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Geometry Characteristic Properties of Similarity
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Characteristic Properties of Similarity
Two triangles are said to be similar if
    (i) their corresponding angles are equal and
    (ii) their corresponding sides are proportional .

Thus two triangles ABC and DEF are similar if
    (i) A = D, B = E, C = F and,
    (ii) AB/DE = BC/EF = CA/FA
Equiangular Triangle: Two triangles are said to be equiangular, if their corresponding angles are equal.

Characteristic Property 1:
AAA Similarity: if two triangles are equiangular, then triangles are similar.
Characteristic Property 2:
SSS Similarity: if the corresponding sides of two triangles are proportional, then they are similar.
Characteristic Property 3:
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.
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Example: In fig , AO/OC = BO/OD = 1/3 and AB = 10 cm. find the value of DC.
Solution: in AOB and COD, we have 
 AOB = COD [vertically opposite angles]
  AO / OC = OB / OD (given)
  So by SAS - Criterion of similarity
  AO/OC = BO/OD = AB/ DC
  1 / 3 = 10 / DC
  DC = 30 cm
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