Only in the case of the square (squares are rectangles).
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A square or rectangle.
If you split a rectangle into three parts one of those halves is a third of a rectangle.
Yes, two equal right-angle triangles.
The square has four lines of symmetry. The rectangle has only two, as it can be folded in half horizontally or vertically: students should be encouraged to try to fold the rectangle in half diagonally to see why this does not work.
You split the water evenly into cups