Here's why:
* The final "e": This is a common rule in English. The "e" at the end of a word often makes the preceding vowel long (like in "bike" or "time"). In "rhyme," the "e" doesn't change the sound of the "i," but it does indicate that the "i" has a long vowel sound.
* The "h": While the "h" is usually silent in "rhyme," it can sometimes be pronounced in cases of strong emphasis.
Therefore, the "e" is always silent, and the "h" is usually silent in "rhyme."