Simple middle school Parallelogram problem

[Selby8]

In the bottom Parallelogram it is given that AD=20 and that the perimeter is 140.
(Ok so therefore AB=DC=50).
It is also given that AS=AM+18.
We need to find AM, therefore AS and then to calculate the area of the Parallelogram which derives from finding AM.
Now, I managed to find the right answer but not the way to prove it, I just played around with Pythagoras.
If I could prove that SC=CB then the rest follows and AM=12, but I don't know how.
Please help, thanks!

 

[MarkFL]

We see that angle \(\displaystyle ADM\) = angle \(\displaystyle ABC\) and so right triangles \(\displaystyle ADM\) and \(\displaystyle ABS\) are similar. This allows us to state:

\(\displaystyle \displaystyle \frac{\overline{AM}}{\overline{AD}}= \frac{\overline{AS}}{\overline{AB}}\)

Can you continue?

 

[Selby8]

We see that angle \(\displaystyle ADM\) = angle \(\displaystyle ABC\) and so right triangles \(\displaystyle ADM\) and \(\displaystyle ABS\) are similar. This allows us to state:

\(\displaystyle \displaystyle \frac{\overline{AM}}{\overline{AD}}= \frac{\overline{AS}}{\overline{AB}}\)

Can you continue?

I did, Thank you so much!

I looked at similar triangles, just not at this pair...