Let the quadrilateral be ABCD. Diagonals AC and BD iare equal and they ntersect (bisect) at right angles at O.
So AO = BO = OC = OD = AC/2 = BD/2
In ΔAOB and ΔBOC, angle AOB = 90 deg = BOC. AO = BO = CO
Hence both triangles are congruent. So AB = BC
Similarly in ΔBOC and Δ COD, they are congruent and BC = CD
Similarly in ΔCOD and Δ DOA, they are congruent and CD = DA
In Δ AOB, angle OAB = OBA as the sides OA and OB are equal.
angle OAB = angle OBA = (180 - AOB)/2 = 90/2 = 45 deg
Similarly in ΔBOC, ΔCOD and ΔDOA, we get
angle OBC = OCB = OCD = ODC = ODA = OAD = 45 deg.
Hence in quadrilateral ABCD, angle A = angle OAD + OAB = 45+45=90 deg.
Angle B = angle OBA + OBC = 45+45 = 90 deg.
Similarly other angles C = D = 90 deg.
Hence ABCD is a square.