There exists a number measure for each line segment. 2. Only one measure is permissible.

which means that the figures have the same shape, and =, which means that the correspon- ding parts of the figures have the same measure. In Figure 1.35, , but meaning that and are not congruent. Does it appear that ? CD ⬵EF EF AB AB ⬵ EF AB ⬵CD A B C D E F Figure 1.35 EXAMPLE 3 In the U.S. system of measures, 1 foot = 12 inches. If AB = 2.5 feet and CD = 2 feet 6 inches, are and congruent? Solution Yes, because 2.5 feet = 2 feet + 0.5 feet or 2 feet + 0.512 inches or 2 feet 6 inches. 쮿 AB ⬵CD CD AB The midpoint of a line segment is the point that separates the line segment into two con- gruent parts. DEFINITION In Figure 1.36, if A, M, and B are collinear and , then M is the midpoint of . Equivalently, M is the midpoint of if AM = MB. Also, if , then is described as a bisector of . If M is the midpoint of in Figure 1.36, we can draw these conclusions: AM = MB MB = AB AB = 2 MB AM = AB AB = 2 AM 1 2 1 2 AB AB CD AM ⬵MB AB AB AM ⬵MB A B C D M Figure 1.36 Discover Assume that M is the midpoint of in Figure 1.36. Can you also conclude that M is the midpoint of ? ANSWER CD AB EXAMPLE 4 GIVEN: M is the midpoint of not shown. EM = 3x + 9 and MF = x + 17 FIND: x and EM Solution Because M is the midpoint of , EM = MF. Then By substitution, EM = 34 + 9 = 12 + 9 = 21. 쮿 x = 4 2x = 8 2x + 9 = 17 3x + 9 = x + 17 EF EF No In geometry, the word union is used to describe the joining or combining of two figures or sets of points. Ray AB , denoted by , is the union of and all points X on such that B is be- tween A and X. Í AB AB AB DEFINITION In Figure 1.37, , , and are shown; note that and are not the same ray. BA AB BA AB Í AB Opposite rays are two rays with a common endpoint; also, the union of opposite rays is a straight line. In Figure 1.39a, and are opposite rays. The intersection of two geometric figures is the set of points that the two figures have in common. In everyday life, the intersection of Bradley Avenue and Neil Street is the part of the roadway that the two roads have in common Figure 1.38. BC BA In Figure 1.40, and n are parallel; in symbols, 储 n and ¨ n = ⭋. However, and m intersect and are not parallel; so ¨ m = A and m. 储 AB has no endpoints A B Line AB A B AB has endpoint A Ray AB BA has endpoint B A B Ray BA Figure 1.37 Another undefined term in geometry is plane. A plane is two-dimensional; that is, it has infinite length and infinite width but no thickness. Except for its limited size, a flat surface such as the top of a table could be used as an example of a plane. An upper- case letter can be used to name a plane. Because a plane like a line is infinite, we can show only a portion of the plane or planes, as in Figure 1.41 on page 27. Bradley Ave. Neil St. Figure 1.38 If two lines intersect, they intersect at a point. POSTULATE 4 When two lines share two or more points, the lines coincide; in this situation, we say there is only one line. In Figure 1.39a, and are the same as . In Figure 1.39b, lines and m intersect at point P. Í AC Í BC Í AB A C B a P m b Figure 1.39 Parallel lines are lines that lie in the same plane but do not intersect. DEFINITION EXAMPLE 5 In Figure 1.40, 储 n. What is the intersection of a lines and m? b line and line n? Solution a Point A b Parallel lines do not intersect. 쮿 m B A n Figure 1.40 Exs. 5–12 Planes R and S Planes T and V T V R S T V R S Figure 1.41 A plane is two-dimensional, consists of an infinite number of points, and contains an infinite number of lines. Two distinct points may determine or “fix” a line; like- wise, exactly three noncollinear points determine a plane. Just as collinear points lie on the same line, coplanar points lie in the same plane. In Figure 1.42, points B, C, D, and E are coplanar, whereas A, B, C, and D are noncoplanar. In this book, points shown in figures are assumed to be coplanar unless otherwise stated. For instance, points A, B, C, D, and E are coplanar in Figure 1.43a, as are points F , G, H, J, and K in Figure 1.43b. B D E C A Figure 1.42 Geometry in the Real World A B C E D a K H J G F b Figure 1.43 Through three noncollinear points, there is exactly one plane. POSTULATE 5 On the basis of Postulate 5, we can see why a three-legged table sits evenly but a four-legged table would “wobble” if the legs were of unequal length. Space is the set of all possible points. It is three-dimensional, having qualities of length, width, and depth. When two planes intersect in space, their intersection is a line. An opened greeting card suggests this relationship, as does Figure 1.44a. This notion gives rise to our next postulate. a b c R R S M N S R R S M N S Figure 1.44 © Yuny ChabanShutterstock The tripod illustrates Postulate 5 in that the three points at the base enable the unit to sit level. Because the uniqueness of the midpoint of a line segment can be justified, we call the following statement a theorem. The “proof ” of the theorem is found in Section 2.2. The intersection of two planes is infinite because it is a line. [See Figure 1.44a on page 27.] If two planes do not intersect, then they are parallel. The parallel vertical planes R and S in Figure 1.44b may remind you of the opposite walls of your class- room. The parallel horizontal planes M and N in Figure 1.44c suggest the relationship between ceiling and floor. Imagine a plane and two points of that plane, say points A and B. Now think of the line containing the two points and the relationship of to the plane. Perhaps your con- clusion can be summed up as follows. Í AB If M is the midpoint of in Figure 1.45, then no other point can separate into two congruent parts. The proof of this theorem is based on the Ruler Postulate. M is the point that is located AB units from A and from B. The numbering system used to identify Theorem 1.3.1 need not be memorized. However, this theorem number may be used in a later reference. The numbering system works as follows: 1 3 1 CHAPTER SECTION ORDER where where found in found found section A summary of the theorems presented in this textbook appears at the end of the book. 1 2 AB AB If two distinct planes intersect, then their intersection is a line. POSTULATE 6 Given two distinct points in a plane, the line containing these points also lies in the plane. POSTULATE 7 Exs. 13–16 The midpoint of a line segment is unique. THEOREM 1.3.1 A B M Figure 1.45 Exs. 17–20 Exercises 1.3 In Exercises 1 and 2, complete the statement. In Exercises 3 and 4, use the fact that 1 foot = 12 inches.

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

In Exercises 5 and 6, use the fact that 1 meter ≈ 3.28 feet measure is approximate.

5. Convert meter to feet. 6. Convert 16.4 feet to meters.

1 2 A C B Exercises 1, 2

1. AB

+ BC = ___ ?

2. If AB

= BC, then B is the ___ ? of . AC

7. In the figure, the 15-mile road

from A to C is under construction. A detour from A to B of 5 miles and then from B to C of 13 miles must be taken. How much farther is the “detour” from A to C than the road from A to C?

8. A cross-country runner jogs at a rate of 15 meters per

second. If she runs 300 meters from A to B, 450 meters from B to C, and then 600 meters from C back to A, how long will it take her to return to point A? HINT: See figure for Exercise 7. In Exercises 9 to 28, use the drawings as needed to answer the following questions.

9. Name three points that appear to be

a collinear. b noncollinear. A B C D Exercises 9, 10

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, and .

12. Explain the difference, if any, between

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

13. Name two lines that appear to be

a parallel. b nonparallel. DC CD DC CD Í DC Í CD CD CD Í CD m t A M C D B Exercises 13–17

14. Classify as true or false:

a b c d e AB = BC

15. Given:

M is the midpoint of and Find: x and AM MB = 3x - 2 AM = 2x + 1 AB AB + BC + CD = AD AD - CD = AC AD - CD = AB AB + BC = AD C B A Exercises 7, 8

16. Given: M

is the midpoint of and Find: x and AB

17. Given: AM

= 2x + 1, MB = 3x + 2, and AB = 6x - 4 Find: x and AB 18. Can a segment bisect a line? a segment? Can a line bisect a segment? a line?

19. In the figure, name

a two opposite rays. b two rays that are not opposite. MB = 3 1x - 22 AM = 2 1x + 12 AB A D O B C

20. Suppose that a point C lies in plane X and b point D

lies in plane X. What can you conclude regarding ?

21. Make a sketch of

a two intersecting lines that are perpendicular. b two intersecting lines that are not perpendicular. c two parallel lines.

22. Make a sketch of

a 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 lies

in both planes M and N, and c point B lies in both planes M and N. What can you conclude regarding ?

24. Suppose that a points A, B, and C are collinear and

b AB ⬎ AC. Which point can you conclude cannot lie between the other two?

25. Suppose that points A, R, and V are collinear. If AR

= 7 and RV = 5, then which point cannot possibly lie between the other two?

26. Points A, B, C, and D are

coplanar; B, C, and D are collinear; point E is not in plane M. How many planes contain a points A, B, and C? b points B, C, and D? c points A, B, C, and D? d points A, B, C, and E?

27. Using the number line provided, name the point that

a is the midpoint of . 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 3AC. AE Í AB Í CD M B D E C A –3 –2 –1 1 2 3 4 A B C D E F G H Exercises 27, 28