Question

Determine the size of an interior angle of the polygon.
1. The ratio of its interior angle to the exterior angle is 2:1.
2. The polygon is a regular hexagon.

• 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:

The relationship `Interior Angle + Exterior Angle = 180°` is a powerful tool for problems involving polygon angles. Also, remember that the sum of all exterior angles of any convex polygon is always 360°. For a regular n-sided polygon, each exterior angle is simply 360/n. This is often faster than using the interior angle formula.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for the size of "an interior angle". For this to be a single, unique value, we must assume the polygon is regular (all sides and angles are equal). Both statements will be evaluated to see if they lead to a unique value for the interior angle.
Step 2: Key Formula or Approach:
For any convex polygon:
- Interior Angle + Exterior Angle = 180\(^\circ\)
For a regular polygon with \( n \) sides:
- Each Interior Angle = \( \frac{(n-2) \times 180^\circ}{n} \)
- Each Exterior Angle = \( \frac{360^\circ}{n} \)
Step 3: Detailed Explanation:
Analyzing Statement (1): The ratio of its interior angle to the exterior angle is 2:1.
Let the interior angle be \( I \) and the exterior angle be \( E \).
We are given \( \frac{I}{E} = \frac{2}{1} \), which means \( I = 2E \).
We know that for any convex polygon, \( I + E = 180^\circ \).
Substituting \( I = 2E \) into the second equation:
\[ 2E + E = 180^\circ \] \[ 3E = 180^\circ \] \[ E = 60^\circ \] Now we can find the interior angle:
\[ I = 2E = 2 \times 60^\circ = 120^\circ \] This gives a unique value for the interior angle. Thus, statement (1) is sufficient.
Analyzing Statement (2): The polygon is a regular hexagon.
A hexagon is a polygon with \( n = 6 \) sides. The term "regular" means all its interior angles are equal.
Using the formula for the interior angle of a regular polygon:
\[ I = \frac{(n-2) \times 180^\circ}{n} \] \[ I = \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 4 \times 30^\circ = 120^\circ \] This also gives a unique value for the interior angle. Thus, statement (2) is sufficient.
Step 4: Final Answer:
Since each statement alone is sufficient to determine the size of the interior angle, the correct choice is (D).