Is there a conformal model of hyperbolic plane where the geodesics are straight lines (in the Euclidean sense)? For example, in the Beltrami-Klein model, geodesics are straight lines, but the model is not conformal.
Note: I am of course not asking for the hyperbolic geodesics to have the same parameterization as Euclidean geodesics.
Take your favorite hyperbolic triangle. The sum of its angles will be less than $\pi$.
Now, draw that hyperbolic triangle in some model of the hyperbolic plane where all geodesics are straight lines. The result will look exactly like a Euclidean triangle, and the sum of the angles of this Euclidean triangle will be $\pi$.
So the triangle's angles are not all preserved in the model, meaning that the model is not conformal.