Book 1 – Proposition 23

On a given straight line and a point on it to construct a rectilineal angle equal to a given rectilineal angle.
[Desmos graphs can be found here]

We begin with a line AB (let A be the point on it) and an angle DCE where D and E are selected at random.

The goal is to construct an angle on AB at point A that is equal to angle DCE.

Begin by joining DE. Construct a triangle AFG on AB such that AF, FG, and GA are equal to CD, DE, and EC respectively. (Proposition 22 shows how such a triangle can be constructed with sides equal to those 3 line segments.)

The two sides DC and CE are equal to the two sides FA and AG, and the base DE is equal to the base FG. By Proposition 8 (Two triangles with two sides equal to two sides respectively and equal bases have equal angles between the two sides) angles DCE and FAG are equal.

Thus we have constructed an angle on AB that is equal to the given angle.