Cessna 172 instructor pilot reported a near miss during a training flight at an uncontrolled airport; when entering the traffic pattern. The pilot maneuvered away from the aircraft then landed safely.

Narrative

My student and I were en route to ZZZ from our training area. At 5.5 nm out; we called ZZZ and informed them of our distance (5.5nm); direction (SW); intent (Option) and information (A). Shortly after that; we heard an additional call from a company platform that was 8 nm; SW; coming in on A. As we were preparing to set ourselves up for a 45 degree entry; we heard the aircraft that was behind us calling that they were currently entering a 45 degree for a left downwind for [Runway] XX. Since we were 3.5 nm out from the airport; we sustained an upwind heading to ensure we sustained separation. We maintained a vigilant lookout for the platform entering the downwind with no avail. After 30-45 seconds; I (the CFI) looked forward to ensure there were no obstacles in our direction of travel while telling my student to start looking behind us toward where base would begin to see if he could get eyes on the aircraft. As I looked forward; the aircraft that had called his entrance for a 45 degree was directly in front of us. The platform was close enough for me to give a description of each individual in the bird. We then began our airport infill.Upon reaching the ground; I parked the aircraft and approached the other aircraft. I asked him what his certification was; he informed me that he had his commercial ratings. I then asked him if he understood cardinal directions and began explaining the entirety of the situation from our announcement to the SW to the near-miss. He informed me that he didn't even see us and was surprised that there was even an issue. I've since brought this situation up with our Chief Pilot; the Owner of the Company; said individual and now the reporting system.