How did NASA engineers determine the correct angle of decent to safely re-enter earth's atmosphere?

Question

How did NASA engineers determine the correct angle of decent to safely re-enter earth's atmosphere?

Answer 1

They used mathematics.

Answer 2

Any object that travels through atmosphere creates some drag, or air resistance.  A re-entry requires enough drag to slow the vehicle down so that it can descend (otherwise the craft continues to over-shoot earth).  Too much drag creates too much heat and the vehicle burns up.  The question is, how much drag is enough?  

Space engineers were able to calculate drag by using gas friction physics and measurements about how thick the atmosphere is at various elevations.

Answer 3

The angle of attack when re-entering is determined by the orbital velocity, the aerodynamics of the vehicle, and the ability of the vehicle to shed heat.