## Unit 3 Parallel & Perpendicular Lines Answer Key

### Introduction

Unit 3 of the parallel and perpendicular lines is an essential part of geometry. It covers various concepts and properties related to parallel and perpendicular lines. To help you understand and practice these concepts effectively, we have prepared an answer key for Unit 3. In this article, we will provide a detailed answer key for each exercise and problem in Unit 3, along with explanations and examples.

### Exercise 3.1: Identifying Parallel and Perpendicular Lines

In this exercise, you are given multiple line segments and asked to identify whether they are parallel or perpendicular to each other. To determine this, you need to analyze the slope of each line segment. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals, the lines are perpendicular. Let's look at an example:

Example:

Line a: y = 2x + 3

Line b: y = -1/2x - 2

Line c: y = -2x + 5

Line d: y = 1/2x + 4

In this example, line a and line b have slopes that are negative reciprocals of each other, making them perpendicular. Line c and line d have equal slopes, making them parallel. By using this method, you can identify parallel and perpendicular lines in any given set of line segments.

### Exercise 3.2: Finding Equations of Parallel and Perpendicular Lines

In this exercise, you are given a line segment and asked to find the equation of a line that is either parallel or perpendicular to it. To determine the equation, you need to use the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept. Let's look at an example:

Example:

Line a: y = 2x + 3

Find the equation of a line parallel to line a and passing through the point (4, 5).

To find the equation of a line parallel to line a, we need to use the same slope. So, the slope of the new line will also be 2. By substituting the point (4, 5) into the slope-intercept form, we can find the value of b. The equation of the new line parallel to line a is y = 2x - 3.

### Exercise 3.3: Applying Parallel and Perpendicular Lines

In this exercise, you are given a scenario or problem that requires the application of parallel and perpendicular lines. These problems can range from finding the distance between two parallel lines to determining the equation of a line perpendicular to two given lines. Let's look at an example:

Example:

Line a: y = 2x + 3

Line b: y = -1/2x - 2

Find the distance between line a and line b.

To find the distance between parallel lines, we need to find the perpendicular distance between any point on one line to the other line. We can use the formula for the perpendicular distance between a point and a line to solve this problem. By substituting the values of the given lines into the formula, we can find the distance between line a and line b.

### Exercise 3.4: Proving Parallel and Perpendicular Lines

In this exercise, you are given a set of line segments or angles and asked to prove whether they are parallel or perpendicular. To prove this, you need to apply the properties and theorems related to parallel and perpendicular lines. Let's look at an example:

Example:

Line a: y = 2x + 3

Line b: y = -1/2x - 2

Prove that line a and line b are perpendicular.

To prove that line a and line b are perpendicular, we can use the property that states if the product of the slopes of two lines is -1, then the lines are perpendicular. By calculating the slopes of line a and line b and multiplying them, we can prove that they are perpendicular.

### Exercise 3.5: Solving Real-World Problems

In this exercise, you are given real-world problems that require the application of parallel and perpendicular lines. These problems can include finding the slope of a road, determining the angle between two intersecting lines, or calculating the length of a diagonal. Let's look at an example:

Example:

A ladder is leaning against a wall, making an angle of 60 degrees with the ground. If the bottom of the ladder is 6 feet away from the wall, how long is the ladder?

To solve this problem, we need to use trigonometry and the knowledge of parallel and perpendicular lines. By applying the properties of a right triangle and using the sine function, we can find the length of the ladder.

### Conclusion

Unit 3 of parallel and perpendicular lines is a crucial part of geometry, and understanding its concepts and properties is essential. By using the answer key provided in this article, you can practice and reinforce your knowledge of parallel and perpendicular lines. Remember to apply the various methods, formulas, and theorems discussed in each exercise to solve problems effectively. Geometry is all around us, and by mastering the concepts of parallel and perpendicular lines, you can better understand and appreciate the world of shapes and angles.