Bignose, I may be making a mistake, but after solving the equations you provided, I didn't get obtain a result for [math]x[/math].
Equation one implies
[math] BDC=180-8x[/math],
and equation three implies
[math]ABD=180-12x[/math].
Substituting the result from equation 3 into equation 2 gives
[math]4x+ADB+180-12x=180 \Longrightarrow ADB=8x[/math].
Substituting the result for [math]ABD[/math] plus the result obtained for [math]BDC[/math] into equation four yields
[math] 180-8x+8x=180 \Longrightarrow 180=180[/math], which, while true, is not very revealing.
I think one must make use of the fact that [math]AB=CD[/math]. How this is done is not readily apparent to me.
Anubisboy