We are given the following information.
m∠ROP = 125°
We are asked to find the measure of each arc in the circle.
Arc RP:
The arc RP can be found by applying the central angle theorem.
[tex]\begin{gathered} Central\;angle=Intercepted\;minor\;arc \\ m\angle ROP=mRP \\ 125\degree=125\degree \end{gathered}[/tex]
m∠ROP is a central angle with an intercepted minor arc from A to B.
Therefore, the arc RP is 125°.
Arc QS:
The arc QS must be equal to the arc RP since the circle is divided into halves.
Therefore, the arc QS is also 125°.
Arc QR and SP:
Notice that the central angles m∠QOR and m∠SOP are congruent.
Recall the theorem that congruent central angles have congruent arcs.
Let x represent arc QR and arc SP.
[tex]\begin{gathered} mRP+mQS+QR+SP=360\degree \\ 125\degree+125\degree+x+x=360\degree \\ 250\degree+2x=360\degree \\ 2x=360\degree-250\degree \\ 2x=110\degree \\ x=\frac{110\degree}{2} \\ x=55\degree \end{gathered}[/tex]
Therefore, arc QR = 55° and arc SP = 55°
