Proposition: 1.29: Parallel Lines III

(Proposition 29 from Book 1 of Euclid's “Elements”)

A straight line falling across parallel straight lines makes the alternate angles equal to one another, the external (angle) equal to the internal and opposite (angle), and the (sum of the) internal (angles) on the same side equal to two right angles.

• For let the straight line $EF$ fall across the parallel straight lines $AB$ and $CD$.
• I say that it makes the alternate angles, $AGH$ and $GHD$, equal, the external angle $EGB$ equal to the internal and opposite (angle) $GHD$, and the (sum of the) internal (angles) on the same side, $BGH$ and $GHD$, equal to two right angles. .

Modern Formulation

If a straight line $$EF$$ intersects two parallel straight lines $$(AB\parallel CD)$$ at one and only one point each then:

• alternate angles $$\angle{AGH}, \angle{DHG}$$ are equal1;
• the exterior angle $$\angle{BGE}$$ equals its corresponding interior angle $$\angle{DHG}$$2;
• the sum of the two interior angles $$\angle{DHG} + \angle{HGB}$$ equals two right angles3.

Proofs: 1 Corollaries: 1

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

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non-Github:
@Calahan
@Casey
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References

Adapted from CC BY-SA 3.0 Sources:

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

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