On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle.
Proof
Let AB be the given line and θ the given angle. At A,
construct ∠BAD=θ (I.23). At A, draw AE⊥AD (I.11). At the midpoint F of AB (I.10), draw
FG⊥AB (I.11), meeting AE at G. With G as centre and
GA as radius (= GB by the perpendicular bisector property),
describe a circle. By III.32, the inscribed angle in the alternate
segment to AD on this circle equals θ.