1-3 Points, Lines, Planes

Points, Lines, and Planes

TEKS

(G.1)(A) To develop an

awareness of the structure

of a mathematical system,

connecting definitions and

postulates

(G.3)(D) To use inductive

reasoning to formulate a

conjecture

GO for HelpCheck Skills You’ll Need

1-31-3

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

B

A

C

x2x2

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.

11 Basic Terms of Geometry

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

Many constellations are named

for animals and mythological

figures. It takes some imagination

to join the points representing

the stars to get a recognizable

figure 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 as

many (straight) lines as possible. How many lines can you draw?

2. Mark four points on another circle. How many lines can you draw to

connect 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 to

connect the ten points of the constellation Leo the Lion. 45 lines

3 lines

16 Chapter 1 Tools of Geometry

1-31-3

16

1. Plan

(G.1)(A) To develop an awareness of the structure of a mathematical system,connecting definitions andpostulates

(G.3)(D) To use inductivereasoning to formulate aconjecture

Dana Center TEKS ToolkitClarifying Activities Activities(G.1)(A), (G.3)(D)

Visit: PHSchool.comWeb Code: auq-9045

TEKS 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 Need

Use student page, transparency, or PowerPoint.

Daily Review

Use transparency 3.

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

PowerPoint

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? no

b. 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 ornaming 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

EF4

11Quick 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 Instruction

Hands-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 not

collinear.

Name the plane shown in twodifferent ways. Sample: plane

RST, plane RSTU

page 16 Check Skills You’llNeed

4.A

B

DC

S T

UR

22

A

B

DC

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: verbal

PowerPoint


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.

O

y

x

y ! #2x $ 8

y ! 3x # 7

1 3

2 (3, 2)

5 7

4

#4

#2

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


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

and intersect at C.*BD)*

AE) E

BA

D

C


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

Plane RST and plane STW

intersect 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 Learners

Have 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 true

without proof, but you try to

determine whether a conjecture

is true or false.

L1

L3

VW

XY

Z

44

* HD)

33

EXAMPLEEXAMPLE33

PowerPoint


Finding the Intersection of Two Planes

What is the intersection of plane HGFE and plane BCGF?

Plane HGFE and plane BCGF

intersect 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

C

G

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


Through any three noncollinear points there is exactly one plane.

Practice and Problem Solving

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

ABF and CBF

3. yes; line n

5. yes; line n

9. Answers may vary.

Sample: , , GA4

EC4

AE4

Answers may vary. Sample: , , DF4

CD4

BF4

GO forHelp

Practice by Example

19

3. Practice

Assignment Guide

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

A B 17-45, 61-63, 65-67

C Challenge 74-79

Tune-Up 80-82Mixed Review 83-88

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

Practice 1-3 Points, Lines, and Planes

Refer to the diagram at the right for Exercises 1–15.

1. Name in another way.

2. Give two other names for plane Q.

3. Why is EBD not an acceptable name for plane Q?

Are the following sets of points collinear?

4. and C 5. B and F

6. and A 7. F and plane Q

Are the following sets of points coplanar?

8. E, B, and F 9. and

10. and 11. and

12. F, A, B, and C 13. F, A, B, and D

14. plane Q and 15. and

Find the intersection of the following lines and planes in the figure

at the right.

16. and

17. planes GLM and LPN

18. planes GHPN and KJP

19. planes HJN and GKL

20. and plane KJN

21. and plane GHL

Refer to the diagram at the right.

22. Name plane P in another way.

23. Name plane Q in another way.

24. What is the intersection of planes P and Q?

25. Are A and C collinear?

26. Are D, A, B, and C coplanar?

27. Are D and C collinear?

28. What is the intersection of and ?

29. Are planes P and Q coplanar?

30. Are and plane Q coplanar?

31. Are B and C collinear?

* AB)

* DC)*

AB)

* KM)

* KP)

* LG)*

GK)

* BD)*

FB)*

EC)

* DC)*

AE)*

ED)*

AC)

* FC)*

DB)

* EB)

* AB)

* AB)

Name Class Date

E

A Q

B C

D

F

K

G H

JL

M

NP

P

D

B

A

C

Q

Practice

L3

L4

L2

L1

L3


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. Obj. 10: (8.14)(B) Which three points are not collinear? C

(0, 0), (0, 2), (0, 4) (0, 0), (0, 2), (3, 0)

(0, 0), (3, 0), (5, 0) (2,-2), (2, 2), (2, 3)

*EF),

*BG),

*CD),

Example 4

(page 19)

*VW)*

XT)*

TS)*

QU)

XT4

VW4

Example 3

(page 19)

Example 2

(page 17)

AB

EF

G

CD

H

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

Apply Your Skills

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.

Careers

Exercise 66 Ask: What fact aboutEarth’s surface complicates thework of a surveyor who uses linesand planes? Earth is a sphere

and not flat, so distances are

measured 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.

2

O x

y4

24

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 Signal

A

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 AEHD

and BFGC never intersect.

* HE)

A

D J

B

C

E

F

G

H

PowerPoint

4. Assess & Reteach


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 1

c. and point P 1 d. and point G 1

e. Make a Conjecture What do you think istrue of a line and a point not on the line?

In Exercise 62 and 63, sketch a figure for the given information. Then name the

postulate 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.

*NM)

*EP)*

FG)

*PH)*

EF)

P Q

GH

FE

A

D C

B

WB

NE

A

See margin.

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

A, B, and D

65–67. 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 nline

Homework HelpVisit: PHSchool.com

Web Code: aue-0103

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

Challenge


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. The price of a shirt is decreased by 30% after $10 is subtracted from the originalprice p.Which expression can help you find how much the shirt sells for now? B

(p - 0.3p) - 10 0.3(p - 10)

p- 10 - 0.3(p- 10) (p - 10) - 0.3p

81. The results of each toss of a number cube are shown below. Based on theresults, what is the experimental probability of rolling a 5? H

1 5 5 6 5 2 4 4 3 5

15% 25% 40% 50%

82. The data below show the number of minutes Anna spent exercising each daylast week. Which box-and-whisker plot best represents the data?

0 10 11 12 12 14 16 17 19 20 C

Make an orthographic drawing for each figure. 83–85. See margin.

83. 84. 85.

Simplify each ratio.

86. 30 to 12 5 to 2 87. 88. 13

521

: 57

r2

pr2

2pr

Skills Handbook

Front RightFrontRightFront Right

Lesson 1-2

1

4

CB

D

A

A CB

See margin.

See margin.

GO forHelp

Tune-Up

Mixed Review

200 11 13 17 200 5 10 15

200 5 10 15 200 5 10 15

Objective 2

TEKS (A.3)(A)

Objective 9

TEKS (8.11)(B)

Objective 9

TEKS (8.12)(C)

TEKS (G.6)(C)

22

83. 84. 85.

Right

Top

FrontRight

Top

FrontRight

Top

Front

• TAKS Tune Up, p. 75• TAKS Strategies, p. 70

• TAKS Daily ReviewTransparencies

• TAKS Review and Preparation Workbook

• TAKS Strategies withTransparencies

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.

BC4

AB4

Documents

1-3 Points, Lines, Planes