C172 flight instructor with student reported a near miss while turning onto final at a non-towered airport when another aircraft not communicating on CTAF turned in front of them. The instructor took evasive action and returned to landing.

Narrative

Training a private pilot student; filling in for another instructor. Just started working at this flight school; planes are old with no ADS-B in for traffic. Was doing pattern work at the airport; on about our second lap in the pattern an airplane entered the downwind in front of us. They did not call on the radio that they were entering the downwind; nor did I see them. Airplane on the ground also said no radio call was made that they were entering the downwind. My student was still learning how to fly a proper approach; and I thought we were number one for landing. We turned base; and as we did; an airplane called that they were turning base. This was super confusing; since I thought no one was in front of us. The airport's CTAF picked up multiple other airports; so I thought it maybe was from there as I didn't catch a tail number or what airport; just that they were turning base for the same runway number as us. As we began to turn final I noticed an airplane turning final in front of us. I immediately took evasive action and turned right to the upwind leg where we flew around the pattern and landed; eventually making our way back to home base. The weather at the time was about 5-6 miles visibility; and it was more difficult to see other traffic. We were making proper radio calls (departure leg; crosswind; downwind; base; final) and only heard the other airplane make a 10 mile call; then a base call (their base call was made after we were turning base). I was not aware they had entered the downwind ahead of us and never saw them. My student was struggling with the landing profiles for a short field landing; so I was explaining this in the downwind. If I had landed full stop to debrief instead; I would have been able to be more vigilant about traffic. Our Garmin 430 had TIS-B traffic alerting; which never went off. I was complacent with the fact that no radio calls were made; and I assumed that meant no one else was in the pattern except the airplane holding short of Runway XX; waiting to take-off. Other factors were that even though it was my second flight of the day; I was starting to feel fatigued. The airplane was hot (it had windows at the top of the cabin; similar to sunroofs in a car); and that was also a factor. I had also never flown with this student before; so I was paying extra attention to their flying rather than the outside. As PIC; it's my duty to see and avoid other aircraft; and I failed to do that. My opinion on what caused this event was my complacency that all airplanes will make proper radio calls; at least when entering the pattern. That has been my experience at other airports in the past; and since I never heard a crossing midfield or entering the downwind call; I assumed no one was there. I also believe the haze played a factor in me not seeing the other airplane while we were both on downwind. To remedy this situation; I will now try to break those complacent thoughts and try to be more aware of uncontrolled fields. I will reconsider flying when the visibility is not ideal; and will try to focus less on teaching and more on being situationally aware. We all have to do our part at uncontrolled airports to ensure that traffic stays sufficiently separated and everyone is aware of everyone else's position. I will be more vigilant now; just because my job is to teach doesn't mean I can lapse my duty to fly.