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Proposition: 3.31: Relative Sizes of Angles in Segments

(Proposition 31 from Book 3 of Euclid's “Elements”)

In a circle, the angle in a semicircle is a right angle, and that in a greater segment (is) less than a right angle, and that in a lesser segment (is) greater than a right angle. And, further, the angle of a segment greater (than a semicircle) is greater than a right angle, and the angle of a segment less (than a semicircle) is less than a right angle.

• Let $ABCD$ be a circle, and let $BC$ be its diameter, and $E$ its center.
• And let $BA$, $AC$, $AD$, and $DC$ have been joined.
• I say that the angle $BAC$ in the semicircle $BAC$ is a right angle, and the angle $ABC$ in the segment $ABC$, (which is) greater than a semicircle, is less than a right angle, and the angle $ADC$ in the segment $ADC$, (which is) less than a semicircle, is greater than a right angle.

Modern Formulation

• An angle inscribed in a segment which is a semicircle, is a right angle ($\angle{BAC}$).
• An angle inscribed in a segment which is smaller than a semicircle, is an obtuse angle ($\angle{ADC}$).
• An angle inscribed in a segment which is greater than a semicircle, is an acute angle ($\angle{CBA}$).

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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References

Adapted from (subject to copyright, with kind permission)

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

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016