Measure of each interior angle of a regular polygon can never be:

Options

A. `150^@`
B. `105^@`
C. `108^@`
D. `144^@`

Answer

To determine the measure of each interior angle of a regular polygon and identify which angle can never be an interior angle of such a polygon, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The formula for the measure of each interior angle of a regular polygon with \( n \) sides is given by:
\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \]

2. **Identify Possible Values**: We need to analyze this formula to see what values it can take. The number of sides \( n \) must be an integer greater than or equal to 3 (since a polygon has at least 3 sides).

3. **Calculate Interior Angles for Various \( n \)**:
- For \( n = 3 \) (Triangle):
\[ \text{Interior Angle} = \frac{(3-2) \times 180^\circ}{3} = \frac{180^\circ}{3} = 60^\circ \]
- For \( n = 4 \) (Quadrilateral):
\[ \text{Interior Angle} = \frac{(4-2) \times 180^\circ}{4} = \frac{360^\circ}{4} = 90^\circ \]
- For \( n = 5 \) (Pentagon):
\[ \text{Interior Angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ \]
- For \( n = 6 \) (Hexagon):
\[ \text{Interior Angle} = \frac{(6-2) \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ \]
- For \( n = 8 \) (Octagon):
\[ \text{Interior Angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ \]

4. **General Observation**: As \( n \) increases, the measure of each interior angle approaches but never reaches \( 180^\circ \). Therefore, the interior angles of regular polygons will always be less than \( 180^\circ \).

5. **Conclusion**: Since the interior angle can never be \( 180^\circ \) or more, we conclude that the measure of each interior angle of a regular polygon can never be \( 180^\circ \) or greater.

### Final Answer:
The measure of each interior angle of a regular polygon can never be \( 180^\circ \) or more.