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.

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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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