Show that $M\in [AC] $ or $M\in [BD] $.

Question

Let $ABCD $ a rombhus and $M $ an interior point. If $\angle AMB+ \angle CMD=180^{\circ}$.

Show that $M\in [AC] $ or $M\in [BD] $.

I tried to solve it with sine and cosine but I didn't succed.

Answer 1

Let $NBCM$ be parallelogram.

Thus, $NBMA$ is cyclic, which says $$\measuredangle BAM=\measuredangle BNM=\measuredangle BCM,$$ which says $$\measuredangle CAM=\measuredangle ACM,$$ which gives $$AM=MC,$$ which says $$M\in BD.$$

All this if $M\not\in AC$.

If $M\in AC$ then we are done again.