Exercises 9.4 1. Which of these two polyhedrons is concave? Note that the interior dihedral angle formed by the planes containing AEJF and AKJF is larger than 180°. K H. (a)

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
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**Exercises 9.4**

1. **Which of these two polyhedrons is concave? Note that the interior dihedral angle formed by the planes containing △EJF and △KJF is larger than 180°.**

**Diagrams:**

- **Diagram (a):**
  - Comprises a four-sided polyhedron with vertices labeled A, B, C, and D.
  - Edges are connected as follows: A to B, B to C, C to D, D to A, and diagonals A to C and B to D.

- **Diagram (b):**
  - Features a more complex polyhedron with vertices labeled E, F, G, H, I, J, and K.
  - Edges form the following connections: E to F, F to G, G to H, H to E, E to I, F to J, G to K, H to J, J to K, and a diagonal from F to I.

**Explanation for Diagrams (if required):**

- **Diagram (a):** Depicts a simple polyhedron, possibly a square pyramid or a tetrahedron with a more straightforward geometric configuration.
  
- **Diagram (b):** Illustrates a more intricate polyhedron that seems to contain additional faces or vertices, forming a more complex structure, suggesting that it's less straightforward to determine concavity from a simple inspection.

**Educational Insight:**

In geometry, a polyhedron's concavity or convexity is determined by the interior angles formed by its faces. An angle larger than 180° indicates a concave shape, as the angle bends inward. This exercise asks students to identify such concavity by examining relational angles between specific planes.
Transcribed Image Text:**Exercises 9.4** 1. **Which of these two polyhedrons is concave? Note that the interior dihedral angle formed by the planes containing △EJF and △KJF is larger than 180°.** **Diagrams:** - **Diagram (a):** - Comprises a four-sided polyhedron with vertices labeled A, B, C, and D. - Edges are connected as follows: A to B, B to C, C to D, D to A, and diagonals A to C and B to D. - **Diagram (b):** - Features a more complex polyhedron with vertices labeled E, F, G, H, I, J, and K. - Edges form the following connections: E to F, F to G, G to H, H to E, E to I, F to J, G to K, H to J, J to K, and a diagonal from F to I. **Explanation for Diagrams (if required):** - **Diagram (a):** Depicts a simple polyhedron, possibly a square pyramid or a tetrahedron with a more straightforward geometric configuration. - **Diagram (b):** Illustrates a more intricate polyhedron that seems to contain additional faces or vertices, forming a more complex structure, suggesting that it's less straightforward to determine concavity from a simple inspection. **Educational Insight:** In geometry, a polyhedron's concavity or convexity is determined by the interior angles formed by its faces. An angle larger than 180° indicates a concave shape, as the angle bends inward. This exercise asks students to identify such concavity by examining relational angles between specific planes.
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