# 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 $$CD$$.

Proofs: 1

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

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### References

#### Adapted from CC BY-SA 3.0 Sources:

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