consider a regular \(2n-gon\)
show that \(R_{180}\) commutes with all other rotations and flips.
for example, below two sequence of transformations give the same final effect :
1) a. rotate 180
b. flip over the symmetry line passing through vertex A
2) a. flip over the symmetry line passing through vertex A
b. rotate 180
in other words, show that the order doesn't matter if one transformation is \(R_{180}\)

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