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Lesson 10.5 Polyhedra pp. 434-438

Lesson 10.5 Polyhedra pp. 434-438. Objectives: 1. To classify hexahedra and define related terms. 2. To prove theorems for parallelpipeds. 3. To state and apply Euler’s formula. Definition. A polyhedron is a closed surface made up of polygonal regions. Definition.

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Lesson 10.5 Polyhedra pp. 434-438

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  1. Lesson 10.5 Polyhedra pp. 434-438

  2. Objectives: 1. To classify hexahedra and define related terms. 2. To prove theorems for parallelpipeds. 3. To state and apply Euler’s formula.

  3. Definition A polyhedron is a closed surface made up of polygonal regions.

  4. Definition A parallelepiped is a hexahedron in which all faces are parallelograms. A diagonal of a hexahedron is any segment joining vertices that do not lie on the same face.

  5. AD is a diagonal parallelepiped A D B C

  6. AC is not a diagonal parallelepiped A D B C

  7. A D B C AB is an edge of the cube; AC is a diagonal of the square face of the cube; AD is a diagonal of the cube.

  8. Definition Opposite faces of a hexahedron are faces with no common vertices. Opposite edges of a hexahedron are two edges of opposite faces that are joined by a diagonal of the parallelepiped.

  9. parallelepiped F E A D H G B C ABCD & EFGH are opposite faces

  10. parallelepiped F E A D H G B C ABCD & CDFG are not opposite faces

  11. parallelepiped F E A D H G B C

  12. BC & EF are opposite edges parallelepiped F E A D H G B C

  13. BC & AD are not opposite edges parallelepiped F E A D H G B C

  14. Theorem 10.16 Opposite edges of a parallelepiped are parallel and congruent.

  15. Theorem 10.17 Diagonals of a parallelepiped bisect each other.

  16. Theorem 10.18 Diagonals of a right rectangular prism are congruent.

  17. Euler’s Formula V - E + F = 2 where V, E, and F represent the number of vertices, edges, and faces of a convex polyhedron respectively.

  18. Euler’s formula applies not only to parallelepipeds but to all convex polyhedra.

  19. Tetrahedron V = 4 E = 6 F = 4 V - E + F = 2 V = E = F = V - E + F =

  20. Octahedron V = E = F = V - E + F = V = 6 E = 12 F = 8 V - E + F = 2

  21. Homework pp. 437-438

  22. ►A. Exercises For each decahedron below, determine the number of faces, edges, and vertices. Check Euler’s formula for each. 7.

  23. 7.

  24. ►B. Exercises Each exercise below refers to a prism having the given number of faces, vertices, edges, or sides of the base. Determine the missing numbers to complete the table below. Draw the prism when necessary; find some general relationships between these parts of the prism to complete exercise 18.

  25. ►B. Exercises F V S E Example 14 24 12 36 13. 7 10 15. 7 17. 8

  26. 13. Faces (F) = 7 Vertices (V) = 10 Sides of the base (S) = Edges (E) = 5 15

  27. ►B. Exercises F V n E Example 14 24 12 36 13. 7 10 515 15. 7 17. 8 18. n

  28. 17. Faces (F) = 8 Vertices (V) = Sides of the base (S) = Edges (E) = 12 6 18

  29. ►B. Exercises F V n E Example 14 24 12 36 13. 7 10 515 15. 7 17. 8 12 6 18 18. n

  30. A E D B C ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 24. Find the area.

  31. D E C A B ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 25. Prove that A  B.

  32. ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 26. Find the distance between two numbers a and b on a number line.

  33. ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 27. True/False: Water contains helium or hydrogen.

  34. ■ Cumulative Review Do not solve exercises 24-27 below, but write (in complete sentences) what you would do to solve them. 28. When are the remote interior angles of a triangle complementary?

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