A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: In a triangle ABC, $\angle A = \angle B-\angle C$. Is angle A acute? Statement-I: ABC is not an obtuse-angled triangle. Statement-II: Angle C is acute. Which one of the following is correct in respect of the above Question and the Statements?

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Question

CDS - 2025

A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
In a triangle ABC, $\angle A = \angle B-\angle C$. Is angle A acute?
Statement-I: ABC is not an obtuse-angled triangle.
Statement-II: Angle C is acute.

Which one of the following is correct in respect of the above Question and the Statements?

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CDS 2 2025 Maths Question Paper (14-Sep-2025)

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The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

The Question can be answered by using either Statement alone

The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

The Question can be answered even without using any of the Statements

The correct answer is

The Question can be answered even without using any of the Statements

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.

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