The half-latitude rule is another way of finding east or west. Stars with a northern declination will rise north of due east and set north of due west. Stars with a southern declination will rise south of due east and set south of due west.
When a star passes through your zenith it is at a height equal to your latitude and has moved away from due west or east by an angle equal to half your latitude. Outside the polar regions this is accurate to around five degrees. One advantage of the half-latitude rule is that you do not need to know which zenith stars you are using. Any will do.
Between about 10° north or south you can believe your zenith star rises off due east by an amount equal to its declination and sets off due west by the same amount. It is not true but works.
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