Understanding the Problem:
- We need to analyze what a negative slope implies about a line
- Slope (m) is defined as the tangent of the angle of inclination (θ)
- Mathematically:
Key Concepts:
- Slope sign indicates direction:
- Positive slope: Line rises left to right
- Negative slope: Line falls left to right
- Zero slope: Horizontal line
- Undefined slope: Vertical line
- Angle of inclination (θ) is measured from positive x-axis to the line
Analyzing Negative Slope:
Given , we know:
- Tangent is negative in Quadrant II (90° < θ < 180°) and Quadrant IV (270° < θ < 360°)
- For lines, we consider 0° ≤ θ < 180°
- Therefore, θ must be in Quadrant II: 90° < θ < 180°
Evaluating Options:
- "θ is an acute angle":
- Acute angle means 0° < θ < 90°
- This would give positive slope
- Incorrect
- "θ is an obtuse angle":
- Obtuse angle means 90° < θ < 180°
- This gives negative slope
- Correct
- "Either the line is x-axis or parallel to x-axis":
- This would mean slope m = 0
- Incorrect
- "None of the above":
- Since option 2 is correct
- Incorrect
Conclusion:
A line with negative slope has an obtuse angle of inclination.
Final Answer: The correct statement is .