Geometry for Elementary School/Copying an angle

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

The construction

1. [[../Our_tools:_Ruler_and_compasses# how to draw a line?| Draw a line]] between A and C and get $\mathrm{△}ABC$.
2. [[../Copying a triangle|Copy the triangle]] $\mathrm{△}ABC$ and get $\mathrm{△}CDE$

Claim

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

Proof

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

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