# SAS Similarity

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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. Statements Reasons 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. Statements Reasons 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

From SAS Similarity to Criteria for Similarity of Triangles