"You just called your entry into the pattern at a non-towered airport on a 45 degree to downwind and shortly after another airplane calls a 5 mile final for your runway. Who has the right of way and how do you handle this situation?" - Randy C.
“FAR 91.113 says the following:
When two or more aircraft are approaching an airport for the purpose of landing, the aircraft at the lower altitude has the right-of-way, but it shall not take advantage of this rule to cut in front of another which is on final approach to land, or to overtake that aircraft.
So, it depends on the relative speed of the two aircraft. I would continue on the downwind until I had the aircraft on final in sight and only turn base if it were absolutely clear that the aircraft on final was so far out that it would not create a problem. The aircraft on final has the right of way.
If you don’t have the aircraft on final in sight or are in doubt regarding the situation, continue on the downwind and let the other aircraft in the pattern know what you are doing on the advisory frequency (Common Traffic Advisory Frequency).
Another measure you can always take at a non-towered airport is to break out of the pattern by turning away from the airport and re-entering the pattern. However be sure to transmit your intentions on the advisory frequency so that other traffic will be aware of your action.”