Analyzing the Triangle Angle Condition

The question asks if angle A (\(\angle A\)) in a triangle ABC is acute, given the specific condition that \(\angle A = \angle B - \angle C\). Let's analyze this condition using the fundamental properties of triangles.

Deriving Angle Relationships

We start with the given relationship:

\(\angle A = \angle B - \angle C\)

Rearranging this equation, we get:

\(\angle B = \angle A + \angle C\)

We also know that the sum of the angles in any triangle is always 180 degrees:

\(\angle A + \angle B + \angle C = 180^\circ\)

Now, we can substitute the expression for \(\angle B\) (from the first step) into the triangle angle sum equation:

\(\angle A + (\angle A + \angle C) + \angle C = 180^\circ\)

Combining like terms:

\(2\angle A + 2\angle C = 180^\circ\)

Dividing the entire equation by 2:

\(\angle A + \angle C = 90^\circ\)

Determining if Angle A is Acute

The equation \(\angle A + \angle C = 90^\circ\) tells us that angles A and C are complementary.

Since A, B, and C are angles of a triangle, they must all be positive values:

• \(\angle A > 0^\circ\)
• \(\angle B > 0^\circ\)
• \(\angle C > 0^\circ\)

From the relationship \(\angle A + \angle C = 90^\circ\):

• Because \(\angle C\) must be greater than \(0^\circ\), \(\angle A\) must be less than \(90^\circ\) (\(\angle A = 90^\circ - \angle C < 90^\circ\)).
• Because \(\angle A\) must be greater than \(0^\circ\), \(\angle C\) must be less than \(90^\circ\) (\(\angle C = 90^\circ - \angle A < 90^\circ\)).

Therefore, based solely on the condition \(\angle A = \angle B - \angle C\), we can conclude that \(\angle A\) must be less than \(90^\circ\). This means \(\angle A\) is always an acute angle.

Additionally, we can determine \(\angle B\) using \(\angle B = \angle A + \angle C\). Since \(\angle A + \angle C = 90^\circ\), it means \(\angle B = 90^\circ\). So, the condition forces the triangle to be a right-angled triangle with the right angle at vertex B.

Evaluating the Statements

The question "Is angle A acute?" can be definitively answered as "Yes" using only the initial condition \(\angle A = \angle B - \angle C\). Let's see what the statements add:

• Statement-I: ABC is not an obtuse-angled triangle. This means all angles are \(\le 90^\circ\). As we found, the condition implies \(\angle B = 90^\circ\) and \(\angle A < 90^\circ\), \(\angle C < 90^\circ\). This statement is consistent but doesn't provide new essential information to determine if \(\angle A\) is acute, as we already know it is.
• Statement-II: Angle C is acute. This means \(0^\circ < \angle C < 90^\circ\). As shown above, from \(\angle A + \angle C = 90^\circ\), if \(\angle C\) is acute, then \(\angle A\) must also be acute (\(0^\circ < \angle A < 90^\circ\)). This statement is also consistent but doesn't provide information not already derivable from the initial condition.

Conclusion

Since the initial condition \(\angle A = \angle B - \angle C\) is sufficient on its own to prove that \(\angle A\) is always acute, neither Statement I nor Statement II is required to answer the question. The question can be answered even without using any of the Statements.