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You could prove this by congruent triangles, but here are two simpler arguments:
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Since a square is a rhombus, and the diagonals of a rhombus are perpendicular bisectors of each other, then the diagonals of a square must be perpendicular bisectors of each other
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A square has four-fold rotational symmetry - as you rotate it around the point where the diagonals cross, there are four positions in which it looks the same. This means that the four angles at the centre must be equal. They will each measure 360/4 = 90 degrees, so the diagonals are perpendicular. Also. the four segments joining the centre to a vertex are all equal, so the diagonals bisect each other.
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The diagonals of a square are perpendicular bisectors of each other?
Yes
How are the diagonals of a square?
The diagonals of a square are congruent, bisect each other, perpendicular, and either diagonal's length is sqrt(2) times any side length.
What quadrilaterals Have diangles that are perpendicular to each other?
A square, a rhombus and a kite are three examples of quadrilaterals that have perpendicular diagonals that intersect each other at right angles.
Do the diagonals of a parallelogram bisect each other at a right angle?
Only for a square or rhombus (diamond shape). The diagonals of a rectangle bisect each other, but are not perpendicular and do not bisect the opposite angles they join.
What is quadrilateral whose diagonals are perpendicular to each other but unequal?
It is a kite or a rhombus both of which have unequal diagonals that are perpendicular to each other creating right angles.
