Lesson 4: Parallel and Perpendicular Lines

Property

Parallel lines are lines in the same plane that do not intersect.

• Parallel lines have the same slope and different y-intercepts.
• If $m_1$ and $m_2$ are the slopes of two parallel lines, then $m_1 = m_2$.
• Parallel vertical lines have different x-intercepts.

Examples

• The lines $y = 5x + 1$ and $y = 5x - 3$ are parallel because they both have a slope of $m=5$ but have different y-intercepts.

• To check if $2x + y = 7$ and $y = -2x + 4$ are parallel, rewrite the first equation as $y = -2x + 7$. Both lines have a slope of $m=-2$ and different y-intercepts, so they are parallel.

Property

Two non-vertical lines are perpendicular (forming a 90° right angle) if the product of their slopes is $-1$:

Geometric Proof via Rotation: A 90° counterclockwise rotation about the origin maps any point $(a, b)$ to the new point $(-b, a)$.