There are two widely accepted possibilities for what started the fire: a lit match or a burning cigarette (a third possibility is that a spark from one of the sewing machines started the blaze, but this is less likely). The Ric Burns documentary mentions the lit match or cigarette, and most tours and memorials say the same thing (little bit of a tangent here but on the anniversary, someone lays out 146 yellow roses on Greene Street, each with a victim's name; seeing them all laid out like tiny bodies has a most powerful effect).

I usually do a little write-up too, but as you're writing a paper on this, you know what happened. I do have a few sources to back up the claim for a match or cigarette:

• "The fire is believed to have started with a match or cigarette butt in the overflowing scrap bin under a cutter's table on the eighth floor. Smokers had caused fires at Triangle before, so even though no one enforced the posted signs forbidding smoking, they did keep water buckets on shelves along the walls (that caution didn't extend to making sure that the firehoses in the stairwells worked)." source
• "According to the fire marshal, the fire most likely began when a lit match was thrown in a waste basket near flammable material such as oil cans or cloth. Although smoking was prohibited in the factory, cigarette cases were found by investigators throughout the work environment. Employees later stated that cigarette smoking was widespread throughout the factory because employers were concerned that an enforcement of the smoking ban would harm the morale of the workers and decrease productivity." source
• "[Isidore] Abramowitz was taking his coat and hat from a nearby peg when he noticed the fire in his scrap bin. Perhaps the cutter had been sneaking a smoke while his assistants prepared the table. Or maybe it was another cutter-they were a close-knit group and liked to stand around talking together. Or maybe it was a cutter's assistant. At any rate, the fire marshal would later conclude that someone tossed a match or cigarette butt into Abramowitz's scrap bin before it was completely extinguished." source

If you're interested (or you need it for the paper), the transcripts of the criminal trial against the owners of the Triangle Shirtwaist company are online and they are a fascinating read.

Good luck with your paper! If you have any other questions, feel free to ask.