# Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles

### (Proposition 16 from Book 1 of Euclid's “Elements”)

For any triangle, when one of the sides is produced, the external angle is greater than each of the internal and opposite angles. * Let $ABC$ be a triangle, and let one of its sides $BC$ have been produced to $D$. * I say that the external angle $ACD$ is greater than each of the internal and opposite angles, $CBA$ and $BAC$. ### Modern Formulation

Construct $$\triangle{ABC}$$ and extend any of its sides, e.g. $$\overline{BC}$$, to the segment $$\overline{CD}$$. Then the exterior angle $$\gamma=\angle{DCA}$$ is greater than either of the interior non-adjacent angles $$\alpha=\angle{CBA}$$ and $$\beta=\angle{BAC}$$.

Proofs: 1 Corollaries: 1

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9

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### References

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

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