Paul reasoned, explained and gave Scriptural evidence of Jesus being the Messiah as prophesied by God in the Pentateuch and the Torah (the Old Testament for Christians).
We are not told what Scriptures Paul used but the following are possibilities.

(NASB) Acts 17:3 “explaining and giving evidence that the Christ had to suffer and rise again from the dead…”

Atonement as defined by the NIV NAVE’S Topical Bible page 55 “The divine act of grace in which God accepts an offering as a substitute for the punishment for sin.”

In the book of Leviticus, God established the ritual for the nation of Israel’s atonement through the sacrifice of a goat – once a year to be performed by the high priest.

(NASB) Leviticus 16:15 “Then he shall slaughter the goat of the sin offering which is for the people, and bring its blood inside the veil and do with its blood as he did with the blood of the bull, and sprinkle it on the mercy seat and in front of the mercy seat.”

(NASB) Hebrews 9:11-14 “But when Christ appeared as a high priest of the good things to come, He entered through the greater and more perfect tabernacle, not made with hands, that is to say, not of this creation;
12 and not through the blood of goats and calves, but through His own blood, He entered the holy place once for all, having obtained eternal redemption.
13 For if the blood of goats and bulls and the ashes of a heifer sprinkling those who have been defiled sanctify for the cleansing of the flesh,
14 how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God?”

Jesus Christ, the Messiah, the Anointed One of God.  Jesus Christ, the High Priest you and I need.  Jesus Christ sacrificed for our sins ~ our Savior.  It is the blood of Jesus which saves you and me through faith in Him.
Jesus had to suffer for you and me so we can be saved from our sins.  It is imperative that we choose to have faith in Him because He has done everything needed.

Resurrection prophesied of the Messiah:
(NIV) Psalm 16:9-10 “Therefore my heart is glad and my tongue rejoices; my body also will rest secure, 10because you will not abandon me to the grave, nor will you let your Holy One see decay.”

The Messiah had to suffer and He had to die…to fulfill all the prophecies throughout the Old Testament.  He had to suffer and die for our sins so we can be forgiven.
The Holy Bible reveals Messiah in the Old Testament and the fulfillment is in the New Testament and into the future.  There is much Scriptural evidence for Messiah.

Paul proclaimed to the Thessalonians:
(NASB) Acts 17:3 “…‘This Jesus whom I am proclaiming to you is the Christ.”