Remark
I'm so glad you got a good answer to the central angle problem. Geometry can be that way. What I did was correct and would have worked with any two arcs, but a wonderful answer you got from the other responder.
This problem uses the fact that from an external point, when two secants are drawn (as in your diagram) the difference in the arc sizes divided by two gives the angle between the secants (in this case 23).
Formula
Angle between secants = (large arc - small arc)/2
Givens
Angle between secants = 23
The small arc = 77
The large arc = x
Sub and solve
23 = (x - 77) / 2 Multiply by 2
23*2 = (x - 77)
46 = x - 77 Add 77 to both sides.
46 + 77 = x
123 = x Answer