No, Triangular prisms have two bases that are triangular but these need not be equilateral.
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Yes, triangular prisms have two faces that are equilateral triangles and three faces that are rectangles.
They can be, but they don't have to be equilateral to be a triangular prism.
Sometimes triangular prisms have isosceles triangle bases. It is the most common, but they don't always have isosceles triangles.
An equilateral triangle with 3 equally sized isosceles triangles connected to it by their bases
... whereas one of the "bases" of prisms are vertices.