SAS Similarity
SAS Similarity : If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar.Given : Two triangles ABC and DEF such that ∠A = ∠D
AB AC  =  DE DF 
Prove that : ΔABC ~ ΔDEF
Construction : Let P and Q be two points on DE and DF respectively such that DP = AB and DQ = AC. Join PQ.


1) AB = DP ; ∠A = ∠D and AC = DQ  1) Given and by construction 
2) ΔABC ≅ ΔDPQ  2) By SAS postulate 
3) AB AC  =  DE DF 
3) Given 
4) DP DQ  =  DE DF 
4) By substitution 
5) PQ  EF  5) By converse of basic proportionality theorem 
6) ∠DPQ = ∠E and ∠DQP = ∠F  6) Corresponding angles 
7) ΔDPQ ~ ΔDEF  7) By AAA similarity 
8) ΔABC ~ ΔDEF  8) From (2) and (7) 
Examples
1) In the given figure, if QT / PR = QR / QS and ∠1 = ∠2.
Prove that ΔPQS ~ ΔTQR.
Given : QT / PR = QR / QS and ∠1 = ∠2
Prove that : ΔPQS ~ ΔTQR.


1) QT / PR = QR / QS  1) Given 
2) QT / QR = PR / QS  2) By alternendo 
3) ∠1 = ∠2  3) Given 
4) PR = PQ  4) Side opposite to equal angles are equal. 
5) QT / QR = PQ / QS  5) By substitution from (2) 
6) PQ / QT = QS / QR  6) By alternendo 
7) ∠PQS = ∠TQR  7) Reflexive (common) in Δ PQS and ΔTQR 
8) ΔPQS ~ Δ TQR  8) By SAS postulate 
Criteria for Similarity
• AAA Similarity
• AA Similarity
• SSS Similarity
• SAS Similarity
• Practice on Similarity
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