Proof of AAA similarity criteria

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