Parallel Lines (Leaving Cert Mathematics): Revision Notes

Parallel Lines

What Are Parallel Lines?

Two lines are parallel if they do not intersect, no matter how far they are extended. This can occur when:

1. The lines have the same slope ($m_1 = m_2$).
2. They lie in the same plane but maintain a constant distance apart.

Equation of Parallel Lines

The general equation of a line is $y = mx + c$, where mm is the slope and $c$ is the $y$-intercept.

Parallel lines have:

• Equal slopes: If $l_1$ and $l_2$ are parallel, $m_1 = m_2$
• Different intercepts: The $c$ values differ unless the lines are coincident (identical).

Slope and Parallelism

To determine if two lines are parallel, compare their slopes:

• Lines represented as $y = m_1x + c_1$ and $y = m_2x + c_2$ are parallel if $m_1 = m_2$
• Vertical lines are parallel if their equations are of the form $x = k_1$ and $x = k_2$, with $k_1 \neq k_2$

Geometric Properties

• If two parallel lines are cut by a transversal, the following angle pairs are equal:
  1. Corresponding angles.
  2. Alternate interior angles.
  3. Alternate exterior angles. These angle properties are useful for proving lines are parallel using geometry.

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Worked Examples

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Summary

• Definition: Parallel lines have the same slope and do not intersect.
• Key Property: For parallel lines $y = m_1x + c_1$ and $y = m_2x + c_2$, $m_1 = m_2$
• Angle Properties: Corresponding, alternate interior, and alternate exterior angles are equal when parallel lines are intersected by a transversal.
• Use the slope and point-slope formulas to solve parallel line problems.
• Practice identifying and working with parallel lines in coordinate geometry to master this concept.