Suppose line segments A B\ a n d\ C D in...

Suppose line segments `A B\ a n d\ C D` intersect at `O` in such a way that `A O=O D\ a n d\ O B=O Cdot` Prove that `A C=B D` but `A C` may not be parallel to `B D`

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In `/_\AOC` and `/_\ODB`
`OA=OD` (Given)
`OB=OC` (Given)
`/_AOC=/_DOB`(Vertically Opposite Angle)
`:./_\AOCcong/_\BOD` (SAS)
`AC=BD` (`CPCT`)
Also, `/_OAC=/_ODB` (`CPCT`)
and `/_OCA=/_OBD`(`CPCT`)
Thus, `/_OAC` may not be equal to `/_OBD`
`therefore AC` may not be parallel to `BD`
However, if `OA=OC` then `/_OAC=/_OCA`
Angles opposite to equal sides of `/_\OAC` but `/_OAC=/_ODB`
`therefore /_OCA=/_ODB`
But these angles form a pair of equal alternate angles.
`therefore AC` parallel to `BD`
Hence proved.

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