AAS or Angle Angle Side congruence rule states that if two pairs of corresponding angles along with the opposite or non-included sides are equal to each other then the two triangles are said to be congruent. The 5 congruence rules include SSS, SAS, ASA, AAS, and RHS. Let us learn more about the AAS congruence rule, the proof, and solve a few examples.
| 1. | What is AAS Congruence Rule? |
| 2. | AAS Congruence Rule |
| 3. | Proof of AAS Congruence Rule |
| 4. | FAQs on AAS Congruence Rule |
By definition, AAS congruence rule states that if any two angles and the non-included side of one triangle are equal to the corresponding angles and the non-included side of the other triangle. The angles are consecutive and corresponding in nature while 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 congruence rule or theorem states that if two angles of a triangle with a non-included side are equal to the corresponding angles and non-included 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 non-included sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.
To prove the AAS congruence rule or 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 rule states that if any two consecutive angles of a triangle along with a non-included side are equal to the corresponding consecutive angles and the non-included 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
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.
Listed below are a few topics related to the AAS congruence rule, take a look.
Example 1: From the below image, which triangle follows the AAS congruence rule? Solution: From the above-given pairs, we can see that pair number 4 fits the AAS congruence rule where two consecutive angles with a non-included angle of one triangle are equal to the corresponding consecutive angles with a non-included 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.
