Calculate the angles marked with letters in the diagram.

Angles subtended by the same arc at the circumference are equal.
Angle $\angle RPS = 55^\circ$ is subtended by arc RS.
Angle $\angle RQS = b$ is also subtended by arc RS.
Therefore, these angles are equal.

$$b = \angle RPS$$ $$b = 55^\circ$$

Angle $\angle QPR = 45^\circ$ is subtended by arc QR.
Angle $\angle QSR$ is also subtended by arc QR.
Therefore, these angles are equal.

$$\angle QSR = \angle QPR$$ $$\angle QSR = 45^\circ$$

Consider triangle $\triangle QRS$. The sum of angles in a triangle is $180^\circ$.
The angles in $\triangle QRS$ are $\angle QRS$ (which is $a$), $\angle RQS$ (which is $b$), and $\angle QSR$.

$$a + \angle RQS + \angle QSR = 180^\circ$$

Substitute the values found in Step 1 and Step 2:

$$a + 55^\circ + 45^\circ = 180^\circ$$ $$a + 100^\circ = 180^\circ$$ $$a = 180^\circ - 100^\circ$$ $$a = 80^\circ$$

The angles are:
$b = \boxed{55^\circ}$
$a = \boxed{80^\circ}$