Lesson 3: Writing Equations of Parallel and Perpendicular Lines

Property

Two lines are parallel if their slopes are exactly equal:

Geometric Proof foundation:

• Alternate Interior Angles Theorem: If two lines are parallel, alternate interior angles are congruent.
• SAS Congruence: If horizontal legs ($\Delta x$) and vertical legs ($\Delta y$) of two slope triangles are equal, and they include a 90° angle, the triangles are congruent by SAS, proving the lines share the exact same angle of elevation.

Examples

• Identifying Parallel Lines: The lines $y = 3x - 2$ and $y = 3x + 5$ are parallel because both have a slope of $m = 3$.
• Checking Standard Form: To check if $2x + y = 5$ and $4x + 2y = 12$ are parallel, convert them to $y=mx+b$. The first is $y = -2x + 5$ ($m_1 = -2$). The second is $y = -2x + 6$ ($m_2 = -2$). Since $m_1 = m_2$, they are parallel.

Property

Two lines are perpendicular if the product of their slopes is $-1$, that is, if

or if one of the lines is horizontal and one is vertical. The slope of one perpendicular line is the negative reciprocal of the other:

Examples

• A line with slope $m_1 = 5$ is perpendicular to a line with slope $m_2 = -\frac{1}{5}$ because their product is $5 \times (-\frac{1}{5}) = -1$.
• The lines $y = \frac{3}{4}x + 2$ and $y = -\frac{4}{3}x - 1$ are perpendicular. Their slopes are $m_1 = \frac{3}{4}$ and $m_2 = -\frac{4}{3}$, and their product is $(\frac{3}{4})(-\frac{4}{3}) = -1$.
• The line $y = 5$ is horizontal (slope is 0), and the line $x = 1$ is vertical (slope is undefined). These lines are perpendicular.

Explanation

Perpendicular lines intersect at a perfect right angle, like the corner of a book. Their slopes are opposites in two ways: one is positive while the other is negative, and their fractions are flipped (reciprocals).