How to find the slope of certain lines matching random points

The method employed by us:
  1. Convert the picture issued by the OP to Windows Bitmap format
  2. Load the BMP file into a computer program, developed for the purpose
  3. Map the color content onto real (double precision) numbers
  4. Define a zero level of this mapping; in our case white = -0.2
  5. Define empirically limitations for the area that looks interesting
  6. Use a contouring module for making isolines at level zero
  7. Use all sort of limitations for distinguishing relevant data from the rest
  8. Calculate second order moments (variances) for the relevant isolines
  9. Take a look at the theory in the webpage Two-dimensional Moments
  10. Especially take notice of the formula $\;\tan(2 \theta) = 2 \sigma_{xy}/(\sigma_{xx} - \sigma_{yy})$
  11. Calculate mean value $\pm$ spread of relevant angles $\alpha=-\theta$ and Output
The numerical end result obtained is:
Alpha = 38 +/- 2 degrees
Picture produced:

Note.
A more robust method may be to consider instead the angles of the minor axes of the ellipses of inertia $\;\sigma_{yy}(x-\mu_x)^2-2\sigma_{xy}(x-\mu_x)(y-\mu_y)+\sigma_{xx}(y-\mu_y)^2=\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2\;$ with the x-axis.

Free accompanying source-only software (Delphi Pascal) has been made available at: MSE publications / references 2021
Disclaimer. Anything free comes without guarantee :-(and without referee)