A circle is divided into four sectors such that the proportions of the

Let the areas of the four sectors be $A_1, A_2, A_3, A_4$, and their corresponding central angles be $\theta_1, \theta_2, \theta_3, \theta_4$.
The proportions of the areas are given as $A_1 : A_2 : A_3 : A_4 = 1 : 2 : 3 : 4$.
Since the areas are proportional to the angles, the ratio of the angles is also $\theta_1 : \theta_2 : \theta_3 : \theta_4 = 1 : 2 : 3 : 4$.

The sum of the angles of the sectors in a circle is $360^\circ$.
So, $\theta_1 + \theta_2 + \theta_3 + \theta_4 = 360^\circ$.

The total proportion is $1 + 2 + 3 + 4 = 10$.
The angles can be found by dividing the total angle ($360^\circ$) according to the proportions:
$\theta_1 = \frac{1}{\text{Total Proportion}} \times 360^\circ = \frac{1}{10} \times 360^\circ = 36^\circ$.
$\theta_2 = \frac{2}{10} \times 360^\circ = \frac{1}{5} \times 360^\circ = 72^\circ$.
$\theta_3 = \frac{3}{10} \times 360^\circ = \frac{3}{10} \times 360^\circ = 108^\circ$.
$\theta_4 = \frac{4}{10} \times 360^\circ = \frac{2}{5} \times 360^\circ = 144^\circ$.

The four angles are $36^\circ, 72^\circ, 108^\circ, 144^\circ$.
Check the sum: $36 + 72 + 108 + 144 = 108 + 108 + 144 = 216 + 144 = 360^\circ$. The sum is correct.

The largest sector corresponds to the largest proportion, which is 4.
The angle of the largest sector is $\theta_4 = 144^\circ$.