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 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.

parallelepiped A D B C AD is a diagonal

parallelepiped A D B C AC is not a diagonal

A B C D 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.

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.

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

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

parallelepiped F E A D H G B C

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

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

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

Theorem 10.17 Diagonals of a parallelepiped bisect each other.

Theorem 10.18 Diagonals of a right rectangular prism are congruent.

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.

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

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

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

Homework pp. 437-438

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

7.

►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.

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

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

►B. Exercises F V n E Example 14 24 12 36 13. 7 10 5 15 15. 7 17. 8 18. n

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

►B. Exercises F V n E Example 14 24 12 36 13. 7 10 5 15 15. 7 17. 8 12 6 18 18. n

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

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

■ 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.

■ Cumulative Review 27. True/False: Water contains helium or hydrogen. 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.

■ 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?