Geometry for elementary school/The Side-Angle-Side congruence theorem

Introduction

In this chapter, we will discuss another congruence theorem, this time the Side-Angle-Side theorem. The theorem appears as Based on Book I, prop 4 at the Elements.

The Side-Angle-Side congruence theorem

Given two triangles $\mathrm{△}ABC$ and $\mathrm{△}DEF$ such that their sides are equal, hence:

1. The side $\overline{AB}$ equals $\overline{DE}$.
2. The side $\overline{CA}$ equals $\overline{DF}$.
3. The angle $\mathrm{\angle }CAB$ equals $\mathrm{\angle }FDE$ (These are the angles between the sides).

Then the triangles congruent and their other angles and side are equal too.

Proof

We will use the method of superposition – we will move one triangle to the other one and we will show that they coincide. We won’t use the construction we learned to copy a line or a segment but we will move the triangle as whole.

1. Superpose $\mathrm{△}ABC$ on $\mathrm{△}DEF$ such that A is place on D and $\overline{AB}$ is placed on $\overline{DE}$.
2. It is given that $\overline{AB}$ equals $\overline{DE}$.
3. Hence, B coincides with E.
4. It is given that the angle $\mathrm{\angle }CAB$ equals $\mathrm{\angle }FDE$.
5. Hence, $\overline{CA}$ is placed on $\overline{DF}$.
6. it is given that $\overline{CA}$ equals $\overline{DF}$.
7. Hence, C coincides with F.
8. Therefore, $\overline{CB}$ coincides with $\overline{EF}$.
9. The triangles $\mathrm{△}ABC$ and $\mathrm{△}DEF$ coincide.
10. The triangles $\mathrm{△}ABC$ and $\mathrm{△}DEF$ congruent.