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Exploring Spherical Geometry Concepts and Applications

Learn about lines in plane and spherical geometry, compare Euclidean and spherical geometries, find surface areas and volumes of spheres and hemispheres. Practice identifying geometric relationships.

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Exploring Spherical Geometry Concepts and Applications

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  1. LESSON 12–7 Spherical Geometry

  2. Five-Minute Check (over Lesson 12–6) Then/Now TEKS New Vocabulary Key Concept: Lines in Plane and Spherical Geometry Example 1: Describe Sets of Points on a Sphere Example 2: Real-World Example: Identify Lines in Spherical Geometry Example 3: Compare Plane Euclidean and Spherical Geometries Lesson Menu

  3. Find the surface area of the sphere. Round to the nearest tenth. A. 201.1 in2 B. 223.5 in2 C. 251.9 in2 D. 268.1 in2 5-Minute Check 1

  4. Find the surface area of the sphere. Round to the nearest tenth. A. 1107 ft2 B. 1256.6 ft2 C. 3987.1 ft3 D. 4188.8 ft3 5-Minute Check 2

  5. Find the volume of the hemisphere. Round to the nearest tenth. A. 2237.8 cm2 B. 2412.7 cm2 C. 8578.6 cm3 D. 9006.1 cm3 5-Minute Check 3

  6. Find the volume of the hemisphere. Round to the nearest tenth. A. 680.6 mm2 B. 920.8 mm2 C. 1286.2 mm3 D. 1294.3 mm3 5-Minute Check 4

  7. Find the surface area of a sphere with a diameter of 14.6 feet to the nearest tenth. A. 637.4 ft2 B. 669.7 ft2 C. 1428.8 ft2 D. 1629.5 ft3 5-Minute Check 5

  8. A hemisphere has a great circle with circumference approximately 175.84 in2. What is the surface area of the hemisphere? A. 4926.0 in2 B. 7381.5 in2 C. 19,704.1 in2 D. 39,408.1 in2 5-Minute Check 6

  9. Targeted TEKS G.4(D) Compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle. Mathematical Processes G.1(A), G.1(G) TEKS

  10. You identified basic properties of spheres. • Describe sets of points on a sphere. • Compare and contrast Euclidean and spherical geometries. Then/Now

  11. Euclidean geometry • spherical geometry • non-Euclidean geometry Vocabulary

  12. Concept

  13. Answer:AD and BC are lines on sphere S that contain point R. Describe Sets of Points on a Sphere A. Name two lines containing R on sphere S. Example 1A

  14. Answer:GK Describe Sets of Points on a Sphere B. Name a segment containing point C on sphere S. Example 1B

  15. Describe Sets of Points on a Sphere C. Name a triangle on sphere S. Answer:ΔKGH Example 1C

  16. A.TL B.VM C.PS D.RO A. Determine which line on sphere Q does not contain point P. Example 1A

  17. A.LO B.MS C.NV D.SO B. Determine which segment on sphere Q does not contain point N. Example 1B

  18. C. Name a triangle on sphere Q. A.ΔVSU B.ΔPRU C.ΔNSU D.ΔNOM Example 1

  19. Identify Lines in Spherical Geometry SPORTS Determine whether the line h on the basketball shown is a line in spherical geometry. Explain. Notice that line h does not go through the poles of the sphere. Therefore line h is not a great circle and so not a line in spherical geometry. Answer: No; it is not a great circle. Example 2

  20. SPORTS Determine whether the line X on the basketball shown is a line in spherical geometry. A. Yes, it is a line in spherical geometry. B. No, it is not a line in spherical geometry. Example 2

  21. Compare Plane Euclidean and Spherical Geometries A. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. If two lines are parallel, they never intersect. Answer: True; given a line, the only line on a sphere that is always the same distance from the line is the line itself. Example 3A

  22. Compare Plane Euclidean and Spherical Geometries B. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. Any two distinct lines are parallel or intersect once. Answer: False; two distinct lines on a sphere intersect twice. Example 3B

  23. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. Perpendicular lines intersect at exactly one point. A. This holds true in spherical geometry. B. This does not hold true in spherical geometry. Example 3A

  24. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. It is possible for two circles to be tangent at exactly one point. A. This holds true in spherical geometry. B. This does not hold true in spherical geometry. Example 3B

  25. LESSON 12–7 Spherical Geometry

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