Step 1: Recall the phase shift required for oscillation.
According to the Barkhausen criterion, the total phase shift around the feedback loop must be
\[
360^\circ.
\]
Step 2: Determine the phase shift produced by the RC network.
In an RC phase shift oscillator,
- the transistor amplifier provides
\[
180^\circ,
\]
- the three RC sections together provide another
\[
180^\circ.
\]
Thus,
\[
180^\circ+180^\circ=360^\circ.
\]
Hence, the RC network contributes
\[
\boxed{180^\circ.}
\]
Therefore,
\[
\boxed{(C)}
\]
is the correct answer.