Two pairs of touching sides are the same length. The diagonals meet at right angles. Opposite sides are not parallel.
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a square
A rectangle.
The answer depends on whether the shape is 2-dimensional or 3-d. For a 2-dimensional polygon with n sides, the minimum number is n-3 diagonals.
A triangle. Also, any shape that has no corners, such as a circle or an ellipse.
A Rectangle.