5. how were northern lawmakers convinced to allow missouri into the union as a slave state under the missouri compromise?

Northern lawmakers were convinced to admit Missouri as part of the Union due to the Missouri Compromise because they also admitted Maine to the Union.  There anti-slavery and pro-slavery states in the Union.  Missouri was a slave state and if it was admitted, it would change the balance in favor of those who advocated slavery.  Maine was a free state so if Main was also admitted, it would maintain the balance between the two opposing ideologies.  Aside this, an imaginary boundary was made along the former Louisiana Territory that kept free state and slave state apart  until the Kansas-Nebraska Act of 1854.

The congress completes the diagram