Angle
late Middle English: from Old French, from Latin angulus ‘corner’.
wiktionary
From Middle English angle, angul, angule, borrowed from Middle French angle, from Latin angulus(“corner, remote area”), from Proto-Indo-European*h₂engulos < *h₂eng-(“corner, hirn”). Cognate with Old High German ancha(“nape of the neck”), Middle High German anke(“joint of the foot, nape of neck”). Doublet of angulus.
From Middle English anglen(“to fish”), from Middle English angel(“fishhook”), from Old English angel, angul(“fishhook”), from Proto-Germanic *angulō, *angô(“hook, angle”), from Proto-Indo-European *h₂enk-(“something bent, hook”). Cognate with West Frisian angel(“fishing rod, stinger”), Dutch angel(“fishhook”), German Angel(“fishing pole”), German angeln(“to fish, angle”), Icelandic öngull(“fishhook”).
etymonline
angle (v.1)
"to fish with a hook," mid-15c., from Old English angel (n.) "angle, hook, fish-hook," related to anga "hook," from Proto-Germanic *angul-, from PIE *ankulo-, suffixed form of root *ang-/*ank- "to bend" (see angle (n.)). Compare Old English angul, Old Norse öngull, Old High German angul, German Angel "fishhook." Figurative sense "catch or elicit by artful wiles" is recorded from 1580s. Related: Angled; angling.
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.
Angle
member of a Teutonic tribe, Old English, from Latin Angli "the Angles," literally "people of Angul" (Old Norse Öngull), a region in what is now Holstein, said to be so-called for its hook-like shape (see angle (n.)). Or the name might refer to fishing (with hooks) as a main activity of the people, and Proto-Germanic *anguz is said also to have meant "narrow," so it might refer to shallow coastal waters.
People from the tribe there founded the kingdoms of Mercia, Northumbia, and East Anglia in 5c. Britain. Their name, rather than that of the Saxons or Jutes, may have become the common one for the whole group of Germanic tribes because their dialect was the first committed to writing.
angle (v.2)
"to move at an angle, to move diagonally or obliquely," 1741, from angle (n.). Related: Angled; angling.
