Which of the following angles cannot be constructed using ruler and compasses?

Options

A. `75^(@)`
B. `15^(@)`
C. `135^(@)`
D. `85^(@)`

Answer

To determine which of the given angles cannot be constructed using a ruler and compass, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Angles**:
- We have four angles to consider: 75 degrees, 15 degrees, 135 degrees, and 85 degrees.

2. **Option 1: 75 degrees**:
- To construct 75 degrees, we can start with a 90-degree angle and bisect it to get 45 degrees.
- Then, we can add 45 degrees to 30 degrees (which can be constructed by bisecting 60 degrees) to achieve 75 degrees.
- **Conclusion**: 75 degrees can be constructed.

3. **Option 2: 15 degrees**:
- To construct 15 degrees, we can start with a 60-degree angle and bisect it to get 30 degrees.
- Then, we can bisect the 30-degree angle to get 15 degrees.
- **Conclusion**: 15 degrees can be constructed.

4. **Option 3: 135 degrees**:
- To construct 135 degrees, we can add 90 degrees and 45 degrees together.
- Both 90 degrees and 45 degrees can be constructed using a ruler and compass.
- **Conclusion**: 135 degrees can be constructed.

5. **Option 4: 85 degrees**:
- To construct 85 degrees, we would need to combine angles that can be constructed using a ruler and compass.
- However, 85 degrees cannot be formed by any combination of standard angles (like 30, 45, 60, or 90 degrees) using only a ruler and compass.
- **Conclusion**: 85 degrees cannot be constructed.

### Final Answer:
The angle that cannot be constructed using a ruler and compass is **85 degrees**.

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