In this chapter, we will show how to copy an angle ${\displaystyle \angle ABC}$ to other angle ${\displaystyle \angle CDE}$. The construction is based on Book I, proposition 23.

The construction

1. Draw a line between A and C and get ${\displaystyle \triangle ABC}$.
2. Copy the triangle ${\displaystyle \triangle ABC}$ and get ${\displaystyle \triangle CDE}$

Claim

The angles ${\displaystyle \angle ABC}$ and ${\displaystyle \angle CDE}$ are equal.

Proof

1. The triangles ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle CDE}$ congruence.
2. Therefore the angles of the triangles equal.
3. Hence, ${\displaystyle \angle ABC}$ and ${\displaystyle \angle CDE}$ are equal.

Note that any two points on the rays can be used to create a triangle.