A rectangle is defined as having two pairs of equal-length sides. Therefore, theoretically the answer is yes.
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Yes cutting a rectangle in half (either horizontally or vertically) will yield two smaller rectangles (each of which is the same size and shape as the other).
yes Sometimes those rectangles could be squares, which are by definition a type of rectangle.
No.
By dividing it into two rectangles.
You are describing a triangular prism. It has a rectangular base, two rectangular sides and two triangular ends.