# Totale absolutkrummung in Differentialgeometrie und by Dirk Ferus By Dirk Ferus By Dirk Ferus

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3 4 1 inches long. So, DE is 9 _ 4 c. Find y and QP if P is between Q and R, QP = 2y, QR = 3y + 1, and PR = 21. 3y ϩ 1 2y Q 21 P R Draw a figure to represent this information. QR = QP + PR Betweenness of points 3y + 1 = 2y + 21 Substitute known values. 3y + 1 - 1 = 2y + 21 - 1 Subtract 1 from each side. 3y = 2y + 20 3y - 2y = 2y + 20 - 2y y = 20 Simplify. Subtract 2y from each side. Simplify. Now find QP. QP = 2y Given = 2(20) or 40 y = 20 4. Find x and BC if B is between A and C, AC = 4x - 12, AB = x, and BC = 2x + 3.

40. REASONING Are all right angles congruent? What information would you use to support your answer? CHALLENGE For Exercises 41–44, use the following information. Each figure below shows noncollinear rays with a common endpoint. 2 rays 3 rays 4 rays 5 rays 6 rays 41. Count the number of angles in each figure. 42. Describe the pattern between the number of rays and the number of angles. 43. Make a conjecture about the number of angles that are formed by 7 noncollinear rays and by 10 noncollinear rays.

A plane containing points D and C Example 2 (p. 7) Example 3 (pp. 7–8) B p E D F C Name the geometric term modeled by each object. 3. the beam from a laser 4. a ceiling R q Draw and label a figure for each relationship. 5. A line in a coordinate plane contains X(3, -1), Y(-3, -4), and Z(-1, -3) and a point W that does not lie on XY . 6. Plane Q contains lines r and s that intersect in P. Example 4 (p. 8) For Exercises 7–9, refer to the figure. 7. How many planes are shown in the figure? B C 9. Are points A, C, D, and J coplanar?