Trisecting an angle
Ravi wants to trisect an angle \(AOB\), which has measure \(\theta\). From \(A\), he drops a perpendicular to side \(OB\), intersecting at \(C\). He then constructs an equilateral triangle \(ACD\) on the opposite side of \(AC\) as compared to \(O\). He claims (without any justification) that \(2\angle DOB = \angle DOA\), so \(OD\) will trisect angle \(AOB\). What is the sum of all angles \(\theta\) with \( 0 ^\circ < \theta < 90^\circ\), such that his claim is true?