Finding an angle in a circle

Question

In circle $O$, $PA\perp AO,AE\perp PO,\angle BCO=30^{\circ},\angle BFO=20^{\circ}$,find $\angle DAF$.

It is obvious that $\angle EAD=\angle PAD=\frac{1}{2}\angle AOP$, but I can't get more proposition.

Answer 1

Assuming all the apparent collinearities in the figure are intentional, we have that $\angle DAF = \angle PAF - \angle PAD = 90^\circ + \angle OAF - \frac{1}{2} \angle AOP$.

The tricky part is finding $\angle AOP$. You will need to use trigonometry.

Answer 2

Match $BO$ you will have $\angle FBO = \angle BFO$ since $OF = OB=$ Circle radius for the same reason $\angle OBC = OCB $ Thus $\angle FBC = 50$ and $\angle FOC = 100$ hence $\angle FOA = 80$ so $\angle OAF = \angle OFA = 50$ I think you can move forward