Unit 3 Parallel And Perpendicular Lines Homework 6 Answer Key

Unit 3: Parallel and Perpendicular Lines - Homework 6 Answer Key: A Comprehensive Guide

This comprehensive guide provides detailed solutions and explanations for Homework 6 in Unit 3, focusing on parallel and perpendicular lines. Understanding these concepts is crucial for geometry and higher-level mathematics. This guide aims to not only provide the answers but also to deepen your understanding of the underlying principles. We'll cover various problem types, from identifying parallel and perpendicular lines to applying theorems and postulates. Remember, understanding the why is just as important as getting the what.

Understanding Parallel and Perpendicular Lines

Before diving into the answers, let's refresh our understanding of the core concepts:

Parallel Lines

Parallel lines are lines in a plane that never intersect. They maintain a constant distance from each other. A key characteristic is that they have the same slope. Think of train tracks; they are parallel and never cross (ideally!). We often use symbols like || to denote parallel lines; for example, line AB || line CD.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90°). The slopes of perpendicular lines are negative reciprocals of each other. If one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. Imagine the intersection of two streets forming a perfect 'T' – that's a visual representation of perpendicular lines.

Homework 6 Problems and Solutions: A Step-by-Step Approach

Let's tackle the problems in Homework 6, breaking them down step-by-step for clarity. Note: Since the specific problems from your homework are not provided, I will create example problems covering the key concepts typically found in such an assignment. You can adapt these methods to your specific questions.

Problem 1: Identifying Parallel Lines

Question: Determine whether lines AB and CD are parallel given their coordinates: A(1, 2), B(3, 4), C(-1, 0), D(1, 2).

Solution:

1. Find the slope of line AB: The slope (m) is calculated as (y2 - y1) / (x2 - x1). Therefore, the slope of AB = (4 - 2) / (3 - 1) = 2/2 = 1.

2. Find the slope of line CD: The slope of CD = (2 - 0) / (1 - (-1)) = 2/2 = 1.

3. Compare slopes: Since the slopes of AB and CD are both equal to 1, the lines are parallel.

Problem 2: Identifying Perpendicular Lines

Question: Determine whether lines EF and GH are perpendicular given their equations: EF: y = 2x + 5 and GH: y = -1/2x - 3.

Solution:

1. Identify slopes: The slope of EF is 2. The slope of GH is -1/2.

2. Check for negative reciprocals: The slope of GH (-1/2) is the negative reciprocal of the slope of EF (2).

3. Conclusion: Since the slopes are negative reciprocals, the lines EF and GH are perpendicular.

Problem 3: Finding the Equation of a Parallel Line

Question: Find the equation of a line parallel to y = 3x - 2 that passes through point (1, 4).

Solution:

1. Identify the slope: The slope of the given line is 3. Parallel lines have the same slope.

2. Use point-slope form: The point-slope form of a line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

3. Substitute: Using the point (1, 4) and the slope 3, we get y - 4 = 3(x - 1).

4. Simplify: y - 4 = 3x - 3 => y = 3x + 1. This is the equation of the parallel line.

Problem 4: Finding the Equation of a Perpendicular Line

Question: Find the equation of a line perpendicular to y = -1/4x + 7 that passes through the origin (0, 0).

Solution:

1. Find the slope of the perpendicular line: The slope of the given line is -1/4. The slope of a perpendicular line is the negative reciprocal, which is 4.

2. Use the point-slope form: y - y1 = m(x - x1). Since the line passes through (0, 0), we have y - 0 = 4(x - 0).

3. Simplify: y = 4x. This is the equation of the perpendicular line.

Problem 5: Using the properties of parallel lines cut by a transversal

Question: Two parallel lines are intersected by a transversal. One of the angles formed measures 110°. Find the measure of all other angles.

Solution:

When parallel lines are intersected by a transversal, several angle relationships emerge:

• Corresponding Angles: Corresponding angles are equal.
• Alternate Interior Angles: Alternate interior angles are equal.
• Alternate Exterior Angles: Alternate exterior angles are equal.
• Consecutive Interior Angles: Consecutive interior angles are supplementary (add up to 180°).

Since one angle is 110°, its corresponding angle is also 110°. Its alternate interior angle is also 110°. Its alternate exterior angle is also 110°. The consecutive interior angle to the 110° angle is 70° (180° - 110°). Using these relationships, you can determine the measures of all eight angles formed by the intersection.

Problem 6: Proving Lines are Parallel

Question: Given that angle A and angle B are alternate interior angles, and the measure of angle A is 75° and the measure of angle B is 75°, prove that the lines are parallel.

Solution:

If alternate interior angles are congruent (equal), then the lines are parallel. Since angle A and angle B are congruent alternate interior angles, we can conclude that the lines are parallel.

Problem 7: Real-World Application

Question: A carpenter needs to ensure two beams are parallel for structural integrity. How can they use the concept of parallel lines and transversals to verify this?

Solution: The carpenter can use a square or another tool to measure angles formed by the beams and a transversal (a straight line intersecting both beams). If corresponding angles, alternate interior angles, or alternate exterior angles are equal, the beams are parallel.

Problem 8: Working with Slopes and Equations

Question: Line A has a slope of 2/3 and passes through point (3, 4). Line B is perpendicular to Line A and passes through point (-1, 2). Find the equation of Line B.

Solution:

1. Find the slope of Line B: Since Line B is perpendicular to Line A, its slope is the negative reciprocal of 2/3, which is -3/2.

2. Use the point-slope form: y - y1 = m(x - x1), where m = -3/2 and (x1, y1) = (-1, 2).

3. Substitute: y - 2 = -3/2(x + 1).

4. Simplify: y = -3/2x + 1/2. This is the equation of Line B.

Expanding Your Understanding

This guide provides a solid foundation for understanding parallel and perpendicular lines. To further enhance your comprehension, consider exploring these topics:

• Vectors: Vectors provide another way to represent lines and understand their relationships (parallelism and perpendicularity).
• Coordinate Geometry: Mastering coordinate geometry is crucial for working with lines in the Cartesian plane.
• Proofs: Practice writing geometric proofs to solidify your understanding of theorems and postulates related to parallel and perpendicular lines.

By diligently practicing these problems and exploring related concepts, you'll develop a comprehensive grasp of parallel and perpendicular lines and their applications in various mathematical contexts. Remember, consistent practice is key to mastering these fundamental geometric principles.