Answer
1. If the diagonals are perpendicular and congruent, they will create four triangles, and each triangle will have a 90 degree corner in the corner in the center of the shape.
2. Since the diagonals are congruent and perpendicular, they will bisect each other.
3. By the definition of a bisector, this means that the sides of each triangle that are touching each 90 degree angle will all be equal.
4. By SAS, this means that all of the triangles are congruent.
5. Because CPCTC, this means that the sides opposite the 90 degree angles will all be congruent.
6. If all four sides are congruent, it is a square.
7. Thus, if the diagonals are perpendicular and congruent, then the shape is a square.
