The killings haunted the police officers who responded to the scenes, the authorities would later say, and for decades detectives in the two communities where the murders occurred, Aurora and Lakewood, Colo., pursued leads and developed theories. Some of them retired or died, but the search for answers eluded them — until last month.

On Friday, officials said that the DNA profile of a man in a Nevada prison on unrelated attempted murder charges matched with evidence found at the Colorado murder scenes.

At a news conference to announce the break in the cases, Peter Weir, the district attorney for the first judicial district in Colorado, said: “Justice in this case has been delayed. I am confident that justice is not going to be denied.”

An arrest warrant has been issued for the inmate, Alexander C. Ewing, 57, and the authorities will seek to extradite him to Colorado, where he faces murder, sexual assault, burglary and related charges in connection with the four killings.

Court papers tell a story of what appear to be random home invasions with unclear motives.

The 50-year-old woman who was killed, Patricia Smith of Lakewood, was found wearing a ring with a gold coin. In the killings of the family — Bruce Bennett, 27; his wife, Deborah Bennett, 26; and their 7-year-old daughter — there appeared to be minimal ransacking.