The net you are referring to is called a square pyramid net. It consists of four triangular faces and one square base. When the net is folded along its edges and assembled, it forms a three-dimensional square pyramid shape.
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It could be a cone
Yes. For example, dodecagons, squares and triangles.
A pair of triangles can form a 4 sided quadrilateral
Draw a rectangle made with three squares. Then put a diagonal line through the center one. Tah-Dah