proofs : SSS similarity

# SSS Similarity

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SSS similarity : If the corresponding sides of two triangles are proportional, then the two triangles are similar.

Given : Two triangles ABC and DEF such that
 AB BC CA ---- = ------ = ------ DE EF FD

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        AC     ---- = ------       DE      DF 1) Given 2)   DP       DQ    ---- = ------       DE       DF 2) As AB = DP and AC = DQ. By substitution 3) PQ || EF 3) By converse of basic proportionality theorem 4) ∠DPQ = ∠E and ∠DQP = ∠F 4) Corresponding angles 5) ΔDPQ ~ ΔDEF 5) By AA similarity 6) DP PQ ---- = ------ DE EF 6) By definition of similar triangles 7) AB PQ ---- = ------ DE EF 7) As DP = AB , by substitution 8)  PQ      BC    ---- = ------     EF     EF 8) From (1) (6) and (7) 9) PQ = BC 9) From (8) 10) ΔABC ≅ ΔDPQ 10) By S-S-S postulate 11) ΔABC ~ ΔDEF 11) From (5) and (10)

Examples Statements Reasons 1) AB = 4; AC = 2;CB = 6; DE = 2; DF = 1; FE = 3 1) Given 2) AB BC CA ---- = ------ = ------ DE EF FD 2) Property of proportion 3) ΔABC ~ ΔDEF 3) By S-S-S-similarity

Practice

1) In ΔPQR ~ ΔXYZ, PQ = 5cm, QR = 4cm and PR= 6m ,find XY, YZ and XZ.
2) In ΔABC, D and E are any points on AB and AC respectively, such that DE||BC. If AD=x cm, DB=x-2 cm, AE=x-1 cm, then find the value of x.

Criteria for Similarity

AAA Similarity
AA Similarity
SSS Similarity
SAS Similarity
Practice on Similarity

From SSS Similarity to Criteria for Similarity of Triangles

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