# Proposition: 1.28: Parallel Lines II

### (Proposition 28 from Book 1 of Euclid's “Elements”)

If a straight line falling across two straight lines makes the external angle equal to the internal and opposite angle on the same side, or (makes) the (sum of the) internal (angles) on the same side equal to two right angles, then the (two) straight lines will be parallel to one another.

• For let $EF$, falling across the two straight lines $AB$ and $CD$, make the external angle $EGB$ equal to the internal and opposite angle $GHD$, or the (sum of the) internal (angles) on the same side, $BGH$ and $GHD$, equal to two right angles.
• I say that $AB$ is parallel to $CD$.

### Modern Formulation

If a straight line $$EF$$ intersects two straight lines $$AB$$, $$CD$$ at one and only point point each such that the exterior angle $$\angle{BGE}$$ equals its corresponding interior angle $$\angle{DHG}$$ or if it makes two interior angles on the same side $$\angle{HGB}, \angle{DHG}$$ equal to two right angles, then the two lines are parallel $$(AB\parallel CD)$$.

Tha angles $$\angle{BGE}$$ and $$\angle{DHG}$$ are called corresponding angles.

Proofs: 1

Proofs: 1 2 3 4 5 6
Propositions: 7

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### References

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

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