Parallel and Perpendicular Lines

Identify parallel and perpendicular lines using slopes. Math 10-C Alberta mathematics.

Lesson 6.2 of Linear Relations and Functions in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Parallel Lines. Discover how equal slopes guarantee two lines never meet. Work through examples comparing slopes of multiple lines to identify which are parallel and which are not.
• Perpendicular Lines. Learn the negative-reciprocal rule that makes two lines cross at exactly 90 degrees. Practice identifying perpendicular pairs and handling special cases like horizontal and vertical lines.
• Finding Perpendicular Slopes and Coordinates. Apply perpendicular slope relationships to find missing slopes and coordinates. Work through finding a point on the x-axis that creates a perpendicular line using cross-multiplication.

Inside the lesson — a free preview

What Makes Lines Parallel?

Definition — Parallel Lines. Parallel lines never intersect — they run in the same direction, always the same distance apart.

Since slope measures direction, parallel lines must have the exact same slope.

What Makes Lines Perpendicular?

Definition — Perpendicular Lines. Perpendicular lines intersect at a 90° angle. Their slopes have a very specific relationship: they are negative reciprocals of each other.

How to Find the Negative Reciprocal

The Two-Step Process. Step 1 — Flip the fraction.

Step 2 — Change the sign.

Examples:

Slope (3)/(4) → perpendicular slope: -(4)/(3)

Slope -2 → perpendicular slope: (1)/(2)

Slope 5 → perpendicular slope: -(1)/(5)

Special Case — Horizontal and Vertical Lines. A horizontal line (m = 0) is perpendicular to a vertical line (m = undefined).

These are the only pair where the negative-reciprocal rule does not apply directly — you simply recognize that horizontal ⊥ vertical.

Worked example: Are These Three Lines Parallel?

Line EF passes through E(-3, -2) and F(-1, 8). Line CD passes through C(-1, -3) and D(1, 7). Line AB passes through A(-3, 7) and B(-5, 2).

Slope of Line EF. Formula: m = (y_2 - y_1)/(x_2 - x_1)

Substitute: m_EF = (8 - (-2))/(-1 - (-3)) = (10)/(2)

Result: m_EF = 5

Slope of Line CD. Substitute: m_CD = (7 - (-3))/(1 - (-1)) = (10)/(2)

Result: m_CD = 5

Slope of Line AB. Substitute: m_AB = (2 - 7)/(-5 - (-3)) = (-5)/(-2)

Result: m_AB = (5)/(2)

Conclusion. m_EF = m_CD = 5, so EF ∥ CD (parallel).

m_AB = (5)/(2) ≠ 5, so line AB is NOT parallel to the other two.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.