In the diagram below lines $HA$ and $BC$ are parallel. If $\angle GEC = 135^{\circ},$ what is $\angle DEC$?[asy] unitsize(1cm); void doRay(pair A, pair B) { draw(A -- 1.4*(B-A) A); } real t = 0; pair X=rotate(t)*(0,0); pair A=rotate(t)*(6,0); pair B=rotate(t)*(-2,3); pair C=rotate(t)*(5,3); pair T=rotate(t)*(4,0); pair U=rotate(t)*(1,3); pair Y =1.3*(T-U) U; pair Z =1.3*(U-T) T; dot(A); dot(B); dot(C); dot(T); dot(U);dot(Y);dot(Z); dot(X); doRay(T,U); doRay(U,T); doRay(X,A);doRay(A,X);doRay(B,C);doRay(C,B); draw(X--A); draw(B--C); label("$135^{\circ}$",U, 3*dir(60) ); label("$A$",A,N); label("$B$",B,N);label("$C$",C,N); label("$D$",T,S); label("$E$",U,SW); label("$F$",Y,SW); label("$G$",Z,NE); label("$H$",X,N); draw( C--1.5*(C-B) B, MidArrow(size=0.3cm)); draw( A--1.4*(A-X) X, MidArrow(size=0.3cm)); [/asy]
