Proposition 16

Theorem

Alternate name(s): exterior angle theorem.

If any side (BC ) of a triangle (△ABC) is extended, then the exterior angle (∠ACD) is bigger than either of the non-adjacent interior angles.

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Commentary

Let △ABC be a given triangle. Extend side BC  to D such that ∠ACD is an exterior angle of the triangle.

Then ∠ACD is greater than either of the two interior angles ∠BAC and ∠ABC.

To avoid confusion, we used the term "non-adjacent interior angles" in the statement, while Euclid used the term "interior and opposite angles", which is short for "the interior angles at the ends of the line segment opposite the angle whose vertex is chosen for extending the side".

This proposition is also known as the Exterior Angle Theorem.

One can deduce the next proposition, Book 1 Proposition 17, from this proposition. A later proposition, Book 1 Proposition 32, is a stronger version of this proposition.

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Computable version

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GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

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Additional instances

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