The Angle Angle Side postulate (often abbreviated as AAS) states that if two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent.
Example of Angle Angle Side Proof (AAS)
These two triangles are congruent because 2 angles and the non included side are congruent.
$
\begin{aligned}
\angle A \cong \angle X \red{\text{ (A)}}
\\
\angle C \cong \angle Z \red{\text{ (A)}}
\\
\overline{AB} \cong \overline {XY} \red{\text{ (S)}}
\end{aligned}
$
- Therefore, by the Angle Angle Side postulate (AAS), the triangles are congruent.
Why it Makes sense
Consider the two partially drawn triangles below. At the start of the animation you can see that both triangles have a congruent side that is included between two congruent angles. In both triangles, we are locked into those congruent pieces. There is only 1 way to complete these triangles, and in both cases the resultant triangle must have the same measurements, which the demonstration below shows.
Identify Angle Angle Side relationship
Identify which pair of triangles below does NOT illustrate an angle angle side (AAS) relationship.
Practice Proofs
Proof 1
Prove that $$ \triangle ABC \cong \triangle DEC $$.
Proof 2
Prove that $$ \triangle ABC \cong \triangle DEF $$.
Can you identify the error in the AAS proof below?
The error involves the side needed to prove two triangles congruent by the Angle Angle Side Postulate.
FE and BC are NOT full sides in either triangle. You would need to use the additon property of equality to add segment BE to FE and BE to be able to state that their is a pair of congruent sides in the two triangles.
