Flight Instructor reported an NMAC during landing pattern training when another aircraft flew an improvised pattern entry which caused the Instructor to take evasive action to prevent a collision.

Narrative

After performing a training flight with my student; we needed to do a few laps in the pattern to work on landings. The event occurred on what was going to be our last landing. In the downwind for Runway X at ZZZ and abeam the numbers; my student reduced power; added 10 degrees of flaps and began a descent. Right as we were about to turn base; I noticed an airplane flying directly towards us at what appeared to be a slightly lower altitude. The airplane was around 2 o'clock position flying in our direction; probably less than a mile away. I immediately took controls from my student and performed evasive action to escape the potential conflict. I immediately initiated a climb and then; once confirmed that the traffic passed below us; began a northwest bound turn to reenter the pattern on the 45 to downwind for Runway X. We were making radio calls for each leg of the pattern on CTAF. Either right before or during our evasive maneuver; I believe the other aircraft made a radio call that he was entering the downwind for Runway X. This was totally incorrect; as he never flew in the downwind and completely ignored the traffic pattern. After our evasive action was taken; the other airplane proceeded on an improvised base leg; flew past final; turned around and then rejoined final off of a right base. The other aircraft ended up passing within 200 FT below us. If we had not seen him or if I had not taken controls and stopped our descent; there could have been a conflict. With multiple airplanes in the pattern; it is unacceptable how this other airplane entered the pattern. The pilots of that plane exhibited very poor situational awareness.