The ratio of the numbers of sides of two regular polygons is 1:2. If each interior angle of the first polygon is `120^@`, then the measure of each interior angle of the second polygon is

Options

A. `140^@`
B. `135^@`
C. `150^@`
D. `160^@`

Answer

To solve the problem, we need to find the measure of each interior angle of the second polygon based on the information provided.

### Step-by-Step Solution:

1. **Understanding the Ratio of Sides**:
- The ratio of the number of sides of the two polygons is given as 1:2.
- Let the number of sides of the first polygon be \( n \) and the number of sides of the second polygon be \( 2n \).

2. **Finding the Number of Sides of the First Polygon**:
- We know that each interior angle of the first polygon is \( 120^\circ \).
- The formula for the interior angle \( A \) of a regular polygon with \( n \) sides is:
\[ A = \frac{(n-2) \times 180}{n} \]
- Setting this equal to \( 120^\circ \):
\[ \frac{(n-2) \times 180}{n} = 120 \]

3. **Solving for \( n \)**:
- Cross-multiply to eliminate the fraction:
\[ (n-2) \times 180 = 120n \]
- Expanding the left side:
\[ 180n - 360 = 120n \]
- Rearranging the equation:
\[ 180n - 120n = 360 \]
\[ 60n = 360 \]
- Dividing both sides by 60:
\[ n = 6 \]

4. **Finding the Number of Sides of the Second Polygon**:
- Since the second polygon has twice the number of sides of the first polygon:
\[ \text{Number of sides of the second polygon} = 2n = 2 \times 6 = 12 \]

5. **Calculating the Interior Angle of the Second Polygon**:
- Now, we can find the interior angle of the second polygon using the same formula:
\[ A = \frac{(2n-2) \times 180}{2n} \]
- Substituting \( n = 6 \):
\[ A = \frac{(12-2) \times 180}{12} \]
\[ A = \frac{10 \times 180}{12} \]
\[ A = \frac{1800}{12} = 150^\circ \]

### Final Answer:
The measure of each interior angle of the second polygon is \( 150^\circ \).