Separation is measured in arc seconds.
Alternatively one could observe the pair when it is near the meridian, so that 0 degrees lies towards the bottom of the FOV in an inverting telescope, with east to the right. With practice, it is possible to estimate the PA of a pair within 10 degrees by eye alone. Some more advanced techniques involve using a micrometer to make precise measurements. Some stars are too close to be split visually. In these cases, we know about the double stars because of the double spectra that is apparent when its light is analyzed with a spectroscope.