In circle O, AD ⊥ BC and the m ∠ACB = 65º . What is the measure of angle OBD?

Angle(ACB) is inscribed in minor arc(AB), therefore arc(AB) = 2·angle(ACB).

 

Since angle(ACB) = 65^o, arc(AB) = 130^o.

 

Arc(AB) = central angle(AOB)   --->   angle(AOB) = 130^o.

 

This makes angle(BOC) = 50^o.

 

So, angle(OBD) = 40^o.