An angle which cannot be constructed by using a ruler and compass is

Options

A. `15^(@)`
B. `22(1)/(2)""^(@)`
C. `37(1)/(2)""^(@)`
D. `50^(@)`

Answer

To determine which angle cannot be constructed using a ruler and compass, we will analyze the given options step by step.

### Step-by-Step Solution:

1. **Understanding Constructible Angles:**
- An angle is constructible if it can be formed using a finite number of steps with a compass and straightedge.
- Constructible angles must be of the form \( \frac{360}{2^n} \) or \( \frac{360}{3^n} \) or combinations thereof, where \( n \) is a non-negative integer.

2. **Analyzing the First Option: 15 Degrees**
- 15 degrees can be constructed by bisecting 30 degrees, which can be derived from 60 degrees (which is constructible).
- Therefore, 15 degrees is constructible.

3. **Analyzing the Second Option: 22.5 Degrees**
- 22.5 degrees can be constructed by bisecting 45 degrees, which is derived from 90 degrees (also constructible).
- Therefore, 22.5 degrees is constructible.

4. **Analyzing the Third Option: 37.5 Degrees**
- 37.5 degrees can be constructed by bisecting 75 degrees, which can be obtained from 150 degrees (which is constructible).
- Therefore, 37.5 degrees is constructible.

5. **Analyzing the Fourth Option: 50 Degrees**
- 50 degrees can be derived from 100 degrees, which can be constructed by bisecting 200 degrees (which is constructible).
- Therefore, 50 degrees is constructible.

6. **Conclusion:**
- All the angles listed (15 degrees, 22.5 degrees, 37.5 degrees, and 50 degrees) are constructible.
- However, if we consider angles like 22.5 degrees, which is a half of 45 degrees, we can see that the angle that cannot be constructed by a ruler and compass is actually 22.5 degrees.

### Final Answer:
The angle that cannot be constructed by using a ruler and compass is **50 degrees**.