Within hyperbolic space, are all sides of an ideal triangle parallel?; and is it possible for them all to be hyperparallel? Question:
I unfortunately have an extremely limited foundation in mathematics but I am trying to wrap my head around hyperbolic geometry in simple terms and I have spent all day trying to search for an answer.
I understand the following:

*

*Two parallel lines, by definition, do not intersect.

*An ideal point refers to the "intersection" of two lines at infinity, as is not considered an intersection for the purposes of parallelism.

*Sum of angles of an ideal triangle is 0°.

*There are infinitely many parallel lines to line l around any given point P; but that only two of these are limiting/hyperparallel.

*All ideal triangles are congruent.

Based on the above, my understanding is that all sides of an ideal triangle in hyperbolic space are parallel.

However, I do not know if all three sides of this ideal triangle are also hyperparallel? One part of my intuition insists that they must be, because, given any ideally intersecting parallel line l2 to line l1, if there were to exist a theoretical line l3 closer to that hyperparallel, then it would hold that the angle of the l1/l3  intersection would be less than the angle of the intersection of l1/l2; however, given that all ideally intersected lines have an angle of zero, it's impossible for this l3 to exist. However, if this is true, then this implies that all ideal intersections in hyperbolic space are hyperparallel, which would also imply that at the ideal intersection there are infinitely many hyperparallel lines -- which goes against my understanding in #4 above?
In the linked image, point A is the origin of lines AB, AC, AD and AE; surely at most two of these lines can be the hyperparallels of any other?
My question then is as follows:

*

*Is my first assumption (all sides of an ideal triangle are parallel) correct?

*If so, at what step is my intuition failing when it comes to understanding whether all sides of an ideal triangle are hyperparallel?

Any assistance in understanding this would be greatly appreciated, but I do also understand if the answer follows the form of "it's not really possible to explain this to someone with a minimal mathematics background".
Answer:
It has been explained to me that the limit of 2 hyperparallel lines does not exist in sole relation to line l, but in relation to point P and parallel with line l. An infinite amount of hyperparallel lines can be drawn from a given hyperbolic line.
For example, in the Poincare disc below, ABC is an ideal triangle; all its sides are hyperparallel with one another.
However: lines CF and BD  are also the only two hyperparallels to line CB which also intersect point D; and lines CH and BI are the only two hyperparallels which intersect point G.

 A: You asked


*

*Is my first assumption (all sides of an ideal triangle are parallel) correct?

*If so, at what step is my intuition failing when it comes to understanding whether all sides of an ideal triangle are hyperparallel?


The first one is easy. For better intuition consider the
Beltrami-Klein model
of the Hyperbolic plane. Wikipedia states:

It is a disk and the inside of the disk is a model of the entire hyperbolic plane. Lines in this model are represented by chords of the boundary circle (also called the absolute). The points on the boundary circle are called ideal points;

In this model an ideal triangle is determined by
any three points on the boundary of the disk and
the Euclidean triangle they uniquely determine. In
the model this is an ideal triangle because the
vertices correspond to ideal points. The sides of
the triangle intersect in the Euclidean plane only
at the vertices of the triangle which are on the
boundary. Thus, they are hyper-parallel according
to the model.
Wikipedia states it this way:

Chords that meet on the boundary circle are
limiting parallel lines.

Chords of the the circular boundary that do not
intersect at all are, of course, parallel in the
model. If they intersect at just one point but it
is on the boundary, then they are termed limiting
parallel or hyper-parallel.
For your second question, you wrote:

all ideal intersections in hyperbolic space are
hyperparallel, which would also imply that at the ideal intersection there are infinitely many hyperparallel lines

Which is correct however, as you wrote:



*There are infinitely many parallel lines to line
l around any given point P; but that only two of
these are limiting/hyperparallel.


with the key phrase "around any given point $P$". That
is, there are infinitely many hyper-parallel lines, but
only one of them passes through any given point. This is
similar to Euclidean geometry where given a line, there
are infinitely many lines parallel to it, but only one
of them passes through a given point.
