Which the following cannot be meausre of an interior angle of a regular polygon

Options

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

Answer

To determine which of the given options cannot be the measure of an interior angle of a regular polygon, we will use the formula for the interior angle of a regular polygon:

\[ \text{Interior Angle} = \frac{(n - 2) \times 180}{n} \]

where \( n \) is the number of sides of the polygon.

We will check each option one by one.

### Step 1: Check Option A (150 degrees)

1. Set the equation:
\[ 150 = \frac{(n - 2) \times 180}{n} \]

2. Cross-multiply:
\[ 150n = (n - 2) \times 180 \]

3. Expand the right side:
\[ 150n = 180n - 360 \]

4. Rearrange the equation:
\[ 180n - 150n = 360 \]
\[ 30n = 360 \]

5. Solve for \( n \):
\[ n = \frac{360}{30} = 12 \]

Since \( n = 12 \) is a positive integer, 150 degrees can be an interior angle of a regular polygon.

### Step 2: Check Option B (105 degrees)

1. Set the equation:
\[ 105 = \frac{(n - 2) \times 180}{n} \]

2. Cross-multiply:
\[ 105n = (n - 2) \times 180 \]

3. Expand the right side:
\[ 105n = 180n - 360 \]

4. Rearrange the equation:
\[ 180n - 105n = 360 \]
\[ 75n = 360 \]

5. Solve for \( n \):
\[ n = \frac{360}{75} = 4.8 \]

Since \( n = 4.8 \) is not a positive integer, 105 degrees cannot be an interior angle of a regular polygon.

### Step 3: Check Option C (108 degrees)

1. Set the equation:
\[ 108 = \frac{(n - 2) \times 180}{n} \]

2. Cross-multiply:
\[ 108n = (n - 2) \times 180 \]

3. Expand the right side:
\[ 108n = 180n - 360 \]

4. Rearrange the equation:
\[ 180n - 108n = 360 \]
\[ 72n = 360 \]

5. Solve for \( n \):
\[ n = \frac{360}{72} = 5 \]

Since \( n = 5 \) is a positive integer, 108 degrees can be an interior angle of a regular polygon.

### Step 4: Check Option D (144 degrees)

1. Set the equation:
\[ 144 = \frac{(n - 2) \times 180}{n} \]

2. Cross-multiply:
\[ 144n = (n - 2) \times 180 \]

3. Expand the right side:
\[ 144n = 180n - 360 \]

4. Rearrange the equation:
\[ 180n - 144n = 360 \]
\[ 36n = 360 \]

5. Solve for \( n \):
\[ n = \frac{360}{36} = 10 \]

Since \( n = 10 \) is a positive integer, 144 degrees can be an interior angle of a regular polygon.

### Conclusion

The only option that cannot be the measure of an interior angle of a regular polygon is **105 degrees**.

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