Residuals and the $SS_{R}$ are defined as \begin{align} R &= Y - (\hat{\theta}_{0} + \hat{\theta}_{1}X)\newline SS_{R} &= \sum_{i=1}^{n} R_{i}^{2} = \sum_{i=1}^{n} (Y - \hat{\theta}_{0} - \hat{\theta}_{1}X)^{2}\newline &= \frac{S_{xx}S_{YY} - S_{xY}^{2}}{S_{xx}} \end{align}
$SS_{R}$ is itself a random variable and it can be shown that \begin{align} \frac{SS_{R}}{\sigma^{2}} \sim \chi_{n-2}^{2}\newline E[\frac{SS_{R}}{\sigma^{2}}] = n - 2\newline E[\frac{SS_{R}}{n-2}] = \sigma^{2}\newline \end{align} since $SS_{R}/\sigma^{2}$ is the sum of squares of normally distributed variables ($E[Y] = \theta_{0} + \theta_{1}X$) and two degrees of freedoms are already taken up by the coefficients. Further, $SS_{R}$ is an unbiased estimator of the variance of the error terms $\sigma^{2}$, and is also independent of the coefficients.