Sure they do. Rhyme doesn’t mean a perfect match. If it did, half the poetry in existence that utilizes rhyming couldn’t exist. At least they rhyme in my pronunciation of English.
“Rhyme” rhymes with “dime” and “time” and also “fine.”
That’s my view, anyway. Perhaps you speak in a different American dialect. Maybe you say “pilfa,” but if you did, you’d likely also say “silva.”
It would be interesting to hear what others think.
From Merriam-Webster's 11th Unabridged Distionary (I'm using definition "c"): rhyme 1 a : correspondence in terminal sounds of two or more words, lines of verse, or other units of composition or utterance: such as (1) also rhyme proper : correspondence of the last accented vowels and all succeeding sounds in two lines or units especially (as in English verse) when the sounds preceding the last accented vowel are different in the two rhyming units (2) : assonance 2b (3) : consonance 2d b : one of two or more words thus corresponding in sound c : correspondence of other than terminal word sounds: such as (1) : beginning rhyme (2) : alliteration (3) : internal rhyme d : rhyme scheme