In quadrilateral ABCD sides BC and AD are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles A and B intersect in point P, those of angles B and C intersect in point Q, those of angles C and D intersect in point R, and those of angles D and A intersect in point S. Suppose that PS is parallel to QR. Prove that ∣AB∣=∣CD∣.[asy]
unitsize(1.2 cm);pair A, B, C, D, P, Q, R, S;A = (0,0);
D = (3,0);
B = (0.8,1.5);
C = (3.2,1.5);
S = extension(A, incenter(A,B,D), D, incenter(A,C,D));
Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A);
P = extension(A, S, B, Q);
R = extension(D, S, C, Q);draw(A--D--C--B--cycle);
draw(B--Q--C);
draw(A--S--D);dot("A", A, SW);
dot("B", B, NW);
dot("C", C, NE);
dot("D", D, SE);
dot("P", P, dir(90));
dot("Q", Q, dir(270));
dot("R", R, dir(90));
dot("S", S, dir(90));
[/asy]Attention: the figure is not drawn to scale. geometryangle bisectorequal segmentsparallel