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In ΔABC, ∠A = 100°, ∠B = 50°. Arrange the sides in increasing order of length.

Class 9Triangles (Congruence)Geometry — Triangles
  • AAB < BC < AC
  • BBC < AB < AC
  • CAC < AB < BC
  • DAB < AC < BC

Step-by-Step Solution

∠C = 180 − 100 − 50 = 30°. Angles in order: ∠C(30°) < ∠B(50°) < ∠A(100°). The side opposite each angle: AB is opposite ∠C (smallest angle → shortest side), AC is opposite ∠B, and BC is opposite ∠A (largest angle → longest side). So the increasing order of sides is AB < AC < BC.
Answer: D) AB < AC < BC

Key idea: The four valid congruence criteria

**SSS**: All three sides equal → congruent.

**SAS**: Two sides + included angle equal → congruent.
IMPORTANT: the angle must be BETWEEN the two given sides. SSA (two sides + non-included angle) is NOT a valid criterion in general.

**ASA**: Two angles + included side equal → congruent.

**AAS**: Two angles + a non-included side equal → congruent (since the third angle is forced by 180°).

**RHS** (right triangles only): Right angle + hypotenuse + one side equal → congruent.

**Once congruent, all corresponding parts are equal (CPCT)** — that's the lemma you cite to extract any matching side or angle in the next step of a proof.

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