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In ΔABC, AB = 4.5 cm, BC = 6 cm and CA = 7.5 cm. In ΔDEF, DE = 6 cm, EF = 8 cm and FD = 10 cm. Which criterion establishes their similarity, and in what order are the vertices matched?

Class 10Triangles (Similarity)Geometry — Similarity
  • ASAS, ΔABC ~ ΔDEF
  • BSSS, ΔABC ~ ΔDEF
  • CSSS, ΔABC ~ ΔEDF
  • DAA, ΔABC ~ ΔDEF

Step-by-Step Solution

Only sides are given, so test all three ratios: . All equal, so the triangles are similar by SSS, and the sides were paired AB–DE, BC–EF, CA–FD, giving ΔABC ~ ΔDEF. SAS needs an angle, which is not given; AA needs angles too; "ΔABC ~ ΔEDF" swaps D and E and so breaks the correspondence.
Answer: B) SSS, ΔABC ~ ΔDEF

Key idea: Three similarity criteria

**AA (Angle-Angle)**: Two angles equal → similar. (Third angle is automatic by 180° angle sum.)

**SSS (Side-Side-Side proportionality)**: All three pairs of corresponding sides in same ratio → similar.

**SAS (Side-Angle-Side)**: Two pairs of sides proportional AND included angles equal → similar.

**Important**: SSA does NOT prove similarity (just like for congruence).

AA is by far the most-used in problems: spot two equal angles (often via parallel lines or shared angle) and declare similarity.

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