Two Things:
1) If we can trace straight lines along the two paths of a ray, incident and transmitted, we can read the IOR off a straight line without angle measures.
But how to measure straight lines if the light is being refracted?
Consider a wedge of, say, glass. A short cylinder, like a block of good cheese. Cut it in half along a diameter.
Here is a clever shape! A ray passing through the flat face, and which strikes at the midpoint, will always exit
the other side in a direction perpendicular to the surface. It will experience no refraction, and the ray passed
outside the glass will have the direction of the transmitted ray.
And any beam striking the curved face which is aimed at the circle center will pass into the glass without refracting.
The ray leaving the flat face gives the refraction from glass to the outside medium.
2) To speak of the behavior of light as it passes through an object, we must consider closed surfaces with known dimensions. Here is a gentle introduction (for me!).

But what about beams of light which are not so kind?
What can we say about the paths of light which strike the flat face at positions other than the center, and the curved face at angles that are not perpendicular to the surface?
I think a good next step is to increase the complexity of the light source.