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28 CHAPTER 1 ,~~ LINE AND ANGLE RELATIONSHIPS

If two distinct planes intersect, then their intersection is a line.

The intersection of two planes is infinite because it is a line. [See Figure 1.44(a) onpage 27.] If two planes do not intersect, then they are parallel. The parallel verticalpl anes Rand S in Figure 1.44(b) may remind you of the opposite walls of your class-room. The parallel horizontal planes M and N in Figure 1.44(c) suggest the relationshipbetween ceiling and floor.

Imagine a plane and two points of that plane, say points A and B. Now think of theline containing the two points and the relationship of AS to the plane. Perhaps your con-clusion can be summed up as follows.

Given two distinct points in a plane, the line containing these point'> also lies in the plane.

Exs. 13-16 Because the uniqueness of the midpoint of a line segment can be justified, we callthe following statement a theorem. The "proof" of the theorem is found in Section 2.2.

The midpoint of a line segment is unique.

If M is the midpoint of AB in Figure 1.45, then no other point can separate AB intotwo congruent parts. The proof of this theorem is based on the Ruler Postulate. M is thepoint that is located !(AB) units from A (and from B).

The numbering system used to identify Theorem J .3.1 need not be memorized.However, this theorem number may be used in a later reference. The numbering systemworks as follows:

A M 8

Figure 1.45

JCHAPTER

wherefound

IORDERfound insection

3SECTION

wherefound

A summary of the theorems presented in this textbook appears at the end of the book.

,-,~ !

In Exercises 1 and 2, complete the statement. In Exercises 3 and 4, use thefact that 1foot = 12 inches.

~~---------~ABC

3. Convert 6.25 feet to a measure in inches.4. Convert 52 inches to a measure in feet and inches.

Exercises 1, 2 In Exercises 5 and 6, use thefact that 1 meter= 3.28feet(measure is approximate).

5. Convert! meter to feet.

6. Convert 16.4 feet to meters.

1. AB+BC=.....:L

2. If AB = BC, then B is the.....:L of AC.

_ onrrical

7. In the figure, the IS-mile roadfrom A to C is underconstruction. A detour from A toB of 5 miles and then from B toC of 13 miles must be taken.How much farther is the"detour" from A to C than theroad from A to C? Exercises 7, 8

8. A cross-country runner jogs at a rate of 15 meters persecond. If she runs 300 meters from A to B, 450 metersfrom B to C, and then 600 meters from C back to A, howlong will it take her to return to point A?

B

A

~ ::'5S-

.~ :he

. ~,ln-

(HINT: See figure for Exercise 7.)

In Exercises 9 to 28, use the drawings as needed to answer thefollowing questions.

c :all- . ")

9. Name three points that appear to bea) collinear. b) noncollinear.

A 8

c

o

Exercises 9, 10c .nto_~.he 10. How many lines can be drawn through

a) point A? c) points A, B, and C?b) points A and B? d) points A, B, and D?

11. Give the meanings of CD, CD, CD, and CD.12. Explain the difference, if any, between

a) CD and Dc. c) CD and DC.b) CD and DC. d) CD and DC.

13. Name two lines that appear to bea) parallel. b) nonparallel.

-. -':-.... ~m

~--~:~---.~. --~~

A [\;1 BCD

----------- t

Exercises 13-17

14. Classify as true or false:a) AB + BC = ADb) AD - CD = ABc) AD - CD = ACd) AB + BC + CD = ADe) AB=BC

15. Given: M is the midpoint of ABAM = 2x + 1 and MB = 3x - 2

Find: x and AM

7.3 Early Definitions and Postulates 29

c 16. Given: M is the midpoint of ABAM = 2(x + 1) andMB = 3(x - 2)

Find: x and AB17. Given: AM= 2x+ 1, MB= 3x+2, andAB=6x-4

Find: x and AB18. Can a segment bisect a line? a segment? Can a line bisect

a segment? a line?1 9. In the figure, name

a) two opposite rays.b) two rays that are not opposite.

20. Suppose that (a) point C lies in plane X and (b) po~Dlies in plane X. What can you conclude regarding CD?

21. Make a sketch ofa) two intersecting lines that are perpendicular.b) two intersecting lines that are not perpendicular.c) two parallel lines.

22. Make a sketch ofa) two intersecting planes.b) two parallel planes.c) two parallel planes intersected by a third plane that is

not parallel to the first or the second plane.

23. Suppose that (a) planes M and N intersect, (b) point A liesin both planes M and N, and (c) point B lies ~oth planesM and N. What can you conclude regarding AB?

24. Suppose that (a) points A, B, and C are collinear and(b) AB > AC. Which point can you conclude cannot liebetween the other two?

25. Suppose that points A, R, and V are collinear. If AR =7and RV= 5, then which point cannot possibly lie betweenthe other two?

26. Points A, B, C, and Darecoplanar; B, C, and Darecollinear; point E is not inplane M. How many planescontaina) points A, B, and C?b) points B, C, andD?c) points A, B, C, and D?d) points A, B, C, and E?

27. Using the number line provided, name the point thata) is the midpoint of AE.b) is the endpoint of a segment of length 4, if the other

endpoint is point G.c) has a distance from B equal to 3(AC).

ABC 0 E G H~1~. __ ~~~_~ ,~

-3 -2 -1 0 1 2 3 4

Exercises 27,28

••

II

--30 CHAPTER 1 LINE AND ANGLE RELATIONSHIPS

28. Consider the figure for Exercise 27. Given that B is themidpoint ofAC and C is the midpoint of BD, what can

you conclude about the lengths of- -

a) AB and CD? c) AC and CD?b) AC andBD?

36. Consider noncoplanar points A, B, C, and D. Using threepoints at a time (such as A, B, and C), how many planes

are determ{ned 6y these pOl~{s;;37. Line e is parallel to plane P (that is, it will not intersect P

even if extended). Line m intersects line e. What can youconclude about m and P?

In Exercises 29 to 32, use only a compass and a straightedgeto complete each construction.

29. Given: AB and CD (AB > CD)Construct: MN on line e so that MN =AB + CD m

B,------.A

Exercises 29, 3038. AB and EF are said to be skew lines because they neither

intersect nor are parallel. How many planes aredetermined bya) parallel lines AB and DC?b) intersecting lines AB and BC?c) skew lines AB and EF?d) lines AB, BC, and DC?e) points A, B, and F?f) points A, C, and H?g) points A, C, F, and H?

30. Given:Construct:

31. Given:Construct:

AB and CD (AB > CD)EF so that EF=AB - CD

AB as shown in the figurePQ on line n so that PQ = 3(AB)

A 8$'---0

n

•Exercises 31,32 G F/i--------~~7

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i:H 1I )-----------+--~E1,,/ \,/l vo c

32. Given: AB as shown in the figure- 1

Construct: TV on line n so that TV = 2: (AB)33. Can you use the construction for the midpoint of a

segment to divide a line segment intoa) three congruent parts? c) six congruent parts?b) four congruent parts? d) eight congruent parts?

34. Generalize your findings in Exercise 33.35. Consider points A, B, C, and D, no three of which are

collinear. Using two points at a time (such as A and B),how many lines are determined by these points?

*39. Let AB =a and BC =b. Point M is the midpoint of BC. IfAN = ~(AB), find the length of NM in terms of a and b.

Angle: Sides of Angle,Vertex of Angle

Protractor PostulateAcute, Right, Obtuse,

Straight, and Reflex Angles

Angle-Addition Postulate

Adjacent AnglesCongruent AnglesBisector of an Angle

Complementary AnglesSupplementary AnglesVertical Angles

This section introduces you to the language of angles. Recall from Sections 1.1 and 1.3that the word union means that two sets or figures are joined.