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.
- For let two straight lines $BC$ and $BD$, not lying on the same side, make adjacent angles $ABC$ and $ABD$ (whose sum is) equal to two right angles with some straight line $AB$, at the point $B$ on it.
- I say that $BD$ is straight-on with respect to $CB$.
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 \(CD \).
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7
Propositions: 8
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References
Adapted from CC BY-SA 3.0 Sources:
- Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
- Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
