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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 \(\beta=\angle{BAC}\).

Table of Contents

Proofs: 1 Corollaries: 1

Mentioned in:

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

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

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References

Adapted from CC BY-SA 3.0 Sources:

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

Adapted from (Public Domain)

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

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"