534 CHAPTER 8 Geometry Important Facts and Concepts Examples and Discussion Section 8.1 Point, line, plane, ray, half line, line segment, angle, and related terms and definitions are discussed throughout Section 8.1. Two angles are complementary angles if the sum of their measures is 90 .° Two angles are supplementary angles if the sum of their measures is 180 .° Vertical angles, alternate interior angles, alternate exterior angles, and corresponding angles and related terms are discussed in Section 8.1. Discussion, pages 454 – 455, Examples 1 and 2, pages 455– 457 Examples 3 and 4, pages 458– 459 Discussion, page 460, Example 7, pages 460– 461 Section 8.2 The sum of the measures of the interior angles of an n-sided polygon is n( 2) − 180 .° Two polygons are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion. Discussion, page 466, Example 1, page 466 Discussion, page 467, Examples 2 and 3, pages 468– 469 Section 8.3 Perimeter, P, and Area, A TRIANGLE SQUARE RECTANGLE b (or s3) s1 h s2 s s s s w l = + + = P s s s s b ( ) 1 2 3 3 P s4 = P l w 2 2 = + = A bh 1 2 A s2 = A lw = PARALLELOGRAM TRAPEZOID w h b s2 s1 b2 b1 h P b w 2 2 = + P s s b b 1 2 1 2 = + + + A bh = = + A h b b ( ) 1 2 1 2 Discussion, pages 476– 478, Example 1, page 478, Examples 3–5, pages 480– 482 Summary CHAPTER 8 18. State the theorem concerning the sum of the measures of the angles of a triangle in a) Euclidean geometry. The sum of the measures of the angles of a triangle is 180°. b) Hyperbolic geometry. The sum of the measures of the angles of a triangle is less than 180°. c) Elliptical geometry. The sum of the measures of the angles of a triangle is greater than 180°. Research Activities 19. Poincaré Disk To complete his masterpiece Circle Limit III, M. C. Escher studied a model of hyperbolic geometry called the Poincaré disk. Write a paper on the Poincaré disk and how it was used in Escher’s art. Include representations of infinity and the concepts of point and line in hyperbolic geometry. 20. M.C. Escher To transfer his two-dimensional tiling known as Symmetry Work 45 to a sphere, M. C. Escher used the spherical geometry of Bernhard Riemann. Write a paper on Escher’s use of geometry to complete this masterpiece. 21. Fantastic Fractals Go to the website FantasticFractals. com and study the information about fractals given there. Print color copies of the Mandlebrot set and the Julia set.
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