Flight Instructor with student reported a near miss entering the traffic pattern. Both aircraft took evasive action to avoid a collision.

Narrative

On returning from a cross country training flight from ZZZ to ZZZ1 to ZZZ our aircraft had a near miss with Piper Cherokee attempting to enter the pattern at ZZZ from the south east. Our aircraft had entered the pattern at ZZZ on the left cross wind for runway 25 between 1500 and 1600 feet. We made the appropriate radio calls approaching the pattern and inside the pattern. As we turned for the downwind leg RWY XX; we heard a radio call of a cherokee entering on a 45º downwind entry. We made a down wind radio call; and did not see immediately see the cherokee due to the position of the cherokee and ground clutter of the foothills. Both my student and I were scanning the area looking for the traffic. I spotted the cherokee nearly the same altitude and on a collision course approximately 300-400 feet away. I immediately took the aircraft from my student and took evasive left bank and pulling turn to avoid a collision. The cherokee took a evasive turn to their left nearly at the same time we initiated a bank away. I estimate we came within 200 feet of each other. We continued our landing with no further incidents. The cherokee exited the pattern and re entered without incident.