It is reversible. The definition distinguishes the line segment as a specific part of a line.

4. The definition is reversible.

i A line segment is the part of a line between and including two points. ii The part of a line between and including two points is a line segment. INITIAL POSTULATES Recall that a postulate is a statement that is assumed to be true. Through two distinct points, there is exactly one line. POSTULATE 1 EXAMPLE 1 In Figure 1.33, how many distinct lines can be drawn through a point A? b both points A and B at the same time? c all points A, B, and C at the same time? Solution a An infinite countless number b Exactly one c No line contains all three points. 쮿 Postulate 1 is sometimes stated in the form “Two points determine a line.” See Fig- ure 1.32, in which points C and D determine exactly one line, namely, . Of course, Pos- tulate 1 also implies that there is a unique line segment determined by two distinct points used as endpoints. Recall Figure 1.31, in which points A and B determine . NOTE: In geometry, the reference numbers used with postulates as in Postulate 1 need not be memorized. AB Í CD C D Figure 1.32 A B C Figure 1.33 Geometry in the Real World On the road map, driving distances between towns are shown. In traveling from town A to town D, which path traverses the least distance? Solution A to E, E to C, C to D: 10 + 4 + 5 = 19 C D B E F A 12 10 6 6

5 4

5 Recall from Section 1.2 that the symbol for line segment AB, named by its end- points, is . Omission of the bar from , as in AB, means that we are considering the length of the segment. These symbols are summarized in Table 1.3. AB AB TABLE 1.3 Symbol Words for Symbol Geometric Figure Line AB Line segment AB AB Length of segment AB A number AB Í AB A B A B A ruler is used to measure the length of any line segment like . This length may be represented by AB or BA the order of A and B is not important. However, AB must be a positive number. AB We wish to call attention to the term unique and to the general notion of unique- ness. The Ruler Postulate implies the following:

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

Characteristics 1 and 2 are both necessary for uniqueness Other phrases that may re- place the term unique include One and only one Exactly one One and no more than one A more accurate claim than the commonly heard statement “The shortest distance be- tween two points is a straight line” is found in the following definition. The measure of any line segment is a unique positive number. POSTULATE 2 왘 Ruler Postulate Geometry in the Real World In construction, a string joins two stakes. The line determined is described in Postulate 1 on the previous page. A X B Figure 1.34 As we saw in Section 1.2, there is a relationship between the lengths of the line seg- ments determined in Figure 1.34. This relationship is stated in the third postulate. It is the title and meaning of the postulate that are important The distance between two points A and B is the length of the line segment that joins the two points. AB DEFINITION If X is a point of and A-X-B, then . AX + XB = AB AB POSTULATE 3 왘 Segment-Addition Postulate Technology Exploration Use software if available. 1. Draw line segment . 2. Choose point P on . 3. Measure , , and . 4. Show that XP + PY = XY. XY PY XP XY XY EXAMPLE 2 In Figure 1.34, find AB if a AX = 7.32 and XB = 6.19. b AX = 2x + 3 and XB = 3x - 7. Solution a AB = 7.32 + 6.19, so AB = 13.51. b AB = 2x + 3 + 3x - 7, so AB = 5x - 4. 쮿 Congruent ⬵ line segments are two line segments that have the same length. DEFINITION In general, geometric figures that can be made to coincide fit perfectly one on top of the other are said to be congruent. The symbol ⬵ is a combination of the symbol ~,