One angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite sides in the same ratio. Prove that the triangles are similar.
                                

Given: ΔABC and ΔPQR 
      ∠ A = ∠ P 
AD and PS bisects ∠A and ∠P respectively.
      BD = QS
      DC    SR 
To prove: ΔABC ~ ΔPQR
Proof: In ΔABC and ΔPQR 
AD bisects ∠A
∴ AB =  BD (Angle bisector theorem) -----(1)
   AC     DC 
Similarly in ΔPQR,
PQ =  QS (Angle bisector theorem) ----(2)
PR     SR 
But BD = QS (given) 
     DC    SR 
∴ According to equation (1) and (2) 
     AB = PQ => AB = AC
     AC    PR     PQ    PR 
∠A = ∠P (given)
∴ ΔABC ~ ΔPQR (SAS similarity).