Do Euclidean geometry preservers parallelism of lines and area ratios?

Question

Do the Euclidean geometry preserves the properties parallelism of lines and area ratios for any possible transformation?

I know that the Affine geometry do and I think that Euclidean geometry also do it.

Answer 1

The usual hierarchy is

• Affine transformations (they preserve parallelism and ratios of segments along a line):
  • Shear
  • Stretch
  • Similarity transformations (these also preserve angles and ratios of areas):
    • Dilation (scaling)
    • Euclidean transformations or Isometries (these also preserve distance):
      • Rotation
      • Reflection
      • Translation
      • Glide reflection