By the "physical hyperbolic surface," do you mean tractricoid?
There is lack of context for your question. What have you covered about this subject? What does it mean to "argue" and what is "intrinsically straight"? Do you need to prove algebraically that the curves that run radially across each annular strip are geodesics? Then this questions is definitely not at the pre-university level.
I personally have not learned about these topics until my third year in college, so you want to ask these questions in the Topology and Differential Geometry forum. Otherwise, if you are studying this on an elementary level, you need to provide more information, such as what properties of straightness you covered.