Question

Is A an obtuse angle?
1. A is more than 90°.
2. A is a supplement of an angle B, an acute triangle.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

In Data Sufficiency, be prepared to interpret minor grammatical errors or typos logically. The phrase "an acute triangle" was likely meant to describe angle B as "an acute angle." Base your reasoning on the most plausible meaning.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question is a "Yes/No" question. We need to determine if angle A is obtuse. An obtuse angle is an angle that measures greater than 90° and less than 180°. So, the question is "Is \( 90° < A < 180° \) ?".

Step 2: Key Formula or Approach:
- Definition of an obtuse angle: An angle \( \theta \) is obtuse if \( 90° < \theta < 180° \).
- Definition of supplementary angles: Two angles are supplementary if their sum is 180°.
- Definition of an acute angle: An angle \( \phi \) is acute if \( 0° < \phi < 90° \).

Step 3: Detailed Explanation:
Analyze Statement (1): "A is more than 90°."
This statement directly gives the definition of an obtuse angle (assuming A is an angle within a geometric figure, thus \( < 180° \)). It definitively answers the question with "Yes". Therefore, Statement (1) is sufficient.
Analyze Statement (2): "A is a supplement of an angle B, an acute triangle."
This statement seems to contain a typo and likely means "A is a supplement of angle B, and angle B is an acute angle." Let's proceed with this logical interpretation.
Supplementary angles sum to 180°, so:
\[ A + B = 180° \] We are told that B is an acute angle, which means:
\[ 0° < B < 90° \] We can express A in terms of B:
\[ A = 180° - B \] Since B is less than 90°, A must be greater than 180° - 90°.
\[ A > 180° - 90° \implies A > 90° \] Since B is greater than 0°, A must be less than 180° - 0°.
\[ A < 180° \] So, we have \( 90° < A < 180° \), which means A is definitively an obtuse angle. This statement answers the question with a "Yes". Therefore, Statement (2) is sufficient.

Step 4: Final Answer:
Since each statement alone is sufficient to definitively answer the question, the correct option is (D).