DC – tangent to the circle centered at point O, B – tangent point, triangle BOA is equilateral. Determine the BOA angle.

Since the AOB triangle is equilateral, all of its internal angles are 60.

OB is the radius of the circle, then it is perpendicular to the tangent DC, the angle OBD = OBC = 90.

Point A can be located on opposite sides of point B.

In one case, the angle ABD = OBD – OBA = 90 – 60 = 30.

In another case, the angle A1BD = 180 – CBA1 = 180 – (OBC – OBA1) = 180 – (90 – 60) = 150.

Answer: Angle AOB is 60, Angle ABD is 30 or 150.