In this Grade 9 Saxon Algebra 1 lesson, students learn how to identify and write equations of parallel and perpendicular lines using slope-intercept form and the point-slope formula. The lesson covers the key slope relationships that define these lines — parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. Students practice converting equations, applying these slope rules, and use coordinate geometry to classify triangles by analyzing line segments.
Section 1
📘 Writing Equations of Parallel and Perpendicular Lines
Two nonvertical lines are parallel if they have the same slope and are not the same line.
Next, you’ll use the rules for parallel and perpendicular slopes to write new linear equations and analyze geometric figures on the coordinate plane.
Section 2
Slopes of Parallel Lines
Two nonvertical lines are parallel if they have the same slope and are not the same line. Any two vertical lines are parallel.
Think of parallel lines as perfect twin roads running side-by-side. They never cross because they have the exact same steepness, or slope. The only difference is their starting point, or y-intercept. If they had the same slope and intercept, they would be the exact same line! So, for lines to be parallel, remember: same slope, different y-intercept.
The lines and are parallel because they both have a slope of .
The equations and are parallel because solving the second for gives , showing they share the same slope.
Section 3
Example Card: Writing Equations of Parallel Lines
Let's use one line as a blueprint to build another that runs perfectly alongside it. This example tackles the first key idea: writing equations for parallel lines.
Write an equation in slope-intercept form for the line that passes through (-3, 5) and is parallel to a line with equation .
(-3, 5). We can use the point-slope formula to find the new line's equation.Section 4
Slopes of Perpendicular Lines
Any two lines are perpendicular if their slopes are negative reciprocals of each other. A vertical and horizontal line are also perpendicular.
Perpendicular lines are like two paths that intersect to form a perfect 'T' or a 90-degree angle. For this to happen, their slopes must be mathematical opposites. You take the slope of one line, flip it upside down (that's the reciprocal), and then reverse its sign (that's the negative). This special relationship guarantees a perfect right-angle intersection every time.
The line is perpendicular to because their slopes, and , are negative reciprocals.
A line with slope is perpendicular to a line with slope , since .
A vertical line like is perpendicular to a horizontal line like .
Section 5
Example Card: Writing Equations of Perpendicular Lines
Ready to make a sharp 90-degree turn? This example shows how to build a line that's perfectly perpendicular to another, focusing on the second key idea.
Write an equation in slope-intercept form for the line that passes through (4, -1) and is perpendicular to a line with equation .
(4, -1).Book overview
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Expand to review the lesson summary and core properties.
Section 1
📘 Writing Equations of Parallel and Perpendicular Lines
Two nonvertical lines are parallel if they have the same slope and are not the same line.
Next, you’ll use the rules for parallel and perpendicular slopes to write new linear equations and analyze geometric figures on the coordinate plane.
Section 2
Slopes of Parallel Lines
Two nonvertical lines are parallel if they have the same slope and are not the same line. Any two vertical lines are parallel.
Think of parallel lines as perfect twin roads running side-by-side. They never cross because they have the exact same steepness, or slope. The only difference is their starting point, or y-intercept. If they had the same slope and intercept, they would be the exact same line! So, for lines to be parallel, remember: same slope, different y-intercept.
The lines and are parallel because they both have a slope of .
The equations and are parallel because solving the second for gives , showing they share the same slope.
Section 3
Example Card: Writing Equations of Parallel Lines
Let's use one line as a blueprint to build another that runs perfectly alongside it. This example tackles the first key idea: writing equations for parallel lines.
Write an equation in slope-intercept form for the line that passes through (-3, 5) and is parallel to a line with equation .
(-3, 5). We can use the point-slope formula to find the new line's equation.Section 4
Slopes of Perpendicular Lines
Any two lines are perpendicular if their slopes are negative reciprocals of each other. A vertical and horizontal line are also perpendicular.
Perpendicular lines are like two paths that intersect to form a perfect 'T' or a 90-degree angle. For this to happen, their slopes must be mathematical opposites. You take the slope of one line, flip it upside down (that's the reciprocal), and then reverse its sign (that's the negative). This special relationship guarantees a perfect right-angle intersection every time.
The line is perpendicular to because their slopes, and , are negative reciprocals.
A line with slope is perpendicular to a line with slope , since .
A vertical line like is perpendicular to a horizontal line like .
Section 5
Example Card: Writing Equations of Perpendicular Lines
Ready to make a sharp 90-degree turn? This example shows how to build a line that's perfectly perpendicular to another, focusing on the second key idea.
Write an equation in slope-intercept form for the line that passes through (4, -1) and is perpendicular to a line with equation .
(4, -1).Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter