Question

If the two diagonals of a quadrilateral bisect each other, what can be concluded about the quadrilateral?

The quadrilateral is a rectangle.
The quadrilateral is a parallelogram.
The quadrilateral is a regular polygon.
The quadrilateral is a square.

Answer 1

If the two diagonals of a quadrilateral bisect each other, it is a parallelogram.

To prove this, imagine a quadrilateral ABCD where AC and BD intersects at E; we can say that EA = EC and EB = ED.
Vertically opposite angles in the bisection area are equal (∠AED = ∠BEC); by the SAS congruence theorem we can say that the triangles ΔAED and ΔBEC are congruent and lastly corresponding parts of congruent triangles are congruent (CPCT) then AD and BC are parallel to each other.
Similar situation happens in the horizontal opposite angles where AB and CD are parallel to each other. Thus making the quadrilateral a parallelogram.