And measurements of, say, very precise equilateral triangles, necessarily imply certain interior angles, which you can compare to the actual angles they make. For instance, on a flat plane, you can fit six equilateral triangles sharing one point to make a hexagon. On a sphere if you make them big enough you'll find that they don't quite fit.
I'm not sure how that matters for this purpose, they used these surveys to measure the shape of the Earth (specifically, the circumference, and later the flattening- ie, plenty precise enough to measure the curvature more or less directly).
Is the idea that land is spherical while the oceans are flat? Do this survey on each continent, including Antarctica, again just constructing big triangles and measuring the deviation from flatness. Eventually your model of the flat earth looks like a bunch of pieces of an eggshell separated by flat water that just happens to assemble into an egg- or rather an ellipsoid- if rearranged, up to and including the expected flattening at the poles.
With sufficient elbow grease you could extend a survey from the far north of the Americas down to the south, measuring the curvature as you went, eventually finding that you must be almost upside-down compared to where you'd started.
EDIT: Also, that's to measure distance, not direction.