Angle – Angle – Similarity

If two angles of a triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

Given: Two △ABC and △PQR such that ∠B = ∠Q,    and   ∠C = ∠R 

To prove: △ABC ~ △PQR 

Proof: In △ABC, ∠A + ∠B +∠C =  180⁰….(1) by angle sum property

In △PQR∠P + ∠Q + ∠R =  180⁰….(2) by angle sum property

From (1) and (2)

           ∠A + ∠B + ∠C = ∠P + ∠Q + ∠R

But ∠B = ∠Q  and  ∠C = ∠R (Given)

                           ∴ ∠A = ∠P

Thus, △ABC ~ △PQR 
                       

 ∴ ∠A = ∠P (From 3)                                  

∴ ∠B = ∠Q                            

 and  ∠C = ∠R              

∴ △ABC ~ △PQR (AA similarity criteria)

                                           Hence Proved.