Geometry for Elementary School/The Side-Angle-Side congruence theorem

In this chapter, we will discuss another congruence theorem, this time the Side-Angle-Side theorem. The angle is called the included angle.

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 are congruent and their other angles and sides are equal too. Success!

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 learnt to copy a line or a segment but we will move the triangle as whole.

1. $\text{Superpose }\mathrm{△}ABC\text{ on }\mathrm{△}DEF\text{ such that }A\text{ is placed on }D\text{ and }\overline{AB}\text{ is placed on }\overline{DE}$
2. $\because \overline{AB}=\overline{DE}\text{ (given)}$
3. $\therefore B\text{ coincides with }E$
4. $\because \mathrm{\angle }CAB=\mathrm{\angle }FDE\text{ (given)}$
5. $\therefore \overline{CA}=\overline{FD}$
6. $\because \overline{CA}=\overline{DF}\text{ (given)}$
7. $\therefore C\text{ coincides with }F$
8. $\therefore \overline{CB}=\overline{EF}$
9. $\therefore \mathrm{△}ABC=\mathrm{△}DEF$
10. $\therefore \mathrm{△}ABC\cong \mathrm{△}DEF$