Parallel and Perpendicular 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.

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.

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