear points? Explain.
60. How many planes contain a given line? Explain.
61. 1. Writing Suppose two points are in plane P. Explain why the line containing the points is also in plane P.
    2. Reasoning Suppose two lines intersect. How many planes do you think contain both lines? Use the diagram below and your answer to part (a) to explain your answer.

    

Probability Suppose you pick points at random from A, B, C, and D shown below. Find the probability that the number of points given meets the condition stated.

62. 2 points, collinear
63. 3 points, collinear
64. 3 points, coplanar

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