Show that if the diagonals of quadrilateral are equal and bisect each other at right angles,then it is a square.

Sure, here is the step-by-step proof:

1. Let's consider a quadrilateral ABCD with diagonals AC and BD that are equal in length and bisect each other at right angles at point O.

2. According to the given condition, AO = OC and BO = OD. Also, ∠AOB = ∠COD = 90°.

3. Now, consider triangles AOB and COD. We can see that:

   • AO = OC (Given)
   • BO = OD (Given)
   • ∠AOB = ∠COD = 90° (Given)

   Therefore, by the RHS (Right angle-Hypotenuse-Side) criterion of congruence, we can say that ΔAOB ≅ ΔCOD.

4. Since the triangles are congruent, their corresponding parts are equal. Therefore, AB = CD and BC = AD.

5. Now, consider triangles BOA and DOA. We can see that:

   • AO = AO (Common)
   • BO = OD (Given)
   • ∠BOA = ∠DOA = 90° (Given)

   Therefore, by the RHS criterion of congruence, we can say that ΔBOA ≅ ΔDOA.

6. Since the triangles are congruent, their corresponding parts are equal. Therefore, AB = AD and DA = DC.

7. From steps 4 and 6, we can conclude that AB = BC = CD = DA.

8. Also, ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°.

9. Therefore, ABCD is a square, as all its sides are equal and all its angles are 90°.