AB is a chord in a circle with centre O. AB is produced to C such that BC is equal to the radius of the circle. C is joined to O and produced to meet the circle at D. If ∠ACD = 32°, then the measure of ∠AOD is ______.

96°

In \(\Delta \)OBC

If OB = BC, then

\(\angle\)BOC = \(\angle\)BCO = \({32}^\circ\)

As we know,

\(\angle\)OBA = \(\angle\)BOC + \(\angle\)BCO = 32 + 32 = 64

If OA = OB, then

\(\angle\)OBA = \(\angle\)OAB = 64

In\(\Delta \)AOB

\(\angle\)AOB + \(\angle\)OAB + \(\angle\)OBA = 180

\(\angle\)AOB + 64 + 64 = 180

\(\angle\)AOB = 180 - 128 = 52

\(\angle\)AOD + \(\angle\)AOB + \(\angle\)BOC = 180

\(\angle\)AOD + 52 + 32 = 180

\(\angle\)AOD = 180 - 84 = \({96}^\circ\).