Which of the following angles CANNOT be constructed by using ruler and compass only?

Options

A. `30^(@)`
B. `45^(@)`
C. `70^(@)`
D. `90^(@)`

Answer

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

### Step-by-Step Solution:

1. **Understanding Constructible Angles**:
- An angle is constructible if it can be created using a finite number of steps with a compass and straightedge.
- Constructible angles are typically those that can be expressed in terms of 30 degrees, 45 degrees, and their combinations.

2. **Analyzing Each Option**:
- **Option A: 30 degrees**:
- This angle can be constructed by bisecting a 60-degree angle (which can be constructed).
- **Construction**: Draw a 60-degree angle and bisect it.
- **Conclusion**: 30 degrees is constructible.

- **Option B: 45 degrees**:
- This angle can be constructed by bisecting a 90-degree angle.
- **Construction**: Draw a 90-degree angle and bisect it.
- **Conclusion**: 45 degrees is constructible.

- **Option C: 70 degrees**:
- This angle cannot be constructed using a ruler and compass alone.
- **Reason**: 70 degrees cannot be derived from the basic constructible angles (30, 45, 60, etc.) through any combination of bisection or addition.
- **Conclusion**: 70 degrees is not constructible.

- **Option D: 90 degrees**:
- This angle can be constructed easily.
- **Construction**: Draw a straight line and create a perpendicular line to form a 90-degree angle.
- **Conclusion**: 90 degrees is constructible.

3. **Final Answer**:
- The angle that cannot be constructed using a ruler and compass only is **70 degrees** (Option C).