Studying Euclidean geometry using hyperbolic criteria

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.

But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré's Elements. This postulate is the obvious statement that given a line $L$ and a point $p$ not on $L$ there are at least two lines through $p$ that do not meet $L$. Your friend wonders what it would be like if this assertion were replaced with the following: given a line $L$ and a point $p$ not on $L$, there is exactly one line through $p$ that does not meet $L$.

You begin investigating this Euclidean geometry, but you find it utterly impossible to visualize intrinsically. You decide your only hope is to find a model of this geometry within your familiar hyperbolic plane.

What model do you build?

I do not know if there's a satisfying answer to this question, but maybe it's entertaining to try to imagine. For clarity, we Euclidean creatures have built models like the upper-half plane model or the unit-disc model to visualize hyperbolic geometry within a Euclidean domain. I'm wondering what the reverse would be.

Here's another version of Doug Chatham's answer, but with details.

If you lived in Hyperbolic space, then Euclidean geometry would be natural to you as well. The reason is that you can take what is called a horosphere (in the half-space model for us, this is just a hyperplane which is parallel to our limiting hyperplane) and this surface actually has a Euclidean geometry on it!

So unlike for us, where the hyperbolic plane cannot be embedded into Euclidean 3-space, the opposite is true: the Euclidean plane can be embedded into hyperbolic 3-space! So this is analogous to our understanding of spherical geometry. It's no surprise the spherical geometry is slightly different, however, it fits nicely into our Euclidean view of things, because spherical geometry is somewhat contained in three-dimensional geometry because of the embedding.

Look up "horosphere" (for example, in page 90 of the Princeton Companion to Mathematics). Wikipedia describes it on its Horoball page.

An alternative to the horosphere model ...

In "A Euclidean Model for Euclidean Geometry", Adolf Madur discusses a Disk model of the Euclidean plane. (Madur says that David Gans has priority for discussing this model, so I'll call it the "Gans Disk".) The "lines" consist of diameters of the Disk, and half-ellipses that have a diameter as a major axis; the measure of the angle between two "lines" is defined as the traditional measure of the angle between their respective major axes. With an appropriate metric (which I have forgotten, and which is just missing in the document preview linked), we get all of the Euclidean plane crammed into the Disk.

Overlaying the Gans Disk on the Poincaré Disk (or a sub-disk thereof) provides another way for Hyperbolians to study Euclidean geometry. They just have to agree to treat these half-ellipse paths (which I don't think are ellipses to them) as "lines", and to alter their concept of angle measure and length accordingly.

This model might be considerably harder for Hyperbolians to wrap their minds around than the horosphere model, though.

Edit. Since ellipses are projections of tilted circles, we can "lift" the Gans Disk to a "Gans Hemisphere". (This is actually a middle phase in the derivation of the Gans Disk model.) There, the "lines" are great semi-circles, with angles measured via their diameters in the equatorial plane. Not a major refinement of the Gans Disk, but at least the "lines" are naturally-occurring geometric objects, instead of the contrived ellipse-paths. Of course, the metric would need adjustment; off the top of my head, I don't know how much more (or less?) complicated that metric would be.