The area is about 50 miles northeast of Downtown Dallas.
The deadly incident unfolded after Dawson initiated a traffic stop of the suspect and his vehicle near the 3500 block of Picket Street.
At that time the assailant ran off, and Dawson pursued the suspect on foot, police said.
Dawson chased the suspect into a wooded area behind a home at 3517 Pickett Street, where the assailant “ambushed” him and shot him multiple times.
“Despite his critical injuries, Officer Dawson displayed exceptional courage, returning fire, and striking the suspect,” Greenville police said in a statement.
Officer Cooper Dawson, of the Greenville Police Department, was struck in his side and leg during the incident. Greenville Police Department
Both Dawson and the suspect were transported to Hunt County Regional Hospital, per KLTV.
Dawson was later airlifted to Medical City Plano, where he succumbed to his injuries.
The condition of the suspect, who has yet to be named, has not been released.
Fox 4 News video from the scene last night shows several police vehicles in front of the house along with a white sedan parked outside.
The shooting took place in a wooded area behind homes on Picket Street at 7:40 p.m. Monday. Fox4
It’s unclear if the sedan was associated with the traffic stop. The scene is cordoned off with yellow police tape.
The Greenville Police Department said that Dawson was a highly respected member of the force and had previously served with the Garland Police Department.
“His dedication to public service and his unwavering commitment to protecting the Greenville community will never be forgotten,” the statement reads.
“We are heartbroken over the loss of Officer Cooper Dawson, who selflessly put his life on the line to serve and protect our community,” Greenville Police Chief Chris Smith added, per the release. “We ask that you keep Officer Dawson’s family, our department, and the Greenville community in your thoughts and prayers during this incredibly difficult time.”
Dawson is the first Greenville Police Department officer in more than 100 years to die in the line of duty, the release said.
Police said that the Texas Department of Public Safety’s Texas Rangers will be conducting the investigation as a third-party investigatory agency.
Further details regarding memorial services and ways to support Officer Dawson’s family will be announced in the coming days, the release states.
A squad car is being displayed in front of the police precinct station in Dawson’s honor.
