 Solution

For $n>2$, the usual unbiased estimator of variance isThe fitted mean lies in the two-dimensional span of the constant design vector and $x$. Orthogonal projection of the Gaussian error onto its complement gives $n-2$ independent standardized normal coordinates. HenceThis residual projection is independent of the fitted-coefficient projection. If maximum likelihood for variance is intended instead, its estimator is ${\hat{\sigma }}_{\mathrm{ML}}^2=\mathrm{RSS}/n$, with $n{\hat{\sigma }}_{\mathrm{ML}}^2/{\sigma }^2\sim {\chi }_{n-2}^2$; the denominator differs, not the residual degrees of freedom.