MINNEAPOLIS — They met one evening four summers ago, and she was instantly drawn to his “great, deep, Southern voice.” She gave him her phone number that night, and they became close, exploring the city’s sculpture garden and its vibrant restaurant scene. Soon she was simply calling him “Floyd,” just like his friends did.

For Courteney Ross, a lifelong resident of Minneapolis, George Floyd made her hometown seem new again, undiscovered.

“Floyd was new to the city, so everything was kind of new to him,” Ms. Ross said. “He made it seem like I was new to my own city.”

On the fourth day of testimony in the murder trial of Derek Chauvin, the former officer charged in Mr. Floyd’s death, the prosecution presented a fuller picture of George Floyd the person. In testimony, Ms. Ross, who had been dating Mr. Floyd for almost three years, described how he was a caring partner, a devoted father and passionate about exercise — a guy who loved to ride his bike and play ball with the neighborhood children.

She talked about all these things, as well as the ups and downs of their relationship, his love for his mother and the devastation he felt when she died a few years ago.

And like so many Americans, the couple had a shared struggle: opioid addiction.

“Our story, it’s a classic story of how many people get addicted to opioids,” she said. “We both struggled from chronic pain. Mine was in my neck and his was in his back.”

After three days of emotional testimony from bystanders who witnessed Mr. Floyd’s death in police custody last May, prosecutors on Thursday nudged the trial forward to one of the central aspects of the case: Mr. Floyd’s drug use.

In calling Ms. Ross to the stand, prosecutors both sought to humanize Mr. Floyd and seize the narrative around his struggle with drugs. By showing he had a high tolerance for opioids, prosecutors hope to cushion the blow of what is expected to be Mr. Chauvin’s primary defense — that Mr. Floyd died from a drug overdose, not from Mr. Chauvin’s knee pressing into his neck for more than nine minutes.

Over the nearly three-year, on-and-off relationship between Mr. Floyd and Ms. Ross, there were periods where they were clean, followed by relapse. When they could not obtain prescriptions for opioids from doctors, she said, they bought drugs on the streets.

“Addiction in my opinion is a lifelong struggle,” Ms. Ross said, in sometimes halting, tearful testimony. “It’s something we dealt with every day. It’s not something that just comes and goes.”

On Thursday, prosecutors were trying to establish that Mr. Floyd — who had fentanyl and methamphetamine in his system when he died, according to the toxicology report — had a high tolerance for fentanyl, which would help them rebut the defense’s claims that Mr. Floyd died of an overdose.

The defense’s focus on Mr. Floyd’s drug use, which arose numerous times in pretrial arguments and motions, echoes other high-profile police brutality cases, especially ones like this where police officers are accused of killing Black men.

It is a familiar playbook: In the Rodney King case in Los Angeles in the 1990s, defense lawyers brought up Mr. King’s alleged drug use; after Michael Brown was killed by a police officer in Ferguson, Mo., in 2014, officers said Mr. Brown was high on marijuana; and in the trial of an officer in Chicago for the killing of Laquan McDonald, the defense presented testimony about the victim’s drug use.

Yet as Mr. Chauvin, who is white, went on trial, one of the primary questions was whether this tactic would still work, especially amid shifting attitudes in America about drug use and with the country in the grips of an opioid epidemic.

“Tens of thousands of Americans struggle with self-medication and opioid abuse and are treated with dignity, respect and support, not brutality,” Ben Crump and Antonio Romanucci, lawyers for the Floyd family, said in a joint statement on Thursday. “We fully expected the defense to put George’s character and struggles with addiction on trial because that is the go-to tactic when the facts are not on your side.”

Eric J. Nelson, Mr. Chauvin’s lawyer, approached his cross-examination of Ms. Ross delicately, and started by saying: “I’m sorry to hear about your struggles with opioid addiction. Thank you for sharing that with the jury.”

Ms. Ross told Mr. Nelson that they relapsed together last spring, and that Mr. Floyd was hospitalized for several days in March after she found him doubled over in pain from an overdose. Later that month, she thought they had both managed to quit again, but in the weeks before he died in May, a change in Mr. Floyd’s behavior made her think he had again begun using.

“We got addicted and tried really hard to break that addiction many times,” she said. “When you know someone who suffers from any type of addiction, you can start to kind of see changes when they’re using again.”

While the questioning of Ms. Ross might have appeared spontaneous and off-the-cuff to jurors and the general public, the trial is well choreographed. All the witnesses are coached and prepared beforehand, sometimes in multiple sessions. Minnesota, in fact, has some of the strictest rules about the sharing of evidence and testimony — so both the defense and prosecution know well in advance what each side will present in a trial.

Jurors also heard on Thursday from two paramedics who said that Mr. Floyd was in a dire state by the time they arrived on the scene on May 25. Derek Smith, one of the paramedics, said he could not find a pulse when he felt Mr. Floyd’s neck as police officers remained on top of him.

“In lay terms, I thought he was dead,” Mr. Smith testified. In efforts to get Mr. Floyd’s heart beating, paramedics used a device to administer chest compressions and a defibrillator to provide an electric shock, but nothing worked, Mr. Smith said. Mr. Floyd was brought to a hospital where he was officially pronounced dead at 9:25 p.m.

Mr. Smith’s testimony could bolster prosecutors’ argument that it was Mr. Chauvin’s actions that led to Mr. Floyd’s death. Mr. Chauvin’s lawyer has suggested that the drugs Mr. Floyd had taken may have killed him.

Minutes after Mr. Floyd was taken away in an ambulance, Mr. Chauvin told a supervisor that police officers “had to hold the guy down” because he would not stay in the back of a police car and “was going crazy,” according to new body camera footage played in court.

The supervisor, Sgt. David Pleoger, testified that Mr. Chauvin had not mentioned applying pressure to Mr. Floyd’s neck until later, when they arrived at a nearby hospital and learned that Mr. Floyd was not doing well. Sergeant Pleoger, who has since retired, said that based on body camera videos from the scene, he thought the police officers should have stopped holding Mr. Floyd down once he became unresponsive.

“When Mr. Floyd was no longer offering up any resistance to the officers, they could have ended their restraint,” Sergeant Pleoger said.

In the earlier testimony, Ms. Ross also said that Mr. Floyd referred to her and his own mother, who died in 2018, by the same nickname: “Mama.” Mr. Floyd had called out for “Mama” as Mr. Chauvin knelt on his neck before his death.

Mr. Floyd had moved to Minneapolis from Houston looking for a fresh start, but after his mother died, Ms. Ross said, he changed. “He seemed like a shell of himself,” she said. “Like he was broken. He seemed so sad. He didn’t have the same kind of bounce that he had.”

On Thursday, jurors heard not only about Mr. Floyd’s struggle with drugs, but also details about his relationship with Ms. Ross.

She first met him at a Salvation Army homeless shelter where Mr. Floyd worked as a security guard. One night, he saw her waiting in the lobby to talk with the father of her two children about the birthday of one of their sons. Mr. Floyd sensed that she was upset.

“He was like, ‘Sis, you OK, sis?’” Ms. Ross recounted. He told her she was not OK.

“He said, ‘Can I pray with you?’”

“This kind person just to come up to me, and say can I pray with you, when I felt alone in this lobby,” she said. “It was so sweet at the time. I had lost faith in God.”

Tim Arango reported from Minneapolis, Nicholas Bogel-Burroughs from New York and Julie Bosman from Chicago.