Let's complete the table and then make observations about the relationships between the faces, edges, and vertices of Platonic solids.
1. Complete the Missing Values for the Cube:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& \text{faces} & \text{vertices} & \text{edges} \\
\hline
\text{tetrahedron} & 4 & 4 & 6 \\
\hline
\text{cube} & 6 & 8 & 12 \\
\hline
\text{dodecahedron} & 12 & 20 & 30 \\
\hline
\end{array}
\][/tex]
2. Observations about Platonic Solids:
- Observation 1: The number of edges ([tex]\(E\)[/tex]) is always greater than the number of faces ([tex]\(F\)[/tex]) for the cube.
[tex]\[
\text{For the cube: } E = 12, \; F = 6 \; \Rightarrow \; E > F \; \Rightarrow \; 12 > 6
\][/tex]
Therefore, [tex]\(E > F\)[/tex] holds true for the cube.
- Observation 2: The number of edges ([tex]\(E\)[/tex]) is always less than the sum of the number of faces and the number of vertices ([tex]\(F + V\)[/tex]) for the cube.
[tex]\[
\text{For the cube: } E = 12, \; F = 6, \; V = 8 \; \Rightarrow \; E < F + V \; \Rightarrow \; 12 < 6 + 8 \; \Rightarrow \; 12 < 14
\][/tex]
Therefore, [tex]\(E < F + V\)[/tex] holds true for the cube.
By examining the cube and the given observations, we can see consistent relationships between [tex]\(F\)[/tex], [tex]\(E\)[/tex], and [tex]\(V\)[/tex] for Platonic solids:
- The number of edges is always greater than the number of faces, [tex]\(E > F\)[/tex].
- The number of edges is always less than the sum of the number of faces and vertices, [tex]\(E < F + V\)[/tex].
These relationships are fundamental properties of Platonic solids.
