Calculate the angles marked with letters in the diagram.

Step 1: Identify angles subtended by the same arc. 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. $b = \angle RPS$ $b = 55^\circ$

Step 2: Find another angle using the same property. Angle $\angle QPR = 45^\circ$ is subtended by arc QR. Angle $\angle QSR$ is also subtended by arc QR. Therefore, these angles are equal. $\angle QSR = \angle QPR$ $\angle QSR = 45^\circ$

Step 3: Calculate angle $a$ using the sum of angles in a triangle. 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: $b = \boxed{55^\circ}$ $a = \boxed{80^\circ}$

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