Find the number of sides in a regular polygon, If its each interior angle is :
`(i) 160^@` (ii) `135^@` (iii) `1 1/5` of a right angle.

Answer

To find the number of sides in a regular polygon given its interior angles, we can use the formula for the interior angle of a regular polygon:

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

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

Let's solve each part step by step.

### Part (i): Interior angle = 160°

1. **Set up the equation**:
\[ \frac{(n - 2) \times 180}{n} = 160 \]

2. **Multiply both sides by \( n \)** to eliminate the fraction:
\[ (n - 2) \times 180 = 160n \]

3. **Distribute 180**:
\[ 180n - 360 = 160n \]

4. **Rearrange the equation** to isolate \( n \):
\[ 180n - 160n = 360 \]
\[ 20n = 360 \]

5. **Divide both sides by 20**:
\[ n = \frac{360}{20} = 18 \]

**Conclusion for Part (i)**: The number of sides in the polygon is **18**.

### Part (ii): Interior angle = 135°

1. **Set up the equation**:
\[ \frac{(n - 2) \times 180}{n} = 135 \]

2. **Multiply both sides by \( n \)**:
\[ (n - 2) \times 180 = 135n \]

3. **Distribute 180**:
\[ 180n - 360 = 135n \]

4. **Rearrange the equation**:
\[ 180n - 135n = 360 \]
\[ 45n = 360 \]

5. **Divide both sides by 45**:
\[ n = \frac{360}{45} = 8 \]

**Conclusion for Part (ii)**: The number of sides in the polygon is **8**.

### Part (iii): Interior angle = \( 1 \frac{1}{5} \) of a right angle

1. **Convert \( 1 \frac{1}{5} \) of a right angle to degrees**:
\[ 1 \frac{1}{5} = \frac{6}{5} \text{ of } 90° = \frac{6 \times 90}{5} = 108° \]

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

3. **Multiply both sides by \( n \)**:
\[ (n - 2) \times 180 = 108n \]

4. **Distribute 180**:
\[ 180n - 360 = 108n \]

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

6. **Divide both sides by 72**:
\[ n = \frac{360}{72} = 5 \]

**Conclusion for Part (iii)**: The number of sides in the polygon is **5**.

### Summary of Results:
- (i) 18 sides
- (ii) 8 sides
- (iii) 5 sides