A point P is selected at random from the interior of the pentagon with vertices A \equal{} (0,2), B \equal{} (4,0), C \equal{} (2 \pi \plus{} 1, 0), D \equal{} (2 \pi \plus{} 1,4), and E \equal{} (0,4). What is the probability that ∠APB is obtuse?
[asy]
size(150);
pair A, B, C, D, E;
A = (0,1.5);
B = (3,0);
C = (2 *pi + 1, 0);
D = (2 * pi + 1,4);
E = (0,4);
draw(A--B--C--D--E--cycle);
label("A", A, dir(180));
label("B", B, dir(270));
label("C", C, dir(0));
label("D", D, dir(0));
label("E", E, dir(180));
[/asy]
<spanclass=′latex−bold′>(A)</span>51<spanclass=′latex−bold′>(B)</span>41<spanclass=′latex−bold′>(C)</span>165<spanclass=′latex−bold′>(D)</span>83<spanclass=′latex−bold′>(E)</span>21