How did Jesus reconcile his teachings with Judaism?

God is the Same God. Here is what the Jesus has said.

"Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. (Matthew 5:17)

Notice he said "to fulfill them".

Now the evils and lawbreaking of revenge, God has reserved for himself.

For under the time of the law it is said:

“Whoever sheds human blood, by humans shall their blood be shed; for in the image of God has God made mankind. (Genesis 9:6 NIV)

For those under grace it is said:

It is mine to avenge; I will repay. In due time their foot will slip; their day of disaster is near and their doom rushes upon them." (Deuteronomy 32:35 NIV)

So since God finds glory in fulfilling the law. There is even more reason to fear him. For in the days when the completion of the law was given as a responsibility to man, men then did fail when Gods word is perfect, he had to step in to finish the balance.

He saw that there was no one, he was appalled that there was no one to intervene; so his own arm achieved salvation for him, and his own righteousness sustained him. (Isaiah 59:16)

Since his satisfaction then is determined by his effort in fulfilling the law, and he has decided to fulfill all of it instead of abolishing it.

"Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. (Matthew 5:17)

The Son in seeing his father is given glory by doing that of the work of the father.

Jesus gave them this answer: "Very truly I tell you, the Son can do nothing by himself; he can do only what he sees his Father doing, because whatever the Father does the Son also does. (John 5:19)

Therefore in trusting the righteous judgement of the Father, the Son walked allowing the offense to occur without compensating the law. Knowing full well the Father's love for him.

"What then will the owner of the vineyard do? He will come and kill those tenants and give the vineyard to others. (Mark 12:9)

In Psychology we have learned the it is "the decision to fix the threatening behavior that causes anger". Therefore by faith that God will fulfill the law, we no longer become angry, and our spirit becomes pure, in the same faith that the Son gave us as an example to follow.

So he reconciled his teachings with Judaism by having faith in the Father. As we should do as well. For what good is an example that no one follows?

Jesus answered, "I am the way and the truth and the life. No one comes to the Father except through me. (John 14:6)

It's important to remember that Judaism was a religion instituted by God, intended for God's purposes, but had been co-opted by man who began adding and removing laws as they saw fit.

So enter Jesus: He says he "fulfills" the law. What does he mean by this? Many people seem to think this meant that he added to, and finished creating all of the rules associated with the law, but I don't believe this to be the case. As Paul said in Romans 3:19-20:

Now we know that whatever the law says it speaks to those who are under the law, so that every mouth may be stopped, and the whole world may be held accountable to God. For by works of the law no human being will be justified in his sight, since through the law comes knowledge of sin.

So if the law's purpose wasn't to primarily justify us, but to show us our sin, then why would Jesus simply come and add laws? (Indeed, he does do this at times, but I believe his point is to push people closer to the edge of despair when it comes to their own self-reliance, and push them to a point where they cry out to have mercy on them.) The answer is, he doesn't. He, being God, clarifies the law by showing the true expectation of the law, that our hearts would be turned towards God.

Now, as for fulfilling the law, he fulfills it by himself following the law. He finishes the expectations of the law by living under the law, and thus, he is the only one who manages, on his own merits, to escape the judgement of God. Now here's where it gets crazy, and why we believe in Christ in the first place: he then goes on to die the death of a sinner anyways, and in doing so, he becomes the sacrifice for the sins of those who have failed to obey the law, and thus he frees us from the law (though not himself).

Now, the temporal consequences of our sin will still remain with us. If I kill someone, then I am more than likely going to suffer temporal consequences for doing so, but the death of Christ is sufficient to pay the penalties for this on an eternal level, and my failure to match up to the requirements of the law are forgiven not solely on the basis that God forgives, but on the basis that Christ has already paid the penalty.

So Jesus reconciled himself to Judaism by following all that Judaism required of him.

As for his contemporaries trying to decide if he did or not, keep in mind that they had already co-opted the law and the prophets for their own gain. They added and removed laws to suit their fancy, and viewed Jesus as a glutton and a drunkard because he didn't follow their twisted version of the law. Nevertheless, the law was given by God, and the law, as given by God (not as twisted by the Pharisees and Sadducees) was fulfilled (obeyed) by Christ.

Actually, without Jesus, Judaism is incomplete. The Old Testament, even as far back as Genesis 3, speaks of a Deliverer, a Messiah, a Savior. This Deliverer would come through the line of Abraham and be the blessing to all people that God promised Abraham.

I want to point out that I am distinguishing Biblical Judaism from rabbinical Judaism. I'm defining Biblical Judaism as acknowledging Jesus as the fulfillment of the Biblical prophecies. Messianic Jews fit this definition of Biblical Judaism. Rabbinical Judaism denies Jesus, of course, but still has no Messiah that has yet come. Still culture and traditions are followed.

The Scriptures foretold when and where the Messiah would be born, that He would be called "God with us", that He would die and rise again, that He would be rejected by His people, and on and on. So, without Jesus, Judaism is incomplete, with God being silent and ignoring the Jewish people for 2500 years. Furthermore, within a generation of the Jewish leaders rejection of Jesus as the Messiah, the Jewish nation was conquered and the Temple destroyed. This lasted for nearly 1900 years. (This dispersion and regathering was also prophesied.)

The apostles never saw themselves as anything other than Jewish. Indeed, the most Jewish thing a Jewish person can do is recognize the Jewish Messiah. Jesus fulfilled the promise to be a Deliverer not only for the people of Israel, but for the whole world. (Isaiah 49:6 and others)

Without Jesus, Judaism is without a Messiah. Without Jesus, Judaism has not been a blessing to all people. Without Jesus, the Jewish prophecies remain unfulfilled and must be considered false.

So, Jesus' teachings fit completely within Judaism, and true Judaism recognizes Jesus as the Jewish Messiah prophesied in the Jewish Scriptures.

Clearly Jesus described himself as being within Judaism and not opposing or superceding it. Matt 5:17 “Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill."

Perhaps the easy answer to your question is to turn it around: What did Jesus ever do or say that contradicted the Hebrew scriptures? What did he say that needed to be reconciled? He never said that he was abolishing Judaism. He never said anything that contradicted anything in the Hebrew scriptures. He routinely quoted scripture to back up his statements, and as I quoted above, he affirmed the scriptures.

Yes, he claimed to be the Messiah and God made flesh, and if that was not true , he was the worst of blasphemers. But if it was true, then he was fulfilling the scriptures.