Construction of an Equal Angle

Theorem

At a given point on a given straight line, it is possible to construct a rectilinear angle equal to a given rectilinear angle.

Construction

Let $A$ be the given point on the given straight line $AB$, and let $\angle DCE$ be the given rectilinear angle, in which points $D$ and $E$ are any points on the straight lines bounding the angle (one on each side).

We can then construct $\triangle AFG$ such that $CE = AF$, $CD = AG$, and $DE = GF$, with $F$ on $AB$.

$\angle GAF$ is the required angle.

Proof

Since all three sides of the triangles are equal, the interior angles of the triangles are also equal.

Thus, $\angle GAF = \angle ECD$, with $\angle GAF$ at the point $A$ on the straight line $AB$.

$\blacksquare$

Historical Note

This is Proposition 23 of Book I of Euclid's The Elements.

The extremely careful reader will note that Proposition 22: Construction of Triangle from Given Lengths does not directly create the necessary triangle at point $A$, but rather at a distance $CE$ from point $A$. However, a slight modification of the construction produces the desired triangle at the desired location.