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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