Angle Chasing in a Parallelogram Using Diagonals

\(ABCD\) is a parallelogram. Find \(\angle BDC\), giving your reason(s) clearly.

Source consistency note: The exact angle labels are inconsistent with a parallelogram. The angle chase gives \(\angle BAM=21^\circ,\angle ABM=39^\circ,\angle CBM=83^\circ\). The sine rule gives \(AM/BM=\sin39^\circ/\sin21^\circ\), but \(CM/BM=\sin83^\circ/\sin37^\circ\); these are unequal, contradicting \(AM=CM\). The numerical answers below are conditional angle-chasing results, not angles of a figure satisfying every label.