The sum of the slopes of lines which are parallel, perpendicular to the line passing through the points  $\left(1,3\right)$ and $\left(3,5\right)$ is

The slope of any line parallel to the line passing through the points  $\left(1,3\right)$ and $\left(3,5\right)$ is  $\mathrm{m}=\frac{5-3}{3-1}=1$
The slope of any line perpendicular to the line passing through the points  $\left(1,3\right)$  and  $\left(3,5\right)$ is $-\frac{1}{\mathrm{m}}$ where  $\mathrm{m}=\frac{5-3}{3-1}=1$
Hence, the slope of line perpendicular to the line passing through the points $\left(1,3\right)$  and $\left(3,5\right)$ is  -1
Therefore, sum of slopes is $1+\left(-1\right)=\boxed{0}$