Construction of 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 $$CD = AF$$, $$CE = AG$$, and $$DE = FG$$, with $$G$$ 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 $$AB$$.

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.