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

Given:□ABCDis a quadrilateral.diagAC=diagBD, intersecting atE.ACandBDare perpendicular bisectors of each other.∴∠E=90oTo prove:□ABCDis a square.Solution:A square is a parallelogram with all sides equal and one angle is90∘.First let us prove all sides are equal.In△ABEand△ADE.BE=DEgivenAE=AE...common side∠AEB≅∠AEDeach90o∴△ABE≅△ADE...SAS test of congruence∴AB=AD...corrseponding sides of congruent triangles(c.s.c.t).. (1)Similarly, we can prove△ABE≅△CBE∴AB=CBc.s.c.t. (2)And, from△ADE≅△CDE∴AD=CDc.s.c.t. (3)∴From (1), (3) and (4),AB=CB=CD=AD(4)Therefore, all sides are equal.Now, we prove one angle is90∘InΔABCandΔABDAC=BDgivenBC=AD Opposite sides of parallelogram are equal.AB=AB...common sideThereforeΔACB≅ΔBDA.. By SSS of congruence.So,∠ABC=∠BAD...Corresponding angles of congruent triangles (c.a.c.t)Since,AD||BCandABis transversalSo,∠A+∠B=180∘.interion angles on same side of a transversal is supplementary .Since∠A=∠BSo,∠A+∠A=180∘2A=180A=1802∴A=90∘Thus, in quad.ABCDAB=CB=CD=ADand∠A=90∘∴□ABCDis a square By definition