Which of the following angles cannot be made using a ruler and a pair of compasses ?

Options

A. `105 ^(@)`
B. `75^(@)`
C. `130^(@)`
D. `15 ^(@)`

Answer

To determine which angle cannot be constructed using a ruler and a pair of compasses, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understand the angles to be analyzed**: The angles given are 105°, 75°, 130°, and 15°.

2. **Check the construction of 105°**:
- We can construct a 90° angle using a right angle.
- We can also construct a 120° angle.
- Since 105° is between 90° and 120°, we can construct it by bisecting the angle between 90° and 120°.
- **Conclusion**: 105° can be constructed.

3. **Check the construction of 75°**:
- We can construct a 90° angle.
- We can also construct a 60° angle.
- Since 75° is between 60° and 90°, we can construct it by bisecting the angle between 60° and 90°.
- **Conclusion**: 75° can be constructed.

4. **Check the construction of 130°**:
- We can construct a 120° angle.
- However, 130° is not a standard angle that can be easily constructed using basic geometric methods with a ruler and compass.
- **Conclusion**: 130° cannot be constructed.

5. **Check the construction of 15°**:
- We can construct a 30° angle.
- By bisecting the 30° angle, we can obtain a 15° angle.
- **Conclusion**: 15° can be constructed.

6. **Final Conclusion**:
- Among the angles analyzed, the angle that cannot be constructed using a ruler and a pair of compasses is **130°**.