Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the law of sines. In particular, length 1 is not constructible. As i recall, positive length $x$ is constructible if and only if $\sinh x$ is constructible in the Euclidean plane.

See my 1995 Intelligencer article at http://zakuski.utsa.edu/~jagy/bib.html

Let's see, in the fourth edition of his book, Marvin Jay Greenberg included Robin Hartshorne's proof of this, very different sort of language.

Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the law of sines. In particular, length 1 is not constructible. As i recall, positive length $x$ is constructible if and only if $\sinh x$ is constructible in the Euclidean plane.

See my 1995 Intelligencer article at http://zakuski.utsa.edu/~jagy/bib.html

Let's see, in the fourth edition of his book, Marvin Jay Greenberg included Robin Hartshorne's proof of this, very different sort of language.

EDIT: it seems part of the question was constructible lengths with irrational ratio. $\log 3$ and $\log 2$ should work.

Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the law of sines. In particular, length 1 is not constructible. As i recall, positive length $x$ is constructible if and only if $\sinh x$ is constructible in the Euclidean plane.

See my 1995 Intelligencer article at http://zakuski.utsa.edu/~jagy/bib.html

Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the law of sines. In particular, length 1 is not constructible. As i recall, positive length $x$ is constructible if and only if $\sinh x$ is constructible in the Euclidean plane.

See my 1995 Intelligencer article at http://zakuski.utsa.edu/~jagy/bib.html

Let's see, in the fourth edition of his book, Marvin Jay Greenberg included Robin Hartshorne's proof of this, very different sort of language.