C172 pilot reported a NMAC with another high wing aircraft at a non-towered airport. C172 pilot took evasive action to avoid a collision.

Narrative

I was doing pattern work at ZZZ; making right traffic; (non-standard); for Runway XX. I had extended my downwind leg due to another plane doing a long straight-in approach for XX. I announced over ZZZ CTAF that I was extending downwind to accommodate said traffic; and I said I would be landing behind them. (Assume from this point forward that every announcement is made on ZZZ CTAF.)After giving the traffic plenty of room; I turned base about 3.8nm from the airport; so I announced that I was turning for an extended right base [runway] XX. Subsequently; I turned and announced I was on final for XX. The traffic for which I had extended announced they were clear of the runway and taxiing back to park.As all of this was happening; another plane was approaching the airport via a very extended left base for XX.I continued my approach; and the plane to my left said something over CTAF; (though he was already on an extended base; I think he may have said he was turning base; but he was difficult to understand); but he angled his flight path about halfway between a square base and direct to the numbers; sending his path on a direct course with mine. I was tracking him on my plane's GPS; as he was above me and impossible to see with my high-wing plane. However; I could see via GPS that he was on a collision course with me.I announced that I was still on final for XX. It would have been helpful had I reiterated my specific distance from the runway at this time; but I only said that I was still on final for XX. Regardless; I received no response.The other plane then turned towards final above and to the left of me; and my traffic alert warning system went off. Immediately following this; my GPS showed him as being 100ft above my elevation and directly above me. I announced that he was directly above me; but I received no response.I looked up and saw him pulling in front of me to the point where I finally had visual contact with him in front of my left wing. He was above my plane and slightly to the left of me. At the closest point; his right wing was hanging directly over the top of my left; and he was about 100 feet above me.At this point; it was clear that he had no intention of giving way. His was also a high-wing Cessna; so I don't know how he didn't see me. Yet; had I continued with my landing; we would have collided somewhere between a local road and the threshold of [runway] XX. Likely; he would have landed on top of me.So; I veered right; and I announced I was doing so. I got a safe distance from the runway and then flew parallel to the upwind as he landed; beginning his taxi back to park.I announced I was upwind; then reentered my previous pattern via a right crosswind and subsequent right downwind for XX. I then made the right base; final; landed; and taxied back to park as well.