Etymology
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Words related to quadrangle

*kwetwer- 
Proto-Indo-European root meaning "four."

It forms all or part of: cadre; cahier; carillon; carrefour; catty-cornered; diatessaron; escadrille; farthing; firkin; fortnight; forty; four; fourteen; fourth; quadrant; quadraphonic; quadratic; quadri-; quadrilateral; quadriliteral; quadrille; quadriplegia; quadrivium; quadroon; quadru-; quadruped; quadruple; quadruplicate; quarantine; quarrel (n.2) "square-headed bolt for a crossbow;" quarry (n.2) "open place where rocks are excavated;" quart; quarter; quarterback; quartermaster; quarters; quartet; quarto; quaternary; quatrain; quattrocento; quire (n.1) "set of four folded pages for a book;" squad; square; tessellated; tetra-; tetracycline; tetrad; tetragrammaton; tetrameter; tetrarch; trapezium.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit catvarah, Avestan čathwaro, Persian čatvar, Greek tessares, Latin quattuor, Oscan petora, Old Church Slavonic četyre, Lithuanian keturi, Old Irish cethir, Welsh pedwar.
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angle (n.)
"space or difference in direction between intersecting lines," late 14c., from Old French angle "an angle, a corner" (12c.) and directly from Latin angulus "an angle, a corner," a diminutive form from PIE root *ang-/*ank- "to bend" (source also of Greek ankylos "bent, crooked," Latin ang(u)ere "to compress in a bend, fold, strangle;" Old Church Slavonic aglu "corner;" Lithuanian anka "loop;" Sanskrit ankah "hook, bent," angam "limb;" Old English ancleo "ankle;" Old High German ango "hook").

Figurative sense "point or direction from which one approaches something" is from 1872. Angle-bracket is 1781 in carpentry; 1956 in typography.
quad 

1820 as a shortening of quadrangle (n.) in the building-court sense (in this case "quadrangle of a college," Oxford student slang); also in old slang the quadrangle of a jail or prison, where prisoners take their exercise. By 1880 as short for the printer's quadrat (n.). By 1896 as "a quadricycle, a bicycle for four riders" (quadricycle is attested by 1879, quadruplet in this sense by 1893). As "one of four young at a single birth" by 1951 (in reference to armadillos), short for quadruplet; 1970 as a shortening of quadraphonic (adj.). Related: Quads.

quadrangular (adj.)

"four-cornered, four-sided," early 15c., quadrangulere, from Medieval Latin quadrangularis "having four corners," from Late Latin quadrangulus "having four angles" (see quadrangle). Related: Quadrangularly.