What were the Jews relying on when they rejected Jesus?

The Jews were relying on their physical genealogy as descendants of Abraham, Isaac, and Jacob, and on their possession of the Scriptures, the Word of God, and the priesthood. Despite these privileges, they were wrapped in unbelief and could not comprehend Jesus or believe His truth.

Les explains that the Jews were relying on their genealogy—being children of Abraham, Isaac, and Jacob—as their basis for acceptance with God. Even though they had the unique opportunity of possessing the Scriptures, the Word of God, and the priesthood, they were wrapped in unbelief and could not comprehend Jesus. They rejected Him despite His truth, as He said, 'I tell you the truth, and you believe me not.' The context connects this to Romans verse 16: 'So then it is not of him that willeth,' meaning a person cannot claim child-of-God status merely by genealogy, nationality, or race.

John 8:39

They answered and said unto him, Abraham is our father. Jesus saith unto them, If ye were Abraham’s children, ye would do the works of Abraham.

What were the Jews in John 8 relying on when they argued with Jesus?

They relied on their physical genealogy as descendants of Abraham, Isaac, and Jacob, assuming that heritage guaranteed a right standing before God. Even though they possessed the Scriptures, the Word of God, and the priesthood, they remained wrapped in unbelief and could not comprehend Jesus' words.