Show that if the diagonals of …

ANSWER Given, Diagonals are equal AC=BD                  .......(1) and the diagonals bisect each other at right angles OA=OC;OB=OD           ...... (2) ∠AOB= ∠BOC= ∠COD=  ∠AOD= 900    ..........(3)   Proof: Consider △AOB and △COB OA=OC  ....[from (2)] ∠AOB= ∠COB OB is the common side Therefore, △AOB≅ △COB From SAS criteria, AB=CB Similarly, we prove △AOB≅ △DOA, so AB=AD △BOC≅ △COD, so CB=DC So, AB=AD=CB=DC               ....(4) So, in quadrilateral ABCD, both pairs of opposite sides are equal, hence ABCD is  parallelogram In △ABC and △DCB AC=BD            ...(from (1)) AB=DC            ...(from $$(4)$$) BC is the common side △ABC≅ △DCB So, from SSS criteria, ∠ABC= ∠DCB Now, AB∥CD,BC is the tansversal ∠B+∠C= 180 ∠B+∠B= 180 ∠B= 90 Hence, ABCD is a parallelogram with all sides equal and one angle is 90 So, ABCD is a square. Hence proved.