Let $m$ and $n$ be the bisectors of angles $\angle DAB$ and $\angle DCB$. Let $P=DC\cap m$ and $Q=AB\cap n$. Since $m\parallel n$ and $m$ and $n$ are angle bisectors $$\angle DAP=\angle PAB=\angle CQB$$$$\angle BCQ=\angle QCD=\angle APD$$This means $\triangle ADP\sim\triangle CBQ$ so $\angle ABC=\angle QBC=\angle PDA=\angle CDA$.