How do we know the distance between stars (without assuming flat spacetime)? In this video, Nick Lucid (an astrophysicist) says we believe spacetime is globally flat because when we form a triangle between Earth and two distant stars, the angles of the triangle always add up to 180°.

However, doesn't this require knowing the distance between the two distant stars?  How is this measured? The normal method using trigonometry won't work without pre-assuming that spacetime is flat, which is circular reasoning.
 A: One method is to use group all stars with a similar red-shift together, and then find a star with a similar red shift that turns into a Type A Supernova (aka a "standard candle") which is of a known brightness. It is a known brightness because it occurs when a star accumulates enough mass so you know what the initial conditions are.
Since you know what the brightness should be, you can use how dim you actually observe it to be to figure out the distance and all other stars with a similar redshift are also a similar distance.
These next parts are where it gets fuzzy (and also from Wikipedia) since they hinge on the predictions of models which probably few understand:
Knowing the distances associated with various redshifts gives you the rate of the Hubble expansion at different points in the past. Apparently, the expansion rate changes over time depending on the curvature (or, at least, something that directly affects the curvature) so you can use this to measure the curvature.
A second method, which also hinges on the predictions of models, is that the typical angle between the hot and cool patches of the cosmic background radiation somehow depend on the curvature of the universe.
