We are given rectangle ABCD with `bar(AC) cong bar(BD),` and we are asked to show that ABCD is a square.
There is insufficient evidence to show that ABCD is a square. In every rectangle, the diagonals are congruent.
All rectangles are parallelograms (both pairs of opposite sides are parallel and congruent, both pairs of opposite angles are congruent, and the diagonals bisect each other). Also, the angles are right angles by definition. So, in rectangle ABCD `Delta ABC cong Delta BCD` by SAS so AC=BD by CPCTC.
In order to show that a quadrilateral is a square, we need to show that it is a rectangle and a rhombus.
First, the quadrilateral needs to be a parallelogram. Then, it needs one of the distinguishing characteristics of a rectangle (at least one of the angles is a right angle, or the diagonals are congruent). Then, it needs one of the characteristics of a rhombus (one pair of consecutive sides are congruent, the diagonals are perpendicular, or the diagonals bisect both pairs of opposite angles).
Here we are given a rectangle, so we need to be given (or be able to show from the givens) that there is a pair of congruent consecutive sides, that the diagonals are perpendicular, or that the diagonals bisect the angles.