On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.

Let $AB$ be the given straight line, $A$ the point on it, and the angle $DCE$ the given rectilineal angle;thus it is required to construct on the given straight line $AB$, and at the point $A$ on it, a rectilineal angle equal to the given rectilineal angle $DCE$.On the straight lines $CD$, $CE$ respectively let the points $D$, $E$ be taken at random;let $DE$ be joined,and out of three straight lines which are equal to the three straight lines $CD$, $DE$, $CE$ let the triangle $AFG$ be constructed in such a way that $CD$ is equal to $AF$, $CE$ to $AG$, and further $DE$ to $FG$. [Prop. 1.22]Then, since the two sides $DC$, $CE$ are equal to the two sides $FA$, $AG$ respectively,and the base $DE$ is equal to the base $FG$,the angle $DCE$ is equal to the angle $FAG$. [Prop. 1.8]Therefore on the given straight line $AB$, and at the point $A$ on it, the rectilineal angle $FAG$ has been constructed equal to the given rectilineal angle $DCE$.Q.E.F.