The coordinates of the endpoints of ab and cd are a(x1, y1), b(x2, y2), c(x3, y3), and d(x4, y4). which condition proves that ab||cd?

Two lines are parallel if they have the same slope. Computing for the slope of line segment ab and line segment cd using m=(y2-y1)/x2-x1):
m of ab=(2-1)/(2-1)=1         m of cd=(4-3)/(4-3)=1
Since line segment ab and line segment cd have the same slope which is 1, then they are said to be parallel.