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A triangular prism has 5 faces. Pentagonal prisms and hexagonal pyramids have 7.
A 3d shape with 5 faces can be a square pyramid
A shape that has fewer than 8 faces and fewer than 10 edges is a triangular prism. A triangular prism has 5 faces (2 triangular faces and 3 rectangular faces) and 9 edges. It meets the criteria of having fewer than 8 faces and fewer than 10 edges.
There is no such shape. The only convex shape that has four faces is a tetrahedron, and that has four vertices, not 5.
pyramid