# 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

Proofs: 1

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

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### References

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