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There is no such shape. The only convex shape that has four faces is a tetrahedron, and that has four vertices, not 5.
There is no shape with only four triangular faces and five vertices. If there is also a quadrilateral face then it is a quadrialteral based pyramid.
A rectangular prism is the only shape that is made up from only rectangular faces. But there are countless shapes which contain rectangular faces along with other shapes.
The answer will depend on what is known of the shape. If you know only the number of faces, or only the number of vertices, then you cannot work out the number of edges. If you only know the shapes of the faces you cannot.The answer will depend on what is known of the shape. If you know only the number of faces, or only the number of vertices, then you cannot work out the number of edges. If you only know the shapes of the faces you cannot.The answer will depend on what is known of the shape. If you know only the number of faces, or only the number of vertices, then you cannot work out the number of edges. If you only know the shapes of the faces you cannot.The answer will depend on what is known of the shape. If you know only the number of faces, or only the number of vertices, then you cannot work out the number of edges. If you only know the shapes of the faces you cannot.
Only one; a triangle is a 2D shape, not a 3D shape. Usually only 3D (or more) shapes are considered to have 'faces'.