Finding shaded triangle areas in a parallelogram There is the following parallelogram involving two shaded triangles.
If I found rightly, angles of $AMD$, $BMN$ and $CDM$ are $45$. But I can’t go further.
 

$\angle AMD$, $\angle BMN$ and $\angle CDM$ are not necessarily $45^\circ$.

$[\triangle AMD]=\dfrac12\times\dfrac23\times[ABCD]$
$[\triangle BMN]=\dfrac12\times\dfrac13\times\dfrac13\times[ABCD]$
$[\triangle CND]=\dfrac12\times\dfrac23\times[ABCD]$
$[\triangle DMN]=\left(1-\dfrac13-\dfrac1{18}-\dfrac13\right)\times[ABCD]=\dfrac5{18}[ABCD]$
So, $\dfrac5{18}[ABCD]=\dfrac12\times6\times10$
$[ABCD]=108$
$\textrm{shaded area}=\left(\dfrac13+\dfrac1{18}\right)\times 108 = 42$
