Angle Angle Side
Angle Angle Side or AAS postulate refers to two angles and one side of two triangles to prove its congruency. The AAS is one of the 5 congruency theorems that states that if two angles along with a nonincluded side are equal to the corresponding angles and nonincluded side of another triangle, the two triangles are considered to be congruent. The 5 congruence rules include SSS, SAS, ASA, AAS, and RHS. Let us learn more about the angle angle side theorem and solve a few examples.
1.  Angle Angle Side Definition 
2.  AAS Criterion for Congruence 
3.  Angle Angle Side Congruence Theorem 
4.  Proof of AAS Congruence Theorem 
5.  FAQs on Angle Angle Side 
Angle Angle Side Definition
By definition, angle angle side is a congruence theorem where it involves two angles and a nonincluded side. Hence, the theorem states that if any two angles and the nonincluded side of one triangle are equal to the corresponding angles and the nonincluded side of the other triangle. The angles are consecutive in nature and the sides are not included between the angles but in either direction of the angles. Look at the image below, we can see the two consecutive or next to each other angles of one triangle are equal to corresponding angles of another triangle. The sides of both the triangles are not included between the angles but are consecutive to the angles, hence the sides are also equal.
AAS Criterion for Congruence
AAS Criterion stands for AngleAngleSide Criterion. Under the AAS criterion, two triangles are congruent if any two angles and the nonincluded side of one triangle are equal to the corresponding angles and the nonincluded side of the other triangle.
If Δ ABC ≅ ΔXYZ under AAS criterion, then the third angle (∠ABC) and the other two sides (AC and BC) of Δ ABC is bound to be equal to the corresponding angle (∠XYZ) and the sides (XZ and YZ) of ΔXYZ.
Angle Angle Side Congruence Theorem
AngleAngleSide congruence theorem states that if two angles of a triangle with a nonincluded side are equal to the corresponding angles and nonincluded side of the other triangle, they are considered to be congruent. Let us see the proof of the theorem:
Given: AB = DE, ∠B=∠E, and ∠C =∠F. To prove: ∆ABC ≅ ∆DEF
If both the triangles are superimposed on each other, we see that ∠B =∠E and ∠C =∠F. And the nonincluded sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.
Proof of AAS Congruence Theorem
To prove the AAS congruence theorem, we need to first look at the ASA congruence theorem which states that when two angles and the included side (the side between the two angles) of one triangle are (correspondingly) equal to two angles and the included side of another triangle.
The AAS congruence theorem states that if any two consecutive angles of a triangle along with a nonincluded side are equal to the corresponding consecutive angles and the nonincluded side of another triangle, the two triangles are said to be congruent. We should also remember that if two angles of a triangle are equal to two angles of another, then their third angles are automatically equal since the sum of angles in any triangle must be a constant 180° (by the angle sum property).
To prove the AAS congruence rule, let us consider the two triangles above ∆ABC and ∆DEF. We know that AB = DE, ∠B =∠E, and ∠C =∠F. We also saw if two angles of two triangles are equal then the third angle of both the triangle is equal since the sum of angles is a constant of 180°. Hence,
In ∆ABC, ∠A + ∠B + ∠C = 180  (i)
In ∆DEF, ∠D + ∠E + ∠F = 180 (ii)
From (i) and (ii) we get,
∠A + ∠B + ∠C = ∠D + ∠E + ∠F
Since we already know that ∠B =∠E and ∠C =∠F, so
∠A + ∠E + ∠F = ∠D + ∠E + ∠F
∠A = ∠D
In both the triangles we know that,
AB = DE, ∠A = ∠D, and ∠C =∠F
Therefore, according to the ASA congruence rule, it is proved that ∆ABC ≅ ∆DEF.
Examples on Angle Angle Side

Example 1: From the below image, which triangle follows the AAS congruence rule?
Solution:
From the abovegiven pairs, we can see that pair number 4 fits the AAS congruence rule where two consecutive angles with a nonincluded angle of one triangle are equal to the corresponding consecutive angles with a nonincluded side of another triangle, then the triangles are considered to be congruent. The pairs are of the other congruence rules such as,
Pair 1 = SSS Congruency Rule
Pair 2 = SAS Congruency Rule
Pair 3 = ASA Congruency Rule

Example 2: From the below triangle, we know that ∠Q = ∠R along with right angles on both sides of the triangle. Can we prove that ∆PQS ≅ ∆PRS?
Solution: Given,
∠Q = ∠R and ∠PSQ = ∠PSR = 90°
Since both the triangles share the same perpendicular line making the length of the line the same for both triangles. Hence, the sides of both triangles are also equal. According to the AAS congruence rule, we can say that ∆PQS ≅ ∆PRS.
FAQs on Angle Angle Side
What is Angle Angle Side?
The Angle Angle Side Postulate (AAS) states that if two consecutive angles along with a nonincluded side of one triangle are congruent to the corresponding two consecutive angles and the nonincluded side of another triangle, then the two triangles are congruent.
How Do You Tell if a Triangle is ASA or AAS?
Both the triangle congruence theorems deal with angles and sides but the difference between the two is ASA deals with two angles with a side included in between the angles of any two triangles. Whereas AAS deals with two angles with a side that is not included in between the two angles of any two given triangles.
What is SSS, SAS, ASA, and AAS?
The 4 different triangle congruence rules are:
 SSS: Where three sides of two triangles are equal to each other.
 SAS: Where two sides and an angle included in between the sides of two triangles are equal to each other.
 ASA: Where two angles along with a side included in between the angles of any two triangles are equal to each other.
 AAS: Where two angles of any two triangles along with a side that is not included in between the angles, are equal to each other.
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