We are given the measure of the angle subtended by arc PS as 55 degrees.
Solution:
The measure of arc PM
[tex]\bar{\text{mPM}}=90^0\text{ (given)}[/tex]The measure of arc MR
[tex]\begin{gathered} \bar{\text{mMR}}+90^0+55^{\text{ 0}}=180^0\text{ (angles in a straight line)} \\ \bar{mMR}=180^0-90^0-55^0 \\ =35^0 \end{gathered}[/tex]The measure of arc RQ
[tex]\bar{\text{mRQ}}=55^0\text{ (Vertically opposite angles)}[/tex]The measure of arc QS
[tex]\begin{gathered} \bar{\text{mQS}}+55^0=180^0\text{ (} \\ \bar{\text{mQS}}=180^{0\text{ }}-55^0 \\ =125^0 \end{gathered}[/tex]