`A B\ a n d\ C D` are two parallel chords of a circle whose diameter is `A C` . Prove that `A B=C D`

Explanation

Since, `AB || CD \and\ OL _|_ AB`
`OM_|_CD.`
Therefore, `LOM` is a straight line.
Again, `AB || CD \and\ AC` is a transversal.
`=>/_OAL =/_OCM` (Alternate angles)
In `/_\OLA and /_\OMC`
`=>/_AOL =/_COM` (Vertically opposite angles)
`/_OLA =/_OMA`(Each equal to `90^@`)
and`OA = OC` (Radii of same circle)
`/_\OLA ~=/_\OMC` (By AAS congruency criteria)
`=>AL = CM` (By C.P.C.T)
`=>2AL = 2CM`
(Perpendicular drawn from the center of a circle to a chord bisects the chord)
`AB = CD`