Question

What can be said regarding a line if its slope is negative?

Answer 1

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: \( m = \tan(\theta) \)

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 \( m = \tan(\theta) < 0 \), 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:

1. "θ is an acute angle":
   • Acute angle means 0° < θ < 90°
   • This would give positive slope
   • Incorrect
2. "θ is an obtuse angle":
   • Obtuse angle means 90° < θ < 180°
   • This gives negative slope
   • Correct
3. "Either the line is x-axis or parallel to x-axis":
   • This would mean slope m = 0
   • Incorrect
4. "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 \(\boxed{\theta \text{ is an obtuse angle}}\).