Problem. On a given right line (AB) to describe a segment of a circle which shall contain an angle equal to a given rectilineal angle (X).
Sol.—If X be a right angle, describe a semicircle on the given line, and the thing required is done; for the angle in a semicircle is a right angle.
If not, make with the given line AB the angle BAE equal to X. Erect AC at right angles to AE, and BC at right angles to AB. On AC as diameter describe a circle: it will be the circle required.
Dem.—The circle on AC as diameter passes through B, since the angle ABC is right [xxxi.] and touches AE, since the angle CAE is right [xvi.]. Therefore the angle BAE [xxxii.] is equal to the angle in the alternate segment; but the angle BAE is equal to the angle X (const.). Therefore the angle X is equal to the angle in the segment described on AB.