Why was Jesus born in a manger and died on the cross?
Why did we need a blood sacrifice to atone for our sins?
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Why the cross ?
“Jesus died on a simple death stake: In support of this there speak (a) the then customary usage of this means of execution in the Orient, (b) indirectly the history itself of Jesus’ sufferings and (c) many expressions of the early Church fathers.”—The Cross and Crucifixion, Hermann Fulda.
The question still remains, why did Jesus die in this fashion ?
The apostle Paul states that Christ became a curse to those under the law by being fastened to a stake, since “Accursed is every man hanged upon a stake [xylon].” Paul here is quoting from the law of Moses, which required that the bodies of executed criminals be fastened to a tree or stake as a warning and which meant that they were cursed by God.—Gal. 3:13; Deut. 21:22, 23
Paul explained in Gal 3:13 that Jews that were under obligation to follow the law are under a curse, Jesus removed or fulfilled the law by taking the curse upon himself by been nailed to the torture stake.
Why a manger ?
Caesar Au·gus′tus, ordered a census to be taken in ones city of birth. Since Joseph was born in Beth′le·hem he had to travel there from the city of Nazareth. When he and Mary arrived, there wasn’t room for them anywhere, so they had to lodge in a stable and put Jesus in a manger.
The Bible doesn't seem to ascribe any other meaning to the fact that Jesus was put in a manger.
Why do we need a blood sacrifice ?
God’s legal requirement of perfect justice is evident in his law to Israel: “Soul will be for soul, eye for eye, tooth for tooth......” (Deut. 19:21) Therefore, to redeem man from sin and death, the ransom price of a perfect human soul was required since such a life was forfeited.
Since blood stands for life, Jesus sacrifice of his life involved the spilling of blood. Thus the blood of Jesus figuratively (and literately) was poured out as a sacrifice.