Angle between Lines (Leaving Cert Mathematics): Revision Notes

Angle Between Lines

What Is the Angle Between Two Lines?

The angle between two intersecting lines measures the inclination of one line relative to the other. This angle, denoted by $\theta$, is calculated using the slopes of the two lines.

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Formula for the Angle Between Two Lines

If the slopes of the lines are $m_1$ and $m_2$, the angle $\theta$ between them is given by:

$$\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right|, \quad 1 + m_1m_2 \neq 0$$

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Special Cases

Perpendicular Lines

If the lines are perpendicular, then:

$$m_1m_2 = -1$$

So:

$$\theta = 90^\circ$$

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Parallel Lines

If the lines are parallel, then:

$$m_1 = m_2$$

So:

$$\theta = 0^\circ$$

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Vertical and Non-Vertical Lines

If one line is vertical ($m = \infty$), then the angle is given by:

$$\tan \theta = |m_2|$$

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Worked Examples

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Summary

• Formula:

  $$\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right|$$

• Special Cases:

  • Perpendicular: $m_1m_2 = -1$, $\theta = 90^\circ$
  • Parallel: $m_1 = m_2$, $\theta = 0^\circ$

• Steps to Calculate:

  1. Identify the slopes.
  2. Apply the formula.
  3. Use inverse tangent to find the angle.

• Practise identifying angles between lines in coordinate geometry problems.