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 ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle DEF}$ such that their sides are equal, hence:

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