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Use a table saw with a dado stack to cut the shoulders of the tenon.
Place tenon over the mortising piece to determine the placement of the mortise, mark out the sides of the tenon on the mortising board.
Use a mortiser to cut the mortise. The mortiser has a square chisel with a drill bit inside it. The drill bit removes the material and the chisel gives the mortise its shape. Start working from one end of the mortise and proceed to the other end. Overlapping the cuts makes the process easier on the mortise. When finished, clean out the waste material from inside the mortise.
Put the mortise and tenon together, then drill a hole through the mortise and tenon. Use a drill bit that complements the size of the peg. (For example, for a 1/4" peg, use a 1/4" drill bit). Do not drill all the way through the other side of the mortise.
Use a block plane (Image 1) to chamfer (furrow) the corners of the peg. This will help the square peg fit into the round hole.
Place the end of the peg into the hole and use a mallet to pound the peg in all the way (Image 2). The square shape of the peg compresses the material around the hole and forces it aside.
Use a flush cut saw to cut off the remaining portion of the peg.