in the given figure, O is the centre of circle, OL IS perpendicular to AB and OM is perpendicular to CB, if angle OAB =angle OCB, then prove that AB = CB.

Here AB and CB is the chord.
And as we know that Perpendicular line from centre of circle bisect the cord.
So in chord AB, AL = LB
Similarly in chord CB, CM = MB
Now in triangle AOL and COM, we have
::Angle CLA = Angle AMC = 90⁰
::OA = OC                  (BOTH ARE RADIUS)
::Angle LOA = Angle MOC          (Vertically opposite angles)

Therefore, triangle AOL is congruent to triangle COM by AAS criterion of congruency
=> AL = MC                                      (BY C.P.C.T)
=>2AL = 2MC                                    (Multiplying both the sides by 2)
=>AB = BC                                        (2AL = AB and 2MC = BC)