OK, so how do I get this dimensioned? 5 deg and 55 minutes. I'm modeling a part for fun and ran across this. I suppose I could figure out how to convert this to a decimal, but I am wondering if the program can deal with this.

Perhaps a look at the "properties" for drawings would help you..... There it is, degrees, radians, degrees/minutes, or degrees/minutes/seconds. Since it is an available measurement unit, Alibre should convert it to the default you have set (probably degrees)

You can directly enter it in sketch mode. Dimension the angle and change the value to 5°55'. It works! If you lose the ° degree sign then re-enter it by holding ALT and typing 248 on the numpad