Quick answer: Parallel lines have equal slopes (m₁ = m₂). Perpendicular lines have slopes that are negative reciprocals, so they multiply to −1 (m₁ × m₂ = −1). For example, a line with slope 3 is perpendicular to a line with slope −⅓. The one exception is a horizontal line (slope 0) and a vertical line (undefined slope), which are perpendicular too.
This quiz has 15 multiple-choice questions on parallel and perpendicular lines, each with the answer and a worked explanation. It suits US Algebra 1 and Geometry, UK GCSE Higher maths (where slope is called gradient) and Canadian Grade 10 math. If you need a refresher first, read How to Find the Slope of a Line and Zero vs Undefined Slope.
The rules you need
| Relationship | Slope rule | Example |
|---|---|---|
| Parallel | m₁ = m₂ (and different y-intercepts) | y = 2x + 1 and y = 2x − 5 |
| Perpendicular | m₁ × m₂ = −1, so m₂ = −1/m₁ | y = 2x + 1 and y = −½x + 4 |
| Same line (coincident) | Same slope and same intercept | y = 2x + 1 and 4x − 2y = −2 |
| Neither | Slopes differ and product ≠ −1 | y = 2x and y = 3x |
How to find a negative reciprocal: flip the fraction and change the sign. 3 → −⅓; −⅖ → 5/2; ¾ → −4/3.
Key takeaways
- Always rewrite equations as y = mx + b before comparing slopes.
- To write a parallel or perpendicular line through a point, use point-slope form: y − y₁ = m(x − x₁).
- A perpendicular bisector passes through the midpoint and uses the negative reciprocal slope.
Parallel and perpendicular lines quiz: 15 MCQs
Q1. What is the slope of a line parallel to y = 3x − 5?
- −3
- 3
- −⅓
- ⅓
Show answer
B, 3. Parallel lines share the same slope.
Q2. What is the slope of a line perpendicular to y = 3x − 5?
- 3
- −3
- −⅓
- ⅓
Show answer
C, −⅓. Negative reciprocal of 3. Check: 3 × (−⅓) = −1.
Q3. A line has equation y = −⅖x + 1. What is the gradient of a perpendicular line?
- ⅖
- −5/2
- 5/2
- −⅖
Show answer
C, 5/2. Flip −2/5 to −5/2, then change the sign to get 5/2.
Q4. The lines y = 4x + 1 and y = 4x − 7 are:
- Parallel
- Perpendicular
- The same line
- Neither
Show answer
A, Parallel. Same slope (4), different y-intercepts.
Q5. The lines y = 2x + 3 and y = −½x + 3 are:
- Parallel
- Perpendicular
- The same line
- Neither
Show answer
B, Perpendicular. 2 × (−½) = −1. (They both cross the y-axis at 3, which is where they meet.)
Q6. The lines 2x + y = 6 and y = −2x + 1 are:
- Parallel
- Perpendicular
- The same line
- Neither
Show answer
A, Parallel. 2x + y = 6 becomes y = −2x + 6, so both slopes are −2 and the intercepts differ.
Q7. Which is the equation of the line parallel to y = 2x + 1 through (3, 4)?
- y = 2x + 4
- y = 2x − 2
- y = −½x + 5.5
- y = 2x + 10
Show answer
B, y = 2x − 2. y − 4 = 2(x − 3) → y = 2x − 6 + 4 = 2x − 2.
Q8. Which is the equation of the line perpendicular to y = 2x + 1 through (4, 1)?
- y = −½x + 3
- y = 2x − 7
- y = ½x − 1
- y = −½x − 1
Show answer
A, y = −½x + 3. Slope −½: y − 1 = −½(x − 4) → y = −½x + 2 + 1 = −½x + 3.
Q9. Line L₁ passes through (1, 2) and (3, 8). Line L₂ passes through (0, 0) and (−3, 1). The lines are:
- Parallel
- Perpendicular
- The same line
- Neither
Show answer
B, Perpendicular. m₁ = (8 − 2)/(3 − 1) = 3; m₂ = (1 − 0)/(−3 − 0) = −⅓; 3 × (−⅓) = −1. Check slopes quickly with the slope calculator.
Q10. What is the slope of a line perpendicular to 3x − 4y = 12?
- ¾
- −¾
- 4/3
- −4/3
Show answer
D, −4/3. −4y = −3x + 12 → y = ¾x − 3. Negative reciprocal of ¾ is −4/3.
Q11. For what value of k is y = kx + 2 parallel to 6x − 2y = 5?
- −3
- 3
- 6
- −⅓
Show answer
B, 3. −2y = −6x + 5 → y = 3x − 2.5, so k must equal 3.
Q12. For what value of k is y = kx − 1 perpendicular to y = ¼x + 9?
- 4
- ¼
- −4
- −¼
Show answer
C, −4. k × ¼ = −1 → k = −4.
Q13. The lines y = x and y = −x are:
- Parallel
- Perpendicular
- The same line
- Neither
Show answer
B, Perpendicular. 1 × (−1) = −1. They cross at the origin at 90°.
Q14. What is the slope of any line perpendicular to x = 5?
- 5
- −⅕
- 0
- Undefined
Show answer
C, 0. x = 5 is vertical, so a perpendicular line is horizontal. Here is why.
Q15. Find the perpendicular bisector of the segment joining A(2, 4) and B(6, 8).
- y = x + 2
- y = −x + 10
- y = −x + 6
- y = x − 10
Show answer
B, y = −x + 10. Midpoint = ((2 + 6)/2, (4 + 8)/2) = (4, 6). Slope AB = 4/4 = 1, so the perpendicular slope is −1. y − 6 = −(x − 4) → y = −x + 10. Confirm the midpoint with the midpoint calculator and the segment length with the distance calculator.
How did you score?
| Score | What to do next |
|---|---|
| 13–15 | Excellent. Try the 20-question slope quiz for mixed practice. |
| 9–12 | Good. Review negative reciprocals and point-slope form, then retake. |
| 0–8 | Go back to how to find slope and work through the examples. |
Where parallel and perpendicular lines are used
Architects and builders rely on perpendicular lines for square corners. A quick check is the 3-4-5 rule, which comes from the Pythagorean theorem; try it on the Pythagorean theorem calculator. Road and rail designers use parallel lines for lanes and tracks, and converting plan measurements is easy with the PakMath length converter. In statistics, the slope of a line of best fit comes from a linear regression calculator.
Make it a worksheet: copy this quiz into a Word file and turn it into a printable handout with Word to PDF. If the file is large, shrink it for email with Compress PDF.
Frequently asked questions
How do you know if two lines are parallel?
Write both in y = mx + b form. If the slopes are equal and the y-intercepts are different, the lines are parallel.
How do you know if two lines are perpendicular?
Multiply their slopes. If the product is −1, they are perpendicular. A horizontal line and a vertical line are also perpendicular.
What is a negative reciprocal?
Flip the number and change its sign. The negative reciprocal of 2 is −½, and the negative reciprocal of −¾ is 4/3.
What is the perpendicular gradient in GCSE maths?
It is the negative reciprocal of the original gradient: if a line has gradient m, a perpendicular line has gradient −1/m.
Can parallel lines have different slopes?
No. Lines with different slopes always meet at exactly one point, so they cannot be parallel.
More slope practice on GK19
About the author: Dr. Muhammad Amjad Khadim holds a PhD in Mathematics from COMSATS University Islamabad and founded PakMath, Adzita and PDFWisely.
