That does not make them interchangeable.
English, due to the particular historical circumstances of its development, often has two words for the same thing, Anglo-Saxon and Norman French respectively. Oftentimes this shows up in class distinctions. So we raise cows, but eat beef, as an example.
Righteousness and justice are synonyms, the former is Germanic and the latter is Latin. We have courts of justice, rather than courts of righteousness, for the same reason that we raise cows but eat beef--Norman French was the language of the rulers after the Norman Conquest.
To be justified is to be made right, to be righteous.
In the Greek of the New Testament there's just one word (and variations of that one word), and it is variously translated as righteousness or justice.
This is why in Romans 1:17 we read "by it the righteousness of God is revealed from faith to faith, just as it is written, 'The just shall live by faith'" It would mean the same thing if we translated it as "the justice of God is revealed from faith to faith, just as it is written, 'The righteous shall live by faith'".
In Greek:
δικαιοσύνη γὰρ θεοῦ ἐν αὐτῷ ἀποκαλύπτεται ἐκ πίστεως εἰς πίστιν καθὼς γέγραπται Ὁ δὲ
δίκαιος ἐκ πίστεως ζήσεται
dikaiosyne and dikaios--justice and just or righteousness and righteous--respectively,
What is the righteousness or justice of God revealed in the Gospel? The righteousness or justice by which one is reckoned, declared, righteous/just by faith. It is not the act of believing certain things that renders one righteous, it is the righteousness of God in the Gospel, now given to another through faith, that the other is now righteous.
Romans 5:18b
καὶ δι ἑνὸς δικαιώματος εἰς πάντας ἀνθρώπους εἰς δικαίωσιν ζωῆς
even by the one righteousness has come justification of life unto all men
All condemned by decree because of the one disobedience; and also all justified by decree by the one righteousness.
All condemned and made unrighteous by Adam's disobedience and sin;
All justified and made righteous by Christ's righteous obedience.
How does one receive this? By faith alone.
-CryptoLutheran