To prove that the vertices of a quadrilateral form a square, you can follow these steps:
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Prove that the vertices form a rectangle: A rectangle can be proven by showing that it has four right angles or that the diagonals are equal in length.
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Prove that the vertices form a rhombus: A rhombus can be proven by showing that all four sides are of equal length or that the diagonals are perpendicular to each other.
By successfully proving both that the quadrilateral is a rectangle and that it is a rhombus, you can conclude that the vertices of the quadrilateral form a square, as a square is defined as a quadrilateral that is both a rectangle and a rhombus.