I believe show here means to prove Show that parallelism is

I believe show here means to prove.

Show that parallelism is not transitive in the hyperbolic plane, that is, find three hyperbolic lines l_1, l_2, l_3 such that l_1 is parallel to l_2, l_2 is parallel to l_3, but l_1 is not parallel to l_3. (b) Still in the hyperbolic plane, show that if l_1 is perpendicular to l_2 and l_2 is perpendicular to l_3, then l_1 is parallel to l_3.l_1

Solution

(a)  If two lines are both parallel to a third line in the same direction, then they are parallel to one another

The distance between two parallels diminishes in the direction of parallelism and tends to zero; in the other direction the distance increases and tends to infinity

These similarities do not mean that parallels in Hyperbolic and Euclidean geometry behave in total accordance. Quite the contrary is the case.

While in Euclidean geometry two geodesics can either intersect or be parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to the same plane can either be:

(b) It is clear that if l₁ is perpendicular to l₂ then angle between l₁ and l₂ is 90^o if l₂ is perpendicular to l₃ then angle between l₂ and l₃ is 90^o and consecutively the angle between l₁ and l₃ is 180^o and we know that if the angle between 2 lines is 180^o then they will be parallel to each other.