“God presented Christ as a sacrifice of atonement through the shedding of his blood- to be received by faith. He did this to demonstrate his righteousness, because in his forbearance he had left the sins committed beforehand unpunished- he did it to demonstrate his righteousness at the present time, so as to be just and the one who justifies those who have faith in Jesus.”
Romans 3:25-26 NIV

Jesus demonstrated His righteousness by being willing to die for sins He did not commit. Jesus’ death and resurrections atoned for all of humanities’ sins.

“God made him who had no sin to be sin for us, so that in him we might become the righteousness of God.”
2 Corinthians 5:21 NIV

On the cross, there was an exchange that happened. Jesus takes on our sin and we take on his righteousness. We believe (have faith) in Jesus death and resurrection and He declares us righteous. We are made in right standing with God.

“Wealth is worthless in the day of wrath,
but righteousness delivers from death.”
Proverbs 11:4 NIV

Christ’s righteousness gives us salvation.

“The righteousness of the blameless makes their paths straight,
but the wicked are brought down by their own wickedness.”
11:5 NIV

Christ’s righteousness makes our path align with God’s path for our lives.

“The righteousness of the upright delivers them,
but the unfaithful are trapped by evil desires.”
11:6 NIV

Christ’s righteousness changes our desires for the world and we make choices that align with our values, morals, and integrity.

“The righteous is rescued from trouble,
and it falls on the wicked instead.”
11:8 NIV

Christ’s righteousness rescues us from trouble by His mercy (spared from the true penalty of sin) and grace (unearned favor).

“Truly the righteous attain life,
but whoever finds evil finds death.”
11:19 NIV