Points, Lines, and Planes

Skills Handbook page 760

Algebra Solve each system of equations.

1. y = x + 5 (1, 6) 2. y = 2x - 4 (3, 2) 3. y = 2x (5, 10)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. See margin, p. 17.

New Vocabulary • point • space • line • collinear points • plane• coplanar • postulate • axiom

D

BA

C

x2x2

What You’ll Learn• To understand basic terms

of geometry

• To understand basicpostulates of geometry

. . . And WhyTo explain why aphotographer uses a tripod, as in Exercise 44

Part 1 Basic Terms of Geometry

In geometry, some words such as point, line, and plane are undefined. In order todefine these words you need to use words that need further defining. It is importanthowever, to have general descriptions of their meanings.

1-31-3

11 Basic Terms of Geometry

Hands-On Activity: How Many Lines Can You Draw?

Many constellations are namedfor animals and mythologicalfigures. It takes some imaginationto join the points representingthe stars to get a recognizablefigure such as Leo the Lion. How many lines can you draw connecting the 10 points in Leo the Lion?

• Make a table and look for a pattern to help you find out.

1. Mark three points on a circle. Now connect the three points with asmany (straight) lines as possible. How many lines can you draw?

2. Mark four points on another circle. How many lines can you draw toconnect the four points? 6 lines

3. Repeat this procedure for five points on a circle and then for six points.How many lines can you draw to connect the points? 10 lines, 15 lines

4. Use inductive reasoning to tell how many lines you can draw toconnect the ten points of the constellation Leo the Lion. 45 lines

3 lines

16 Chapter 1 Tools of Geometry

Check Skills You’ll Need GO for Help

1-31-3

16

1. PlanObjectives1 To understand basic terms of

geometry2 To understand basic

postulates of geometry

Examples1 Identifying Collinear Points2 Naming a Plane3 Finding the Intersections of

Two Planes4 Using Postulate 1-4

Math Background

The formal study of geometryrequires simple ideas andstatements that can be acceptedas true without proof. Theundefined terms point, line, andplane provide the simple ideas.Basic postulates about points,lines and planes can be acceptedwithout proof. These form thebuilding blocks for the firsttheorems that students can prove.

More Math Background: p. 2C

Lesson Planning andResources

See p. 2E for a list of theresources that support this lesson.

Bell Ringer Practice

Check Skills You’ll NeedFor intervention, direct students to:Skills Handbook, p. 760

Special NeedsFor Example 4, pair students with visual difficultieswith visual learners to identify and name planes.

Below LevelEncourage students to use pencils and sheets of paperto model Postulates 1-1, 1-2, and 1-3. Students may cutpartially through each sheet and join the sheets tomodel Postulate 1-3.

L2L1

learning style: visual learning style: tactile

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Lesson 1-3 Points, Lines, and Planes 17

You can think of a as a location. A point has no size. It is represented by asmall dot and is named by a capital letter. A geometric figure is a set of points.

is defined as the set of all points.

You can think of a as a series of points that extends in two oppositedirections without end. You can name a line by any two points on the line, such as

(read “line AB”). Another way to name a line is with a single lowercase letter,such as line t (see above). Points that lie on the same line are

Identifying Collinear Points

a. Are points E, F, and C collinear? If so, name the line on which they lie.

Points E, F, and C are collinear.They lie on line m.

b. Are points E, F, and D collinear? If so, name the line on which they lie.

Points E, F, and D are not collinear.

a. Are points F, P, and C collinear? nob. Name line m in three other ways. Answers may vary. Sample: , ,c. Critical Thinking Why do you think arrowheads are used when drawing a line or

naming a line such as ?

A is a flat surface that has no thickness. A plane contains many lines andextends without end in the directions of all its lines. You can name a plane by eithera single capital letter or by at least three of its noncollinear points. Points and linesin the same plane are

Naming a Plane

Each surface of the ice cube represents part of a plane. Name the plane represented by the front of the ice cube.

You can name the plane represented by the front of the ice cube using at least threenoncollinear points in the plane. Some names are plane AEF, plane AEB, and plane ABFE.

List three different names for the plane represented by the top of the ice cube.22Quick Check

H G

C

E F

D

BA

EXAMPLEEXAMPLE22

coplanar.

plane

*EF)

CE4

FC4

EF411Quick Check

F

�

E

P

C

D

m

n

EXAMPLEEXAMPLE11

collinear points.

*AB)

line

A

B

t

Space

point

Arrowheads are used to show that the lineextends in opposite directions without end.

Answers may vary. Sample: HEF, HEFG, FGH

B

CA

P

Plane P Plane ABC

(“line AB”) and (“line BA”) name the same line.

*BA)*

AB)Vocabulary Tip

17

2. Teach

Guided InstructionHands-On ActivitySome students may count lines inboth “directions.” Point out thateach line should be counted onlyonce.

Additional Examples

In the figure below, namethree points that are collinear andthree points that are not collinear.Y, Z, and W are collinear; X, Y,and Z and X, W, and Z are notcollinear.

Name the plane shown in twodifferent ways. Sample: planeRST, plane RSTU

page 16 Check Skills You’llNeed

4.A

BDC

S T

UR

22

A

BDC

11

Advanced LearnersIn Example 4, have students find the number ofdifferent planes named by any three vertices of thecube.

English Language Learners ELLThe word noncollinear in Postulate 1-4 may confusestudents. Discuss how the prefix non- indicates not.So, noncollinear points do not lie on a line.

L4

learning style: verballearning style: visual

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18 Chapter 1 Tools of Geometry

A or is an accepted statement of fact.

You have used some of the following geometry postulates in algebra.For example, you used Postulate 1-1 when you graphed an equation such as y = -2x + 8. You plotted two points and then drew the line through those two points.

In algebra, one way to solve a system of two equations is to graph the two equations. As the graphs of

y = -2x + 8

y = 3x - 7

show, the two lines intersect at a single point, (3, 2). The solution to the system of equations is (3, 2).

This illustrates Postulate 1-2.

There is a similar postulate about the intersection of planes.

When you know two points in the intersection of two planes, Postulates 1-1 and 1-3 tell you that the line through those points is the line of intersection of the planes.

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y

x

y � �2x � 8

y � 3x � 7

1 3

2 (3, 2)

5 7

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axiompostulate

12 Basic Postulates of Geometry

Key Concepts Postulate 1-1

Through any two points there is exactly one line.

Line t is the only line that passes through points A and B. A

B

t

Key Concepts Postulate 1-2

If two lines intersect, then they intersect in exactlyone point.

and intersect at C.*BD)*

AE) E

BA

DC

Key Concepts Postulate 1-3

If two planes intersect, then they intersect in exactlyone line.

Plane RST and plane STWintersect in .

*ST)

R

T WS

There is exactly onemeans “there is one and there is no more than one.”

Vocabulary Tip

18

Guided Instruction

Tactile LearnersHave students use index cards toillustrate Postulates 1-3 and 1-4.

Math Tip

The drawing of a plane isnecessarily finite. Remindstudents that the intersection is a line, not a segment, and that they should use a line symbol above the letters.

Additional Examples

Use the diagram from Example 3. What is theintersection of plane HGCand plane AED?

Shade the plane that containsX, Y, and Z.

Resources• Daily Notetaking Guide 1-3• Daily Notetaking Guide 1-3—

Adapted Instruction

Closure

Explain how a postulate and aconjecture are alike and how they are different. Sample: You accept a postulate as truewithout proof, but you try todetermine whether a conjectureis true or false.

L1

L3

V W

XY

Z

44

* HD)

33

EXAMPLEEXAMPLE33

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Lesson 1-3 Points, Lines, and Planes 19

Finding the Intersection of Two Planes

What is the intersection of plane HGFE and plane BCGF?

Plane HGFE and plane BCGFintersect in

Name two planes that intersect in .

A three-legged stand will always be stable. As long as the feet of the stand don’t lie in one line, the feet of the three legs will lie exactly in one plane.

This illustrates Postulate 1-4.

Using Postulate 1-4

a. Shade the plane that contains b. Shade the plane that containsA, B, and C. E, H, and C.

a. Name another point that is in the sameplane as points A, B, and C. D

b. Name another point that is coplanar with points E, H, and C. B

Practice and Problem Solving

Are the three points collinear? If so, name the line on which they lie.

1. A, D, E no 2. B, C, D yes; line n

3. B, C, F 4. A, E, C yes; line m

5. F, B, D 6. F, A, E no

7. G, F, C no 8. A, G, C yes; line m

9. Name line m in three other ways.

10. Name line n in three other ways.

FB

E

A

CG

m

nD

Example 1(page 17)

44Quick Check

H G

C

E F

DBA

H G

C

E F

DBA

EXAMPLEEXAMPLE44

*BF)

33Quick Check

*GF).

H G

C

E F

DBA

EXAMPLEEXAMPLE33

Key Concepts Postulate 1-4

Through any three noncollinear points there is exactly one plane.

Practice and Problem SolvingFor more exercises, see Extra Skill, Word Problem, and Proof Practice.EXERCISES

Practice by ExampleAA

ABF and CBF

3. yes; line n

5. yes; line n

9. Answers may vary.Sample: , , GA

4EC4

AE4

Answers may vary. Sample: , , DF4

CD4

BF4

GO forHelp

19

3. PracticeAssignment Guide

A B 1-16, 46-60, 64, 68-73

A B 17-45, 61-63, 65-67C Challenge 74-79

Test Prep 80-84Mixed Review 85-90

Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 5, 18, 50, 62, 66.

Error Prevention!

Exercises 9, 10 Check thatstudents use a line symbol andnot a segment or ray symbol asthey name lines.

2

1

Guided Problem SolvingGPS

Enrichment

Reteaching

Adapted Practice

Name Class Date

Practice 1-3 Segments, Rays, Parallel Lines, and Planes

Write true or false.

1. is the same as . 2. is the same as .

3. If and are opposite rays, 4. If two rays have the same endpoint,then they are collinear. then they form a line.

5. If the union of two rays is a line, 6. If and are the same rays, thenthen the rays are opposite rays. Q and R are the same point.

Refer to the diagram at the right.

7. Name all segments parallel to .

8. Name all segments parallel to .

9. Name three pairs of skew lines.

Refer to the diagram at the right.

10. Which pair(s) of planes is (are) parallel?

11. Which pair(s) of planes intersect?

12. Which planes intersect in ?

13. Which planes intersect in ?

Refer to the diagram at the right.

14. Name in another way.

15. How many different segments can be named?

16. Name a pair of opposite rays with E as an endpoint.

17. Name in two different ways the ray opposite .

18. Name in two other ways.

19. Are and the same segment?

Draw each of the following.

20. parallel planes S, T, and U

21. planes R and W intersecting in * PQ)

GEEG

GE)

FG)

EF)

* RS)

* MN

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Practice

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20 Chapter 1 Tools of Geometry

Name the plane represented by each surface of the box.

11. the bottom ABCD 12. the top EFHG

13. the front ABHF 14. the back EDCG

15. the left side EFAD 16. the right side BCGH

Use the figure at the right for Exercises 17–37.First, name the intersection of each pair of planes.

17. planes QRS and RSW 18. planes UXV and WVS

19. planes XWV and UVR 20. planes TXW and TQU

Name two planes that intersect in the given line.

21. 22. 23. 24.

Copy the figure. Shade the plane that contains the given points.

25. R, V, W 26. U, V, W 27. U, X, S 28. T, U, X 29. T, V, R

Name another point in each plane.

30. plane RVW 31. plane UVW 32. plane UXS 33. plane TUX 34. plane TVR

Is the given point coplanar with the other three points?

35. point Q with V, W, S 36. point U with T, V, S 37. point W with X, V, R

Postulate 1-4 states that any three noncollinear points lie in one plane. Find the plane containing the first three points listed, then decide whether the fourth point is in that plane.Write coplanar or noncoplanar to describe the points.

38. Z, S, Y, C coplanar 39. S, U, V, Y coplanar

40. X, Y, Z, Unoncoplanar 41. X, S, V, U coplanar

42. X, Z, S, V noncoplanar 43. S, V, C, Y noncoplanar

44. Photography Photographers and surveyors use a tripod, or three-legged stand,for their instruments. Use one of the postulates to explain why. See margin.

45. Open-Ended Draw a figure with points B, C, D, E, F, and G that shows and with one of the points on all three lines. See margin.

If possible, draw a figure to fit each description. Otherwise write not possible.

46. four points that are collinear 47. two points that are noncollinear

48. three points that are noncollinear 49. three points that are noncoplanar

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

50. (3,-3), (2,-3), (-3, 1) 51. (2, 2), (-2,-2), (3, 2)

52. (2,-2), (-2,-2), (3,-2) 53. (-3, 3), (-3, 2), (-3,-1)

54. Multiple Choice Which three points are not collinear? C(0, 0), (0, 2), (0, 4) (0, 0), (3, 0), (5, 0)(0, 0), (0, 2), (3, 0) (2,-2), (2, 2), (2, 3)

*EF),

*BG),

*CD),

Apply Your SkillsBB

Example 4(page 19)

*VW)*

XT)*

TS)*

QU)

XT4

VW4

Example 3(page 19)

Example 2(page 17)

AB

EF

G

CDH

X W

S

U V

TRQ

Exercises 17–37

VU

X

Y

C

Z

S

ConnectionReal-World

Careers The photographeruses a tripod to help assure a clear picture.

17. RS4

UV4

QUX and QUV

S X R Q X

XTS and QTS

25–29. See back of book.

no yes no

50–53. See back of book.

46–49. See margin.

UXT and WXT UVW and RVW

GPS

20

Exercises 38–43 These exercisesintroduce the term noncoplanar.Discuss with students how theycan derive its meaning from whatthey already know about theterms collinear and noncollinear.

Exercises 47–50 Discuss theexercises for which students thinkthat drawing a figure for thedescription is not possible. Havestudents explain their reasoning.This is a good way to clarify theideas that form the basis of aformal proof.

Error Prevention!

Exercises 56–61 These exercisesreinforce the vocabulary andpostulates in the lesson. Havestudents work with partners todiscuss any unclear terms.Emphasize the mathematicalimportance of the phrase exactlyone in Exercise 57. Also point out in Exercise 61 that two linesmeans two distinct lines.

CareersExercise 66 Ask: What fact aboutEarth’s surface complicates thework of a surveyor who uses linesand planes? Earth is a sphereand not flat, so distances aremeasured along curves.

47. not possible

48.

49. not possible

62. Post. 1-4:Through threenoncollinearpoints there isexactly oneplane.

63. Answers may vary.Sample:

M

NL

PK

VA

BC

NA B

C

44. Through any threenoncollinear pointsthere is exactly oneplane. The ends of thelegs of the tripodrepresent threenoncollinear points, sothey rest in one plane.Therefore, the tripodwon’t wobble.

45. Answers may vary.Sample:

46.A B C D

n

E

FG

C

DB

21

65. Post. 1-1: Through anytwo points there isexactly one line.

66. Post. 1-3: If two planesintersect, then theyintersect in exactly one line.

67. The end of one leg mightnot be coplanar with theends of the other threelegs. (Post. 1-4)

73.

2O x

y4

�4

Alternative Assessment

Have students write Postulate 1-4,illustrate it, and explain it in their own words. Then have them explain how changing theword three to four or the wordplane to line makes the postulateunreasonable.

75.

By Post. 1-1, points Dand B determine a lineand points A and Ddetermine a line. Thedistress signal is onboth lines and, by Post.1-2, there can be onlyone distress signal.

DB

Distress SignalA

no

Lesson Quiz

Use the diagram above.

1. Name three collinear points.D, J, and H

2. Name two different planesthat contain points C and G.planes BCGF and CGHD

3. Name the intersection of plane AED and plane HEG.

4. How many planes contain thepoints A, F, and H? 1

5. Show that this conjecture is false by finding onecounterexample: Two planesalways intersect in exactly oneline. Sample: Planes AEHDand BFGC never intersect.

* HE)

A

D J

BC

E

FG

H

PowerPoint

4. Assess & Reteach

Lesson 1-3 Points, Lines, and Planes 21

Use always, sometimes, or never to make a true statement.

55. Intersecting lines are 9 coplanar. always

56. Two planes 9 intersect in exactly one point. never

57. Three points are 9 coplanar. always

58. A plane containing two points of a line 9 contains the entire line. always

59. Four points are 9 coplanar. sometimes

60. Two lines 9 meet in more than one point. never

61. How many planes contain each line and point?a. and point G 1 b. and point E 1c. and point P 1 d. and point G 1e. Make a Conjecture What do you think is

true of a line and a point not on the line?

In Exercise 62 and 63, sketch a figure for the given information. Then name thepostulate that your figure illustrates. 62–63. See margin, pp. 20–21.

62. The noncollinear points A, B, and C are all contained in plane N.

63. Planes LNP and MVK intersect in .

64. Optical Illusions The diagram (right) is an optical illusion.Which three points are collinear: A, B, and C or A, B, and D? Are you sure? Use a straightedge to check your answer.

Writing Use postulates to explain each situation.

65. A land surveyor can always find a straight line from the point where she stands to any other point she can see.

66. A carpenter knows that a line can represent the intersection of two flat walls.

67. A furniture maker knows that a three-legged table is always steady, but a four-legged table will sometimes wobble.

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

68. (1, 1), (4, 4), (-3,-3) 69. (2, 4), (4, 6), (0, 2) 70. (0, 0), (-5, 1), (6,-2)

71. (0, 0), (8, 10), (4, 6) 72. (0, 0), (0, 3), (0,-10) 73. (-2,-6), (1,-2), (4, 1)

74. How many planes contain the same three collinear points? Explain. See left.

75. Navigation Rescue teams use Postulates 1-1 and 1-2 to determine the location of a distress signal. In the diagram,a ship at point A receives a signal from the northeast. A ship at point B receives the same signal from due west.Trace the diagram and find the location of the distress signal.Explain how the two postulates help locate the distress signal.

ChallengeCC

*NM)

*EP)*

FG)

*PH)*

EF)

P Q

GHFE

A

D C

B

WB

NE

A

See margin.

A line and a point not on the line are always coplanar.

A, B, and D65–67. See margin.

68–72. See back of book. 73. See margin.

74. Infinitely many;explanations may vary.Sample: Infinitely manyplanes can intersect inone line.

GO nlineHomework HelpVisit: PHSchool.comWeb Code: aue-0103

lesson quiz, PHSchool.com, Web Code: aua-0103

22 Chapter 1 Tools of Geometry

76. a. Open-Ended Suppose two points are in plane P. Explain why it makes sensethat the line containing the points would be in the same plane.

b. Suppose two lines intersect. How many planes do you think contain both lines? You may use the diagram and your answer in part (a) to explain your answer.

Probability Points are picked at random from A, B, C, and D, which are arranged as shown.Find the probability that the indicated number of points meet the given condition.

77. 2 points, collinear 1 78. 3 points, collinear 79. 3 points, coplanar 1

80. In the figure at the right, which points are collinear with C and H? AA. B, FB. E, F, GC. A, D, G, ID. A, D, E, H

81. A solid chunk of cheese is to be cut into 4 pieces. What is the least number of slices needed? JF. 5 G. 4 H. 3 J. 2

82. Ronald is making a table. What is the least number of legs that the tableshould have so that it will not wobble? BA. 4 B. 3 C. 2 D. 1

83. At most, how many lines can contain pairs of the points P, Q, and R? HF. 1 G. 2H. 3 J. 4

84. Use the figure at the right. a–b. See margin.a. Name all the planes that form the figure.b. Name all the lines that intersect at D.

Make an orthographic drawing for each figure. 85–87. See margin.

85. 86. 87.

Simplify each ratio.

88. 30 to 12 5 to 2 89. 90. 13

521 : 57

r2

pr2

2pr

Skills Handbook

Front RightFrontRightFront Right

Lesson 1-2

Short Response BD

C

A

E F

BA

H I

GDC

Multiple Choice

14

CB

D

A

A CB

Test Prep

Mixed ReviewMixed Review

Exercise 84

See margin.

See margin.

GO forHelp

22

85. 86. 87.

Right

Top

FrontRight

Top

FrontRight

Top

Front

Test Prep

ResourcesFor additional practice with avariety of test item formats:• Standardized Test Prep, p. 75• Test-Taking Strategies, p.70• Test-Taking Strategies with

Transparencies

76. a. Since the plane is flat,the line would have tocurve so as to containthe 2 points and notlie in the plane; butlines are straight.

b. One plane; Points A,B, and C arenoncollinear. By Post.1-4, they are coplanar.Then, by part (a),

and are coplanar.

84. [2] a. ABD, ABC, ACD,BCD

b.

[1] one part correct

AD4

, BD4

, CD4

BC4

AB4