In this chapter, we will discuss another congruence theorem, this time the Angle-Side-Angle theorem, usually shortened as ASA. The side in this theorem is called the included side.

The Angle-Side-Angle congruence theorem

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

1. The angle ${\displaystyle \angle BAC}$ equals ${\displaystyle \angle EDF}$.
2. The side ${\displaystyle {\overline {BA}}}$ equals ${\displaystyle {\overline {ED}}}$.
3. The angle ${\displaystyle \angle ABC}$ equals ${\displaystyle \angle DEF}$ (These are the angles between the sides, or the included angle).

Then the triangles are congruent and their other angles and sides are equal too. Note that the side must be the included side of the two angles.