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Begin the process by determining your longest wall. If you have a piece of baseboard molding that is longer than the actual wall, you can do this first wall with just one baseboard. If this is the case, measure carefully and then cut each end of the baseboard at 90 degrees to run directly into each side's perpendicular wall.
If you need two pieces for the first wall, join the two pieces with a scarf joint. A scarf joint marries two 45-degree cuts on different boards (Image 1). If possible, cut and situate the two boards so that the scarf joint can be secured to the wall at a stud location.
Start with two pieces of baseboard that have 90-degree cuts at opposite ends. To create the scarf joint in the middle, place the first board in place and mark a 45-degree cut at a stud location. Using the miter saw, cut this end of the first board at a 45-degree angle and sand it smooth. Don't oversand. Hammer it in place with a few finish nails, keeping the heads exposed. Measure the other board, being very careful to draw exactly the 45-degree cut from the board already in place. Cut the end of the second board at the proscribed 45-degree cut and fit the two pieces together (Image 2). If there's a slight bow, re-cut the end of the 90-degree cut.
If the two baseboards fit snugly together, run a bead of wood glue on the angled joint and push them together. Clean off any excess glue. To secure the scarf joint, drive one nail toward the top of the base on a raised portion of the profile (Image 3) -- this will make it easier to fill later. Then drive the second nail toward the bottom and angle it down into the floor. Finish sinking the other exposed nails with a nail set.
A coped joint is used where the baseboards meet an inside corner joint.
To make a coped joint, butt one piece of baseboard flush against the wall at a 90-degree angle (as in the previous step). Next, lay the other baseboard molding (that will join the one already in place) face-down on the floor. Hold a scrap piece of baseboard molding perpendicular to the face-down board and trace the profile of the baseboard with a pencil. This will give you a reference point to make it easier to complete the coped joint.
Cut along the profile to make a back bevel cut (at least at a 90-degree angle) just short of the profile -- about 1/16". Finish taking off the remaining back bevel with your half-round and three-square files until the piece sits flush against your first piece with no gap.
Continue to measure, cut and install the baseboard molding around the room. When you get to an outside corner, set the first piece to that it extends past the outside corner; mark it where it will meet the other piece of the outside corner by setting your try square or combination square against the surface of the wall it meets. Using a miter box or a power miter saw, cut the baseboard at a 45-degree angle. Before nailing it in place, mark the second piece the same way, and test for fitting. Use 1-1/4" or 1-1/2" brads to close the corner.
When you encounter the doorway, measure the distance between the door casing and the wall. This piece will butt against the door casing with a 90-degree angle cut.
Use wood filler (Image 1) to fill in holes and gaps in baseboards. Let the wood filler dry completely, then lightly sand off excess. Use caulk along the top of baseboards to fill the crack between the baseboard and the wall (Image 2).