The definition is reversible.

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 ~, 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