Geometry for Elementary School/Copying a triangle

In this chapter, we will show how to copy a triangle $\mathrm{△}ABC$ to other triangle $\mathrm{△}CDE$ . The construction is an excellent example of the reduction technique – solving a problem by solution to a previously solved problem.

The construction

1. [[../Constructing a triangle|Construct a triangle]] from the sides of $\mathrm{△}ABC$ : $\overline{AB}$, $\overline{AC}$, $\overline{BC}$ and get $\mathrm{△}CDE$.

Claim

The triangles $\mathrm{△}ABC$ and $\mathrm{△}CDE$ are congruent.

Proof

1. $\overline{AB}$, $\overline{AC}$, $\overline{BC}$ are sides of the triangle $\mathrm{△}ABC$ and therefore obey the triangle inequality.
2. Therefore one can build a triangle whose sides equal these segments.
3. The sides of the triangle $\mathrm{△}ABC$ and $\mathrm{△}CDE$ are equal.
4. Due to [[../The Side-Side-Side congruence theorem/]] the triangles $\mathrm{△}ABC$ and $\mathrm{△}CDE$ congruence.