Can anyone help with this geometry question (angles)

Question

For the problem above, I need to find angle $\angle ABQ$, but for some reason, I can't seem to find it.

I've (think I have) found every angle except for $\angle QBA$ and the angle in question, $\angle ABQ$.

Here is what I've found so far:

• $\angle QPA$ = 150-a$^\circ$
• $\angle BQA$ = 90$^\circ$
• $\angle APQ$ = 30$^\circ$

I would really appreciate any help.

Answer 1

Connect $B$ and $P$. Then $m\angle PBQ= m\angle PAQ=a$, since they subtend the same arc. Also, $m\angle PBA=30^\circ$ because arc $AP$ must measure $60^\circ$, being one-third of a semi-circle. Therefore $m\angle ABQ=a+30^\circ$.