The answer is 69.

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Because $\angle AOB = 153$ , the other side of $\angle AOB = 360 - 153 = 207$ . Point $P$ trisects the major arc $AB$ thus $OP$ trisects $\angle AOB$ . Therefore $\angle BOP = 69$ , and $\angle POA = 138$ . By central angle theorem we can tell that $\angle AQP = \frac{1}{2} \angle POA$ . $\angle AQP = \boxed{69}$