The boat is outside of the semicircle. So how can his angle help him decide when he is entering the dangerous semicircle? is there a certain theory that this pertains to? I still don't understand. Please, help again? thanks
bobpursley answered this question for you a few hours ago.
The answer is based on the fact that in any semicircle, the angle subtended by the diameter anywhere on the circle is 90º
Your diagram probably has a line joining the two lighthouses, that is the diameter of the circle.
Observe that if the angle falls outside the circle, the angle will be less than 90º and if the angle falls inside the circle, it will be greater than 90º
So as long as the angle formed by the lighthouses is less than 90, the boat will be outside the semicircle and safe.