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Cut a 9" section from the 1" x 4" board, then cut a 3" section from the 1" x 4". The remaining piece of 1" x 4" will be used as a wall support.
Draw a diagonal line between two corners on the 3" piece, and cut along the line to create shelf brackets. The 9" section will be used later.
Mark the locations for the pegs down the center of the wall support, about 3" from each end, and space several pegs evenly between the two outer marks. Drill all the way through the wall support, using a spade bit sized to match the pegs (Image 1). After you've drilled all the holes, sand off any burrs.
Apply glue to the bases of the Shaker pegs (Image 2) and slide them into the holes (Image 3). Twist the pegs as you push them to spread the glue evenly.
Stand the wall support on its side and mark it on the top edge, 6" from each side. Run a bead of wood glue along the top of the wall support between the two 6" marks.
Center the 1" x 6" shelf on top of the wall support so the back edges are flush. Use the 9" piece of 1" x 4" to help hold up the shelf while the glue dries (Image 1). You might want to secure the boards with nails or screws for additional stability. If you use screws, countersink pilot holes so the screw heads won't show (Image 2).
Place the triangular brackets between the shelf and the support, 3" from each end of the shelf. Secure the brackets with nails or screws.
Stain the Shaker shelf.
Locate and mark the studs behind the wall where the shelf will be located. Transfer the marks to the support, and countersink pilot holes at those marks. Secure the shelf over the studs with wood screws, and cover the holes with wooden buttons.