Proposition: 1.14: Combining Rays to Straight Lines

Euclid's Formulation

If two straight lines, not lying on the same side, make adjacent angles (whose sum is) equal to two right angles with some straight line, at a point on it, then the two straight lines will be straight-on (with respect) to one another.


Modern Formulation

If at the endpoint of a ray \(\overline BA \) there exist two other rays \(\overline BC \), \(\overline BD \) standing on opposite sides of that ray such that the sum of their adjacent angles is equal to two right angles $\angle{DBA} + \angle{ABC}=2\cdot 90^\circ,$ then these two rays build a straight line \(CD \).

Proofs: 1

Proofs: 1 2 3 4 5 6 7
Propositions: 8

Thank you to the contributors under CC BY-SA 4.0!



Adapted from CC BY-SA 3.0 Sources:

  1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

  1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"