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By James F. Ulrich

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Postulate 8: The Ruler Postulate a. To every distinct pair of points there corresponds exactly one positive number. This number is the distance between the two points. b. To every real number there corresponds a point on the number line. To every point on the number line there corresponds a real number. In Example 1, you found that DE= 1 and EG = 4. Similarly, you could find DG = 5. Thus, DE + EG = DG. This leads to a formal definition of betweenness for points. Definition: and C if For three distinct points A, 8, and C.

The nonadjacent angles formed when two lines intersect. Some of these angles pairs are related to each other. For example, two angles that form a linear pair are always supplementary. However, supplementary angles do not always form a linear pair. Example 1 points out some of these relationships.

Example l Find the measure of each given angle. J F C B A Measure Angle a. LFBC ISO - DI= 30 or 10 - 301 = 30 b. LFBG ISO - 301 = 50 or 130 - 801 = 50 c. LHBF I104 - d. LGBJ 1135-801=55 or IB0-1351=55 301 = 74 or 130 - 1041 = 74 Segments and Angles 49 Example 1 is based on the following postulate. Postulate 9: The Protractor Postulate a. To every angle there corresponds exactly one real number between O and 180. This number is the measure of the angle. b. For half-plane Hand the point Q on its edge, p, the rays in H or p with vertex Q can be placed in one-to-one correspondence with the real numbers n, 0 :5 n :5 180, such that a coordinate system for rays is established.

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Geometry by James F. Ulrich

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