Assume I put a board (in red) behind the black-outlined area. I draw a line against the top-right angled side so I can get cut the triangle off the board so that it will fit flush inside the outline.
If I cut out the triangle I am left with (pic 2). The cut-out board is x too short. I know why this is, but I don't know how to calculate the green line. There are various reasons why I need to do it this way, none pertinent to the question (it involves scribing in carpentry).
Should the green line be equal to the blue line? I've laid it out in a grid in photoshop with perfect angles but, as you can see, it doesn't appear to be the case in the illustrations (but maybe my drawing is off). I feel like I should be taking the hypotenuse into account.
I have drawn a figure below. You are asking if the distance between lines $JI$ and $ED$ is the same as $FA$. It is not. If $\theta=\angle DEL$ we have the distance between $JI$ and $ED$ is $FA \sin \theta$. Draw a perpendicular from E and you get a right triangle that shows it.