Question

In triangle \(ABC\), is \(\angle A\) obtuse? Statement (I): \(\angle C\) is acute. Statement (II): \(\angle B\) is obtuse.

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

A triangle can never contain two obtuse angles because the angle sum is only \(180^\circ\).

Answer 1

The Correct Option is B

Concept: The sum of interior angles of a triangle is \[ 180^\circ. \] A triangle can contain at most one obtuse angle.

Step 1: Analyze Statement (I). Given: \[ \angle C \text{ is acute}. \] This tells us only that \[ \angle C<90^\circ. \] Both \(\angle A\) and \(\angle B\) may still be acute or one of them may be obtuse. Hence Statement (I) is insufficient.

Step 2: Analyze Statement (II). Given: \[ \angle B>90^\circ. \] A triangle can have only one obtuse angle. Therefore, \[ \angle A<90^\circ. \] Hence \(\angle A\) cannot be obtuse. The answer to the question is definitively "No". Statement (II) alone is sufficient.