In a survey experiment, three independent measurements $29.5^{\circ}$, $30.5^{\circ}$, $120.5^{\circ}$ are obtained from the three angles $\alpha,\beta,\gamma$ of a triangle. Formulate the appropriate linear model, and find the least squares estimates $\hat{\alpha}, \hat{\beta},\hat{\gamma}$ of the respective angles.
I'm not quite sure how to represent this. What I had done was say that $$180 = \alpha + \beta + \gamma + \varepsilon$$ where $\varepsilon$ is the error term. Then, the sum of squares of errors, $SS$, is simply $$SS= (180-\alpha-\beta-\gamma)^2.$$ Differentiating with respect the each of the parameters, setting the equation to $0$, and thus dividing out $-2$, we get the normal equations as \begin{align*} &\left\{ \begin{array}{l} (180-\alpha-\beta-\gamma)\alpha = 0 \\ (180-\alpha-\beta-\gamma)\beta = 0 \\ (180-\alpha-\beta-\gamma)\gamma = 0 \end{array} \right. \\[0.25cm] \implies &\left\{ \begin{array}{l} \hat{\alpha} = 180-\hat{\beta}-\hat{\gamma} \\ \hat{\beta} = 180-\hat{\alpha}-\hat{\gamma} \\ \hat{\gamma} = 180-\hat{\alpha}-\hat{\beta} \\ \end{array} \right. . \end{align*} This seems to make intuitive sense, but is it right? I'm just not sure if I set up the model correctly. There is, then, 1 observation, but 3 parameters. A subsequent question asks to calculate an unbiased estimator for the variance. How would this be done?
Hint
I wonder if the problem is not to minimize $$SS=(\hat{\alpha}-\alpha)^2+ (\hat{\beta}-\beta)^2+(\hat{\gamma}-\gamma)^2$$ subject to the condition $\hat{\alpha}+\hat{\beta}+\hat{\gamma}=180$.
If this is the case, extract $\hat{\gamma}=180-\hat{\alpha}-\hat{\beta}$ from the constraint to get $$SS=(\hat{\alpha}-\alpha)^2+ (\hat{\beta}-\beta)^2+(180-\hat{\alpha}-\hat{\beta}-\gamma)^2$$ Compute $SS'_{\hat\alpha}$ and $SS'_{\hat\beta}$ and set them equal to zero. This will give you two linear equations for two unknowns $\hat{\alpha}$ and $\hat{\beta}$.