AAA similarity theorem or criterion:
If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and the triangles are similar




In ΔABC and ΔPQR, ∠A = ∠P , ∠B = ∠Q , and ∠C = ∠R  then  AB PQ =  BC QR =  ACPRand ΔABC ∼ ΔPQR.

Given: In ΔABC and ΔPQR, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

To prove:   AB PQ =  BC QR =  ACPR

Construction : Draw LM such that    PL AB = PM AC .

Proof: In ΔABC and ΔPLM,

AB = PL and AC = PM (By Contruction)

∠BAC = ∠LPM (Given)

∴ ΔABC ≅ ΔPLM (SAS congruence rule)

∠B = ∠L (Corresponding angles of congruent triangles)

Hence ∠B = ∠Q (Given)

∴ ∠L =  ∠Q 

LQ is a transversal to LM and QR.

Hence  ∠L =  ∠Q (Proved)

∴ LM ∥ QR

  PL LQ =  PM MR 

  LQ PL =  MR PM   (Taking reciprocals)

  LQ PL + 1 =  MR PM + 1  (Adding 1 to both sides)

  LQ+PL PL =  MR+PM PM 

  PQ PL =  PR PM

  PQ AB =  PR AC   (AB = PL and AC =PM)

  AB PQ =  AC PR  (Taking Reciprocals) ............... (1)

  AB PQ =  BC QR  

  AB PQ =  AC PR =   BC QR 

∴ ΔABC ~ ΔPQR

AAA Criteria in Triangles:

Introduction:
In geometry, AAA (Angle-Angle-Angle) is a criteria used to determine if two triangles are congruent. Congruent triangles have exactly the same shape and size. The AAA criteria states that if the three angles of one triangle are congruent to the three angles of another triangle, then the triangles are congruent.

Explanation of AAA Criteria:
The AAA criteria is based on the fact that the sum of the angles in a triangle is always 180 degrees. Therefore, if two triangles have the same three angles, then the measures of their corresponding angles are equal. This implies that the corresponding sides of the triangles are also proportional, which leads to congruence.

Proof of AAA Criteria:
To prove the AAA criteria, we can use the following steps:

1. Let's consider two triangles, ΔABC and ΔDEF, where ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.
2. We need to show that the corresponding sides of the triangles are proportional.
3. From the given information, we have:
∠A = ∠D (Given)
∠B = ∠E (Given)
∠C = ∠F (Given)
4. Since the sum of the angles in a triangle is always 180 degrees, we have:
∠A + ∠B + ∠C = 180 degrees (Sum of angles in ΔABC)
∠D + ∠E + ∠F = 180 degrees (Sum of angles in ΔDEF)
5. By substituting the corresponding angles from step 3 into the above equations, we get:
∠D + ∠E + ∠F = 180 degrees (Sum of angles in ΔABC)
∠D + ∠E + ∠F = 180 degrees (Sum of angles in ΔDEF)
6. Since both equations are equal, we can conclude that the corresponding sides of the triangles are proportional.
7. Therefore, ΔABC and ΔDEF are congruent by the AAA criteria.

Conclusion:
The AAA criteria states that if the three angles of one triangle are congruent to the three angles of another triangle, then the triangles are congruent. This criteria is based on the fact that the sum of angles in a triangle is always 180 degrees. By showing that the corresponding sides of the triangles are proportional, we can prove the congruence using the AAA criteria.