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Parallel and Perpendicular Lines

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Parallel and Perpendicular Lines

Study Guide

Key Definition

Parallel lines are lines in a plane which never meet; they have the same slope. Perpendicular lines intersect at a $90^\circ$ angle.

Important Notes

Parallel lines have the same slope: $m_1 = m_2$
Perpendicular lines' slopes are negative reciprocals: $m_1 \times m_2 = -1$
Perpendicular lines form a $90^\circ$ angle
Parallel lines never intersect
Perpendicular lines intersect at exactly one point

Mathematical Notation

$\angle$ means angle

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Understanding slopes helps determine if lines are parallel or perpendicular. For parallel lines, $m_1 = m_2$, and for perpendicular lines, $m_1 \times m_2 = -1$.

Remember

Parallel lines: $m_1 = m_2$; Perpendicular lines: $m_1 \times m_2 = -1$.

Quick Reference

Parallel Lines:$y = mx + b$ where $m_1 = m_2$

Perpendicular Lines:Slopes are negative reciprocals: $m_1 \times m_2 = -1$

Understanding Parallel and Perpendicular Lines

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Parallel lines have the same slope: $m_1 = m_2$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which line below is parallel to $y = 2x + 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a skate park. You want two ramps that are perpendicular to each other. If one ramp is expressed by the line $y = \frac{1}{3}x + 2$, what should be the slope of the other ramp?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given two lines, $y = -2x + 5$ and $y = \frac{1}{2}x - 3$, are these lines parallel, perpendicular, or neither?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following pairs of lines are perpendicular? $y = x + 1$ and $y = -x + 4$

Recap

Watch & Learn

Review key concepts and takeaways