I'm trying to picture your circle as you see it. I see the circle with centre P. There's a line PD from P to the circumference, and another line PA from P to the circumference. The angle APD at the centre between the radial arms is either 138 or 128 degrees, depending on the whether the problem is described correctly in the title or in the context. There's a point B on the circumference and it's joined to A and D. That's the angle we're trying to find. The answer depends on whether both the V-shaped angles APD and ABD point in the same direction or in opposite directions. Let's say the shape ABDP looks like an arrowhead. In this case ABD will be half APD, so it's 69 or 64 degrees (half of 138 or 128). Now take the case where ADBP is a quadrilateral shaped roughly like a diamond. This time we have to take the reflex angle APD. This angle is 360 less 138 or 128 so it's 222 or 232 degrees and ABD is half of it: 111 or 116 degrees. Whichever answer is selected, the same theorem applies: in a circle, the angle at the centre is twice the angle at the circumference. I hope this helps.
