# AAA Similarity

This section explains you the proof on AAA Similarity.

Statement: If in two triangles, the corresponding angles are equal, i.e., if the two triangles are equiangular, then the triangles are similar. Given : Triangles ABC and DEF such that ∠A = ∠D; ∠B = ∠E; ∠C = ∠F

Prove that : Δ ABC ~ ΔDEF

Construction : We mark point P on the line DE and Q on the line DF such that AB = DP and AC = DQ, we join PQ.

There are three cases :

Case ( i ) : AB = DE, thus P coincides with E.
 Statements Reasons 1) AB = DE 1) According to 1st case 2) ∠A = ∠D 2) Given 3) ∠B = ∠E 3) Given 4) ΔABC ≅ ΔDEF 4) By ASA postulate

⇒ AB = DE, BC = EF and AC = DF

Consequently, Q coincides with F.

 AB BC CA ---- = ------ = ------ DE EF FA

Since the corresponding angles are equal, we conclude that Δ ABC ~ Δ DEF.

Case( ii ) : AB < DE. Then P lies in DE

In triangles ABC and DPQ,

 Statements Reasons 1) AB = DP 1) By construction 2) ∠A = ∠D 2) Given 3) AC = DQ 3) By construction 4) ΔABC ≅ ΔDPQ 4) By SAS postulate 5) ∠B = ∠DPQ 5) CPCTC 6) ∠B = ∠E 6) Given 7) ∠E = ∠DPQ 7) By transitive property ( from above) 8) PQ || EF 8) If two corresponding angles are congruent then the lines are parallel 9) DP/DE = DQ/DF 9) By basic proportionality theorem 10) AB/DE = BC/EF 10) By construction 11) AB/DE = AC/DF 11) By substitution property 12) Δ ABC ~ Δ DEF 12) By SAS postulate
Case ( iii ): If AB > DE. Then P lies on DE produced. Proof for this case is same as above case ( ii ).

Criteria for Similarity

AAA Similarity
AA Similarity
SSS Similarity
SAS Similarity
Practice on Similarity

From AAA similarity to Criteria for Similarity of Triangles