Four L-shaped members: Mechanical Analysis Problem

[Steve4Physics]

I used basic trigo and drew a triangle with adjacent and opposite sides both a to get 45 degrees

Note that this is only justified because section DG is in equlibrium with only 2 forces acting on it.
So:
a) the 2 forces must have equal magnitudes and opposite directions;
b) the 2 forces must have a common line-of-action (dashed line, below).so that these forces have zero net moment.

Edit: attempted simplifications/improvements.