3.9Parallel & Perpendicular Lines
Two lines can be parallel (same slope, never meet), perpendicular (slopes are negative reciprocals, meet at 90°), or neither. Slope is the key to telling them apart and writing their equations.
Why This Matters
Parallel and perpendicular relationships appear everywhere in geometry, architecture, and coordinate geometry problems on the SAT/ACT. Understanding slope relationships lets you write equations for lines without graphing.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do the slopes of parallel and perpendicular lines relate to each other, and how do you use these relationships to write equations of lines?
Lesson Overview
Two lines in the same plane are parallel if they never intersect — they have the same slope (m₁ = m₂) but different y-intercepts. Two lines are perpendicular if they intersect at a right angle (90°) — their slopes are negative reciprocals: m₁ × m₂ = −1, or equivalently m₂ = −1/m₁. To find the negative reciprocal of a slope, flip the fraction and change the sign. If neither condition is met, the lines are neither parallel nor perpendicular. To write an equation of a parallel or perpendicular line through a given point, use the appropriate slope and substitute the point to find b.
Slope Relationships at a Glance
Parallel Lines
m₁ = m₂
Same slope, different y-intercepts. Lines never meet.
Example: y = 2x + 1 and y = 2x − 4
Perpendicular Lines
m₁ × m₂ = −1
Slopes are negative reciprocals. Lines meet at 90°.
Example: y = 2x + 1 and y = −(1/2)x + 3
Neither
m₁ ≠ m₂
Different slopes, not negative reciprocals. Lines intersect but not at 90°.
Example: y = 2x + 1 and y = 3x − 2
Parallel and Perpendicular Lines — Visual
Parallel lines — both have slope m = 2
Same slope (m = 2), different y-intercepts. Lines never intersect.
Perpendicular lines — slopes 2 and −1/2
Slopes 2 and −1/2 are negative reciprocals: 2 × (−1/2) = −1. Lines meet at 90°.
Finding the Negative Reciprocal
| Slope m | Flip fraction | Change sign | Perp. slope |
|---|---|---|---|
| 2 | 1/2 | −1/2 | −1/2 |
| −3 | −1/3 | 1/3 | 1/3 |
| 3/4 | 4/3 | −4/3 | −4/3 |
| −2/5 | −5/2 | 5/2 | 5/2 |
| 1 | 1/1 | −1/1 | −1 |
Rule: flip the fraction AND change the sign. Always verify: m × (negative reciprocal) = −1.
Worked Examples
Are y = 3x + 1 and y = 3x − 4 parallel, perpendicular, or neither?
Identify the slopes: m₁ = 3 (from y = 3x + 1) and m₂ = 3 (from y = 3x − 4).
Check for parallel: m₁ = m₂? → 3 = 3 ✓
Check y-intercepts: b₁ = 1 and b₂ = −4. They are different.
Same slope, different y-intercepts → the lines are parallel.
They will never intersect.
Are y = 2x + 1 and y = −(1/2)x + 3 parallel, perpendicular, or neither?
Identify the slopes: m₁ = 2 and m₂ = −1/2.
Check for parallel: 2 ≠ −1/2. Not parallel.
Check for perpendicular: m₁ × m₂ = 2 × (−1/2) = −1 ✓
Product of slopes equals −1 → the lines are perpendicular.
They intersect at a 90° angle.
Write the equation of a line parallel to y = 4x − 1 that passes through (2, 3).
Parallel lines have the same slope. m = 4.
Use slope m = 4 and point (2, 3) to find b.
Substitute into y = mx + b: 3 = 4(2) + b
3 = 8 + b → b = 3 − 8 = −5
Write the equation: y = 4x − 5.
Check: y = 4(2) − 5 = 3 ✓. Slope = 4 (same as original) ✓.
Write the equation of a line perpendicular to y = 3x + 2 that passes through (0, 1).
Perpendicular slope is the negative reciprocal of 3.
Negative reciprocal of 3: flip → 1/3, change sign → −1/3. So m = −1/3.
The point (0, 1) is the y-intercept, so b = 1.
Write the equation: y = −(1/3)x + 1.
Check: slopes 3 × (−1/3) = −1 ✓.
Classify: y = 5x + 2 and y = −(1/5)x − 3. Are they parallel, perpendicular, or neither?
Identify slopes: m₁ = 5 and m₂ = −1/5.
Check parallel: 5 ≠ −1/5. Not parallel.
Check perpendicular: m₁ × m₂ = 5 × (−1/5) = −5/5 = −1 ✓
Product equals −1 → perpendicular.
Guided Practice
Guided Practice Video: Parallel and Perpendicular Lines
Watch the guided practice walkthrough for parallel and perpendicular lines, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Are y = −2x + 5 and y = −2x − 3 parallel, perpendicular, or neither?
Hint: Both have slope m = −2. Same slope, different y-intercepts → parallel.
Are y = (1/4)x + 2 and y = −4x + 1 parallel, perpendicular, or neither?
Hint: m₁ = 1/4, m₂ = −4. Check: (1/4) × (−4) = −1 → perpendicular.
Write the equation of a line parallel to y = −3x + 7 through (1, 2).
Hint: Same slope m = −3. Substitute (1,2): 2 = −3(1) + b → b = 5. y = −3x + 5.
Write the equation of a line perpendicular to y = 2x − 5 through (4, 1).
Hint: Perpendicular slope: negative reciprocal of 2 is −1/2. Substitute (4,1): 1 = −(1/2)(4) + b → b = 3. y = −(1/2)x + 3.
Classify y = 3x + 1 and y = 4x − 2. Are they parallel, perpendicular, or neither?
Hint: m₁ = 3, m₂ = 4. Not equal (not parallel). 3 × 4 = 12 ≠ −1 (not perpendicular). Neither.
Key Vocabulary
Parallel Lines
Two lines in the same plane that never intersect. Parallel lines have equal slopes (m₁ = m₂) but different y-intercepts.
Example: y = 2x + 3 and y = 2x − 1 are parallel (both have slope 2).
Perpendicular Lines
Two lines that intersect at a right angle (90°). Their slopes are negative reciprocals: m₁ × m₂ = −1.
Example: y = 3x + 1 and y = −(1/3)x + 2 are perpendicular: 3 × (−1/3) = −1.
Negative Reciprocal
The negative reciprocal of m is −1/m. To find it: flip the fraction and change the sign.
Example: Negative reciprocal of 4 is −1/4. Negative reciprocal of −2/3 is 3/2.
Slope Relationship
The mathematical connection between the slopes of two lines that determines whether they are parallel, perpendicular, or neither.
Example: Same slope → parallel. Product = −1 → perpendicular. Otherwise → neither.
Interactive Practice — 5 Questions
Which line is parallel to y = 5x − 3?
What is the slope of a line perpendicular to y = 4x + 1?
Are y = −2x + 3 and y = (1/2)x − 1 perpendicular?
Write the equation of a line parallel to y = −3x + 2 through (0, 5).
Classify y = 7x + 1 and y = 7x + 1.
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Classify y = 6x − 2 and y = 6x + 9. Are they parallel, perpendicular, or neither?
Classify y = (3/5)x + 1 and y = −(5/3)x − 4. Are they parallel, perpendicular, or neither?
Write the equation of a line parallel to y = −x + 8 through (3, 1).
Write the equation of a line perpendicular to y = (1/2)x − 3 through (2, 5).
Challenge: Line A passes through (1, 3) and (4, 9). Line B passes through (0, 5) and (3, 4). Determine if A and B are parallel, perpendicular, or neither. Show all work.
ChallengeCommon Mistakes
Thinking the negative reciprocal of 3 is −3 (just changing the sign without flipping).
The negative reciprocal requires BOTH flipping the fraction AND changing the sign. Negative reciprocal of 3 = −1/3. Negative reciprocal of −2/3 = 3/2.
Saying two lines with the same slope are always parallel — including when they are the same line.
Parallel lines must have the same slope AND different y-intercepts. If both slope and y-intercept are equal, the equations describe the same line, not two parallel lines.
Confusing the slope of a perpendicular line with the slope of a parallel line.
Parallel: use the SAME slope. Perpendicular: use the NEGATIVE RECIPROCAL slope. These are two different operations.
Forgetting to verify the perpendicular slope by checking that m₁ × m₂ = −1.
Always multiply the two slopes as a check. If the product is −1, the lines are perpendicular. If not, recheck your negative reciprocal.
Math Tips
Parallel slope shortcut: copy the slope exactly. Change only the y-intercept (find b using the new point).
Perpendicular slope shortcut: flip the fraction and change the sign. Then verify: original slope × new slope = −1.
Horizontal lines (slope 0) are perpendicular to vertical lines (undefined slope) — even though the product rule does not apply directly.
On the SAT/ACT, parallel and perpendicular questions almost always require you to identify slopes from y = mx + b form first. Convert to slope-intercept before classifying.