Vine's Expository Dictionary of NT Words
denotes "one of four rulers" (tetra, "four," arche, "rule"), properly, "the governor of the fourth part of a region;" hence, "a dependent princeling," or "any petty ruler" subordinate to kings or ethnarchs; in the NT, Herod Antipas, Matthew 14:1; Luke 3:19; 9:7; Acts 13:1 .
"to be a tetrarch," occurs in Luke 3:1 (thrice), of Herod Antipas, his brother Philip and Lysanias. Antipas and Philip each inherited a fourth part of his father's dominions. Inscriptions bear witness to the accuracy of Luke's details.
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Vines, W. E., M. A. Entry for 'Tetrarch'. Vine's Expository Dictionary of NT Words. https://www.studylight.org/dictionaries/eng/ved/t/tetrarch.html. 1940.