30

To find the angle at which the swimmer should swim across the river to reach a point directly opposite their starting point, we can use vector addition to solve the problem.

Let:
- $v_s$ = velocity of the swimmer relative to still water = 12 km/h
- $v_r$ = velocity of the river = 6 km/h

The swimmer needs to swim with a velocity that has a component directly opposing the river's flow. The swimmer's effective velocity across the river (perpendicular to the river bank) is $v_s \sin(\theta)$ and the downstream velocity due to the river is $v_r$. Thus, to reach the opposite bank directly,
$v_s \sin(\theta) = v_r$.
Substituting the values, we have:
$12 \sin(\theta) = 6$.
This simplifies to:
$\sin(\theta) = \frac{6}{12} = 0.5$.
The angle $\theta$ can be calculated as $\theta = \arcsin(0.5) = 30^{\circ}$.
Therefore, the swimmer should swim at an angle of 30 degrees with respect to the flow of the river.