why do sororities and fraternities use greek letters

To shed their boozy reputation. Clubs have existed on America's college campuses since 1750. Many began as literary debate clubs, and some took Latin names because it was the language of the scholars. Take William and Mary's popular F. H. C. societyБwhich may have stood for the Latin
Fraternitas, Hilaritas, Cognitoque (people knew it better as the "Flat Hat Club").

Ostensibly a debate club, in practice it was a group of drunken rabble-rousers. So on December 5, 1776, five William and Mary students met at a tavern to start their own debate club. The new club wanted a fresher, more serious image so they penned their motto, charter, and eventual name in Latin and Greek to differentiate it from all thoseб drunky clubs (and to keep their motives secret).

Today, we know this group as the prestigious honor societyб Phi Beta Kappa. In the late 18thб century,б ннн expanded to other schools, inspiring students at Union College to form spinoff groups in 1825. They, too, chose Greek lettersБand the social clubs quickly spread like a keg party to other campuses. б Yes, but in the middle ages in Christian western Europe, which was where the precursors of modern universities emerged, hardly anyone could read Greek at all, and Latin was the universal language of scholarship.

Surely it was the lingering effects of this tradition that led to Latin still being considered fundamental in high schools in the 19th century, and often still taught as an option in the 20th. (Both I and my wife studied Latin in high school in the 1960s and '70s, me in an English state grammar school, and she at a public school in Los Angeles. ) It is my impression that Greek was never anywhere near as widely taught at high school level as Latin.

I find your claim about Los Angeles High surprising, and I suspect it was very unusual.

It seems to me that the use of Greek by these organizations (no real equivalents of which exist outside the USA, so far as I am aware) was probably connected with some sort of 19th century American movement to revive Greek and the reputation of the ancient Greeks vis-a-vis the Romans, and to replace the study of the more traditional Latin with that of Greek.