WASHINGTON — The U.S. Supreme Court’s justices put themselves in the driver’s seat Tuesday, hearing arguments in two cases involving vehicle searches, but it was unclear what routes the justices will take to resolve the cases.

The justices heard arguments in a case involving Pennsylvania state troopers’ stop of a rental car driven by a man who wasn’t on the rental agreement. They also heard a case from Virginia involving a policeman’s search for a stolen motorcycle.

Chief Justice John Roberts and Justice Stephen Breyer in particular seemed to be searching for a way to give easy-to-understand directions in resolving the Pennsylvania case. That case is about police’s ability to search a rental car they stop when the driver isn’t listed on the rental agreement.

“One of the things that I think is very important in these types of cases is the ability to give clear guidance not only to the courts but to the police,” Roberts said. Breyer, when trying to describe a resolution to the case, at one point said he was “looking for something simple.”

The case the justices were discussing involves Terrence Byrd, who was driving his fiancee’s rental car on a Pennsylvania highway when a state trooper pulled him over for an alleged minor traffic violation. He acted nervously during the stop, at one point telling troopers he had a marijuana cigarette in the car, and officers eventually asked to search the vehicle.

But because the rental agreement didn’t authorize Byrd to drive the gray Ford Fusion, troopers told him they didn’t actually need his consent for the search. And when troopers opened the trunk, they found body armor and about 2,500 little bags of heroin. Byrd later acknowledged he planned to sell the drugs for roughly $7,000, and a court sentenced him to 10 years in prison.

Byrd’s attorneys argue his case has potential consequences for the 115 million car rentals that take place annually in the United States. They say that if the government wins, police will have an incentive to pull over a rental car driver who commits a traffic violation because police will know they can search the car if the driver isn’t on the rental agreement.

Byrd tried to get the evidence from the search excluded from his case. But a court ruled that because Byrd was an unauthorized driver of the car he had no reasonable expectation of privacy in the car and therefore couldn’t challenge the search using the Fourth Amendment, which protects against unreasonable searches. The Trump administration and courts in several parts of the country agree that’s the right outcome. Other courts disagree.

On Tuesday, Roberts and Justice Samuel Alito seemed to be the most willing to side with the government while Justice Sonia Sotomayor seemed strongly sympathetic to Byrd’s argument.

“If we rule that … someone has no expectation of privacy even when the renter has given it to them, then what we’re authorizing is the police to stop every rental car and search every rental car, without probable cause, that might be on the road,” Sotomayor said.

Byrd’s case dates to 2014, when his fiancee Latasha Reed, with whom he has five children, rented a car from a Budget rental office in New Jersey. The government disputes their relationship, calling her Byrd’s girlfriend, but both sides agree that the rental agreement didn’t cover Byrd.

Even so, Reed handed him the keys as soon as she left the rental office. He was later pulled over while driving alone near Harrisburg. The reasons the trooper who pulled him over gave for finding Byrd’s driving suspicious included that he was driving as most people are taught to drive, with his hands placed at 10 and 2 on the wheel, and he spent too long in the left lane making a passing maneuver.

The outcome also was unclear in the second case the justices heard Tuesday, about whether police need a warrant before searching a vehicle on private property outside a home.

The case, involving a Virginia motorcyclist who eluded police at speeds up to 140 mph, tests the limits of a long-established exception to the need for a warrant when it comes to searching vehicles. The exception stems from the fact that vehicles can be moved before police are able to obtain a warrant to search them.