`A B C D` is a parallelogram. `A B` is produced to `E` so that `B E=A B` . Prove that `E D` bisects `B Cdot`

Explanation

Given: `ABCD` is a parallelogram.
To prove: `ED` bisects `BC`
Proof: `AB` parallel to `CD`
`=>AE` parallel to `CD`
`CB` is the transversal
`=>angle3=angle4` (alternate `angles` are equal)
In`triangleOCD` and `triangleOBE`,
`CD=AB=BE`
`angle2=angle1` (Vertically opposite `angles`)
`angle3=angle4` (Proved above)
`∴ triangleOCD~=triangleOBE` (AAS)
`=>BO=OC`
`=>O` is the mid-point of `BC`
`therefore ED` bisects `BC`