# Proposition: 1.13: Angles at Intersections of Straight Lines

### (Proposition 13 from Book 1 of Euclid's “Elements”)

If a straight line stood on a(nother) straight line makes angles, it will certainly either make two right angles, or angles whose sum is) equal to two right angles. * For let some straight line $AB$ stood on the straight line $CD$ make the angles $CBA$ and $ABD$. * I say that the angles $CBA$ and $ABD$ are certainly either two right angles, or (have a sum) equal to two right angles.

### Modern Formulation

If the straight line $$AB$$ intersects the straight line $$CD$$ at one and only one point ($$B$$), then either $$\angle{ABC}$$ and $$\angle{DBA}$$ are right angles or the sum $$\angle{ABC}+\angle{DBA}$$ equals the sum of two right angles.

Proofs: 1 Corollaries: 1 2

Proofs: 1 2 3 4 5 6 7 8 9 10 11

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### References

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

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