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1.2 Using Segments and Congruence Types of rules used in Geometry: Definitions Property/Postulate/axiom: Rule accepted without proof. Theorem: Rule that must be proven to be used. Ruler Postulate The points on a line can be assigned real number coordinates. (think of numbers on a number line.) ! The distance between any two points is their difference. ! (Subtract to find out how far apart they are.) ! ! Your book reminds you to take the absolute value because you want the distance, which must be a positive value. EXAMPLE 1 Apply the Ruler Postulate Measure the length of ST to the nearest tenth of a centimeter. SOLUTION Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align S with 2, T appears to align with 5.4. ST = 5.4 – 2 = ANSWER The length of ST is about 3.4 centimeters. or ST = 3.4 cm. Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. (Not all rules work both directions.) 2 segment pieces add to equal the whole segment. Any time a segment is named without its symbol, we are referring to the number measurement, or length of the segment. Map The cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri. SOLUTION Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate. LS = LT + TS = 380 + 360 = 740 ANSWER The distance from Lubbock to St. Louis is about 740 miles. Congruent: Same size and shape. Congruent segments have the same length. Use the diagram to find GH. SOLUTION Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. FH = FG + GH Segment Addition Postulate. 36 = 21 + GH Substitute 36 for FH and 21 for FG. 15 = GH Subtract 21 from each side. Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JK and LM are congruent. SOLUTION To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints. JK = 2 – (– 3) Use Ruler Postulate. To find the length of a vertical segment, find the absolute value of the difference of the ycoordinates of the endpoints. LM = – 2 – 3 = ANSWER JK and LM have the same length. So, JK ~ = LM. Geometry Page 12 ( 1-10, 12, 16-20ev, 21-29 odd, 32, 33, 40-45) Sophomore Math Day one ! Page 12 ( 1-12, 21-23, 32, 43-45) ! Day two ! Page 12 (13-19, 24-26, 33, 40-42) !