Because in the word “meter/metre”, the ending ‘r’ sound follows a consonant, but in “wire” and “lyre”, it follows a vowel. The presence of a consonant there requires an additional syllable, which American English represents by putting the ‘e’ before the ‘r’.
Another contributing factor is the rule for a silent ‘e’ in American English, which is that vowel-consonant-e (such as ire or yre) at the end of a word makes the ‘e’ silent. The pattern consonant-consonant-e (such as tre) actually means that the ‘e’ should not be silent (again, in American English–obviously, it is silent in British English).
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