Angle Measurements

Let the center of the circle be $O$ . Given that arc $BC$ and arc $DA$ are $30^\circ$ and $80^\circ$ respectively, this means that the central angles, $\angle BOC = 30^\circ$ and $\angle DOA = 80^\circ$ . Since the angle at the circumference is half that of the central angle extended by the same arc, $\angle BAC = 15^\circ$ and $\angle DCA = 40^\circ$ . And since $\angle P + \angle BAC = \angle DCA$ , $\implies \angle P = \angle DCA - \angle BAC = 40^\circ - 15^\circ = \boxed{25^\circ}$ .

$P=\angle {APD}=180\degree-(30\degree+70\degree+55\degree)=\boxed {25\degree}$ .