Two segments `A C` and `B D` bisect each other at `O` . Prove that `A B C D` is a parallelogram.

Explanation

`AB \and\CD`bisect each other at O
`AO=BO \and\ CO=DO`
in`/_\COA and /_\DOB`
Given`CO=OD,/_\COA=/_BOD`(vertically opp angles)
`AD=BD`
`/_\COA~=/_\BOD`
(i)`AC=BD`(C.P.CT)
(ii)` /_CAB=/_ABD`(C.P.CT)
In `/_\COB \and\ /_\AOD`
`CO=OD`(given)
`BO=AO`(given)
`/_COB=/_AOD`(vertically opp angles)
`/_\COB~=/_\AOD`(iii)
`/_CBA=/_BAD`(C.P. C.T)
`AD∣∣CD(/_CBA=/_BAD)`(which are altanate angles)
`AD=CB`(C.P.C.T)