Lesson 3: Parallel and Perpendicular Lines

If a given line is vertical, what is the slope of any line parallel to the given line? perpendicular to the given line?

What is the relationship between the slopes of two perpendicular lines? What is the product of the slopes?

Slope
y-intercept
Point
horizontal intercept
parallel lines
perpendicular lines

S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

SAT questions related to writing equations: 5-4-11,4-3-8,8-4-7,1-3-12,7-3-19
functions: 8-4-4, 7-4-18, 8-4-18, 2-4-14, 3-4-20, 1-4-5, 7-4-21,
Solving Equations: 5-3-17, 6-4-32, 3-4-7, 8-3-1, 3-3-17, 6-3-17, 1-3-1, 3-3-2, 2-3-1, 7-3-16, 6-3-6,

Lesson Description :This lesson focuses on identifying and constructing equations for lines based on their geometric orientation to other lines. Key topics include: Parallel Lines: Understanding that parallel lines have the same slope (m_1 = m_2) but different y-intercepts. They move in the same direction and never intersect. Perpendicular Lines: Learning that perpendicular lines have slopes that are negative reciprocals of each other (m_1 times m_2 = -1). For example, if one slope is 2/3, the other is -3/2.Algebraic Construction: Writing the equation of a new line that passes through a specific point and is either parallel or perpendicular to a given line. Classification: Analyzing pairs of equations to categorize them as parallel, perpendicular, or neither (intersecting but not at a right angle).
Purpose
The purpose of Section 10.3 is to connect algebraic values to geometric properties. In fields like construction, architecture, and engineering, maintaining parallel or perpendicular relationships is essential for structural integrity. By mastering these slope relationships, students learn how to "copy" the direction of a line or create a perfectly "square" corner using only algebra. This section also reinforces the importance of the slope (m) as the defining characteristic of a line's identity.
Depth of Knowledge (DOK) Level
DOK Level 2 & 3
Level 2 (Skill/Concept): Identifying parallel and perpendicular slopes from given equations. Finding the negative reciprocal of a fraction. Writing a basic equation when given a point and a parallel/perpendicular requirement.
Level 3 (Strategic Thinking): Solving multi-step problems where students must first identify the slope of an existing line (often by converting it to y = mx + b form), determine the necessary new slope, and then solve for a new y-intercept. Students must also justify why two lines are perpendicular by showing that the product of their slopes is -1.

On the plans for a treehouse, a beam represented by line QR has endpoints Q(-6,2) and R(-1,8). A connecting beam represented by line ST has endpoints S(-3,6) and T(-8, 5). Are the beams perpendicular? Explain.

Encourage students to understand the process used instead of simply the mechanics. For example, what do they know from the given information? What do they know about perpendicular lines? What do they need to know to write a new equation?

Students may be familiar with the terms parallel and perpendicular. However, before covering the examples, you may want students to use rulers to draw parallel and perpendicular lines on graph paper.

Extension: Write A(-4, -1), B(1, 4), C(4, 1), and D(-1, -4) on the board. Ask students to determine what geometric figure is made if they were to connect these points to get a polygon. Ask students to justify their answer.

Use "you try" pg 238 and practice problems to assess students' understanding of the lesson concepts