Question

Is the triangle \(ABC\) equilateral? Statements: (I) \(\angle A\) is an acute angle. (II) \(\angle B, \angle C\) are acute angles and \(\angle B > \angle C\).

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

To disprove an equilateral triangle, it is enough to show that any two angles are unequal.

Answer 1

The Correct Option is B

Concept: In an equilateral triangle: \[ \angle A=\angle B=\angle C=60^\circ \] So all three angles must be equal. If any two angles are unequal, the triangle cannot be equilateral.

Step 1: Checking Statement (I).
Given: \[ \angle A \text{ is acute} \] This means: \[ \angle A<90^\circ \] But an acute angle can be many values like: \[ 30^\circ,45^\circ,60^\circ,80^\circ \] It does not guarantee: \[ \angle A=60^\circ \] So Statement (I) alone is not sufficient.

Step 2: Checking Statement (II).
Given: \[ \angle B,\angle C \text{ are acute} \] and \[ \angle B>\angle C \] For an equilateral triangle: \[ \angle B=\angle C \] But here: \[ \angle B\neq\angle C \] Thus, the triangle cannot be equilateral. The answer is definitely No. So Statement (II) alone is sufficient. Hence, the correct answer is (B).