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In ΔLMN, LM = 20 cm, MN = 21 cm and NL = 29 cm. A student says the right angle is at N, because N is a letter of the longest side NL. Which angle is actually the right angle?

Class 10Triangles (Similarity)Geometry — Similarity
  • A∠L
  • B∠N
  • CThere is no right angle
  • D∠M

Step-by-Step Solution

, so by the converse of Pythagoras the triangle is right-angled, with the right angle OPPOSITE the longest side NL. The vertex not lying on NL is , so ∠°. The right angle is never at a vertex of the hypotenuse — that is exactly the student’s mistake.
Answer: D) ∠M

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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