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In ΔABC and ΔPQR, ∠B = ∠Q, AB = 3 cm, BC = 4.5 cm, PQ = 5 cm and QR = 7.5 cm. Which criterion proves ΔABC ~ ΔPQR?

Class 10Triangles (Similarity)Geometry — Similarity
  • AAA, because one pair of angles is equal
  • BSSS, because all three pairs of sides are proportional
  • CSAS, because two pairs of sides are proportional and the included angles are equal
  • DNo criterion applies — this is the SSA pattern

Step-by-Step Solution

and , and ∠ lies between and while ∠ lies between and . Two proportional sides with the INCLUDED angle equal is exactly SAS similarity. would need a second pair of equal angles; SSS would need the third pair of sides; and the angle here IS included, so it is not the SSA pattern.
Answer: C) SAS, because two pairs of sides are proportional and the included angles are equal

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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