There are many reasons to learn how to divide ague into syllables. Separating a word like ague into syllables is mainly to make it easier to read and pronounce. The syllable is the smallest sound unit in a word, and the separation of the ague into syllables allows speakers to better segment and emphasize each sound unit.
Knowing how to separate ague into syllables can be especially useful for those learning to read and write, because it helps them understand and pronounce ague more accurately. Furthermore, separating ague into syllables can also be useful in teaching grammar and spelling, as it allows students to more easily understand and apply the rules of accentuation and syllable division.
In the case of the word ague, we find that when separating into syllables the resulting number of syllables is 1. With this in mind, it's much easier to learn how to pronounce ague, as we can focus on perfecting the syllabic pronunciation before trying to pronounce ague in full or within a sentence. Likewise, this breakdown of ague into syllables makes it easier for us to remember how to write it.