Is it possible to have a regular polygon whose each exterior angle is :
(i) `80^@` (ii) 40% of a right angle

Answer

To determine if it is possible to have a regular polygon with the specified exterior angles, we will follow these steps:

### Step 1: Understanding Exterior Angles
The exterior angle of a polygon is formed by one side of the polygon and the extension of an adjacent side. The sum of all exterior angles of any polygon is always 360 degrees.

### Step 2: Analyzing the First Case (80 degrees)
1. **Given**: Each exterior angle = 80 degrees.
2. **Formula**: To find the number of sides (n) of the polygon, we use the formula:
\[ n = \frac{360}{\text{each exterior angle}} \]
3. **Calculation**:
\[ n = \frac{360}{80} = 4.5 \]
4. **Conclusion**: Since the number of sides (n) cannot be a fraction, it is **not possible** to have a regular polygon with each exterior angle of 80 degrees.

### Step 3: Analyzing the Second Case (40% of a Right Angle)
1. **Given**: Each exterior angle = 40% of a right angle.
2. **Calculate the degree measure of a right angle**: A right angle is 90 degrees.
3. **Calculation**:
\[ \text{Each exterior angle} = 40\% \text{ of } 90 = \frac{40}{100} \times 90 = 36 \text{ degrees} \]
4. **Using the formula for number of sides**:
\[ n = \frac{360}{36} = 10 \]
5. **Conclusion**: Since n is a whole number, it is **possible** to have a regular polygon with each exterior angle of 36 degrees.

### Final Summary
- For the first case (80 degrees), it is **not possible** to have a regular polygon.
- For the second case (36 degrees), it is **possible** to have a regular polygon.