Angled holes are often drilled when building chairs or other furniture. They typically form holes for dowels or pilot holes for screws. The problem with drilling angled holes is the tendency for the drill bit to bounce or slide when introduced to the wood at an angle. The solution is to use a drill press with a jig, a drilling template clamped to the drill press table to hold the wood so it doesn't move after the angle has been established.
Set the mitre gauge on a table saw and cut a scrap piece of plywood at a 15-degree angle.
Stand the angled piece up on its edge on the strip of scrap plywood. Glue together, and shoot pin nails into the scrap piece from the bottom so that it is stands up like a shark's fin. You have created a jig that will hold the wood firmly during drilling.
Lay the strip with shark's fin on the drill press table.
Lay the board you wish to drill up against the angled shark's fin. Face the side into which you will drill away from the fin. Slide the strip and the board underneath the drill bit together and position the board under the bit where you want to drill the angled hole. Clamp the strip to the table with a hand clamp.
Hang on to the board with one hand to hold it steady against the shark's fin. Use your other hand to turn on the drill press. Pull down on the drill press handle down slowly but steadily to drill an angled hole in the board. You can slide the board into another position with the jig to drill the same angled hole over and over again.
Do a few test holes at any given angle to test your angle. You can also clamp the board to the drill press table if you need to. The measurements here are for examples. Use different measurements and angles for whatever you need.
Always wear safety glasses when working with wood.