With the help of a ruler and a compass it is not possible to construct an angle of

Options

A. `37.5^(@)`
B. `40^(@)`
C. `22.5^(@)`
D. `67.5^(@)`

Explanation

To determine which angle cannot be constructed using a ruler and a compass, we need to understand the principles of angle construction. Here’s a step-by-step solution to the question:

### Step 1: Understand Constructible Angles
Constructible angles are those that can be formed using a finite number of steps with a compass and straightedge. The angles that can be constructed are typically based on dividing a circle into equal parts.

### Step 2: Identify Known Constructible Angles
Common constructible angles include:
- 0 degrees
- 30 degrees
- 45 degrees
- 60 degrees
- 90 degrees
- 120 degrees
- 135 degrees
- 180 degrees

Additionally, we can construct angles that are bisectors of these angles, such as:
- 15 degrees (half of 30 degrees)
- 22.5 degrees (half of 45 degrees)
- 37.5 degrees (half of 75 degrees)
- 67.5 degrees (half of 135 degrees)

### Step 3: Analyze the Given Angles
We need to check which angles can and cannot be constructed. The angles mentioned in the question are:
- 90 degrees (constructible)
- 60 degrees (constructible)
- 45 degrees (constructible)
- 22.5 degrees (constructible)
- 37.5 degrees (constructible)
- 40 degrees (not constructible)

### Step 4: Conclusion
From the analysis, we find that while many angles can be constructed, the angle of 40 degrees is not constructible using just a ruler and compass. Therefore, the answer to the question is that it is not possible to construct an angle of 40 degrees.

### Final Answer
**It is not possible to construct an angle of 40 degrees with the help of a ruler and a compass.**

---