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If a short run of molding is damaged, replace it all; for longer runs replace only the damaged area with a patch. Take a sample of your baseboard to a lumber yard to match it.
To replace a damaged section of baseboard, pry the damaged section away from the wall. Place wooden shims between the molding and wall, on each side of the damaged area.
Draw pencil guide lines vertically down the baseboard on each side of the damaged area, marking the section that will be cut away.
Using a miter box, carefully cut down each guide line, using short, accurate strokes. Cut at the angles shown below.