Drilling angled holes (also called pocket holes) can be accomplished with or without a jig. The problem with attempting to "start" the drill bit at an angle is that it will "walk" along the board.
The quick solution is to start the hole by drilling straight down into the wood. After the depth of the hole is roughly equal to the width of the bit change the angle of the drill and continue the hole.
Other methods for drilling an angled hole involve the use of special tools or jigs. Commercial pocket hole jigs can be purchased for under $15
These jigs are made from metal for longer life and some models have integrated clamps to secure the jig to the wood.
A simple pocket hole jig can be made by 1) drilling a straight hole into a block of wood. 2) Then cut the board at the appropriate angle and clamp it to the workpiece. You may need to notch out a section of the jig to clamp it to the workpiece.