解：過(guò)點(diǎn)?$A$?作?$AE⊥CD，$?垂足為點(diǎn)?$E，$?過(guò)點(diǎn)?$B$?作?$BF⊥DC，$?
交?$DC$?的延長(zhǎng)線于點(diǎn)?$F，$?過(guò)點(diǎn)?$A$?作?$AG⊥BF，$?交?$FB$?于點(diǎn)?$G$?
則?$AE=FG，$??$∠BFC=∠AGB=90°$?
∵?$∠BCD=120°$?
∴?$∠BCF=180°-∠BCD=60°$?
∴?$∠FBC=90°-∠BCF=30°$?
在?$Rt△BCF$?中，?$BC=4\ \mathrm {cm}$?
∴?$BF=BC ·sin {60}°=4×\frac {\sqrt 3}2=2\sqrt 3(\mathrm {cm})$?
∵?$∠ABC=85°$?
∴?$∠ABG=180°-∠ABC-∠FBC=65°$?
在?$Rt△ABG$?中，?$AB=6\ \mathrm {cm}$?
∴?$BG=AB ·cos {65}°≈6×0.423=2.538(\mathrm {cm})$?
∴?$GF=BG+BF=2.538+2\sqrt 3≈6.002\ \mathrm {cm}$?
故點(diǎn)?$A$?到?$ CD$?的距離約為?$6.002\ \mathrm {cm}$?
