3 Parallel and Perpendicular Lines

3 Parallel and Perpendicular Lines

Parallel and perpendicular lines are fundamental concepts in geometry, essential for understanding the relationships between lines in a coordinate plane. This resource explores key theorems, including the Corresponding Angles Theorem and the Alternate Interior Angles Theorem, which are crucial for identifying parallel lines. It also covers methods for constructing perpendicular lines and finding distances from points to lines. Ideal for high school geometry students, this guide provides clear explanations, examples, and exercises to reinforce learning. Topics include slope calculations, angle relationships, and practical applications in real-life scenarios.

Key Points

  • Explains the properties of parallel and perpendicular lines in geometry.
  • Covers key theorems such as the Corresponding Angles Theorem and Alternate Interior Angles Theorem.
  • Includes step-by-step methods for constructing perpendicular lines and finding distances from points to lines.
  • Provides exercises and examples for high school geometry students to practice and reinforce concepts.
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3
Parallel and
Perpendicular Lines
3.1 Pairs of Lines and Angles
3.2 Parallel Lines and Transversals
3.3 Proofs with Parallel Lines
3.4 Proofs with Perpendicular Lines
3.5 Equations of Parallel and Perpendicular Lines
Tree House (p. 130)
Kiteboarding (p. 143)
Crosswalk (p. 154)
Bike Path (p. 161)
Gymnastics (p. 130)
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(p. 130)
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SEE the Big Idea
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123
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Finding the Slope of a Line
Example 1 Find the slope of the line shown.
Let
(
x
1
, y
1
)
= (−2, −2) and
(
x
2
, y
2
)
= (1, 0).
slope =
y
2
− y
1
—
x
2
− x
1
Write formula for slope.
=
0 − (−2)
—
1 − (−2)
Substitute.
=
2
—
3
Simplify.
Find the slope of the line.
1.
x
y
3
1
1−1−3
(−1, 2)
(3, −1)
−3
2.
x
y
4
2
−2
4
2−2−4
(−2, 2)
(−3, −1)
3.
x
y
4
2
−4
4
2−2−4
(−3, −2)
(1, −2)
Writing Equations of Lines
Example 2 Write an equation of the line that passes through the point (−4, 5)
and has a slope of
3
—
4
.
y = mx + b Write the slope-intercept form.
5 =
3
—
4
(−4) + b Substitute
3
—
4
for
m
, −4 for
x
, and 5 for
y
.
5 = −3 + b Simplify.
8 = b Solve for
b
.
So, an equation is y =
3
—
4
x + 8.
Write an equation of the line that passes through the given point and
has the given slope.
4. (6, 1); m = −3 5. (−3, 8); m = −2 6. (−1, 5); m = 4
7. (2, −4); m =
1
—
2
8. (−8, −5); m = −
1
—
4
9. (0, 9); m =
2
—
3
10. ABSTRACT REASONING Why does a horizontal line have a slope of 0, but a vertical line has
anundefi ned slope?
2
x
y
4
2
4
2−2−4
3
(−2, −2)
(1, 0)
Dynamic Solutions available at BigIdeasMath.com
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124 Chapter 3 Parallel and Perpendicular Lines
Mathematical Mathematical
PracticesPractices
Characteristics of Lines in a Coordinate Plane
Mathematically profi cient students use technological tools to explore concepts.
Monitoring ProgressMonitoring Progress
Use a graphing calculator to graph the pair of lines. Use a square viewing window. Classify the lines
as parallel, perpendicular, coincident, or nonperpendicular intersecting lines. Justify your answer.
1. x + 2y = 2 2. x + 2y = 2 3. x + 2y = 2 4. x + 2y = 2
2x − y = 4 2x + 4y = 4 x + 2y = −2 x − y = −4
Classifying Pairs of Lines
Here are some examples of pairs of lines in a coordinate plane.
a. 2x + y = 2 These lines are not parallel b. 2x + y = 2 These lines are coincident
x − y = 4 or perpendicular. They 4x + 2y = 4 because their equations
intersect at (2, −2). are equivalent.
6
−4
−6
4
6
−4
−6
4
c. 2x + y = 2 These lines are parallel. d. 2x + y = 2 These lines are perpendicular.
2x + y = 4 Each line has a slope x − 2y = 4 They have slopes of m
1
= −2
of m = −2. and m
2
=
1
—
2
.
6
−4
−6
4
6
−4
−6
4
Lines in a Coordinate Plane
1. In a coordinate plane, two lines are parallel if and only if they are both vertical lines
or they both have the same slope.
2. In a coordinate plane, two lines are perpendicular if and only if one is vertical and the
other is horizontal or the slopes of the lines are negative reciprocals of each other.
3. In a coordinate plane, two lines are coincident if and only if their equations
are equivalent.
Core Core ConceptConcept
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FAQs of 3 Parallel and Perpendicular Lines

What are the key properties of parallel lines?
Parallel lines are defined as lines in the same plane that never intersect, maintaining a constant distance apart. In a coordinate plane, two distinct nonvertical lines are parallel if they have the same slope. This means that if one line has a slope of m, any line parallel to it will also have a slope of m. Additionally, any two vertical lines are considered parallel, as they run in the same direction and do not meet.
How do you determine if two lines are perpendicular?
Two lines are considered perpendicular if the product of their slopes equals -1. This means that if one line has a slope of m1, the other line must have a slope of m2 such that m1 * m2 = -1. For example, if one line has a slope of 2, the perpendicular line must have a slope of -1/2. Additionally, horizontal lines are perpendicular to vertical lines, as their slopes are 0 and undefined, respectively.
What is the significance of the Alternate Interior Angles Theorem?
The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. This theorem is significant because it provides a method for proving lines are parallel based on angle relationships. For example, if you can show that the alternate interior angles formed by a transversal are equal, you can conclude that the lines are parallel, which is a fundamental concept in geometry.
How can you find the distance from a point to a line?
To find the distance from a point to a line, you first need to determine the equation of the line perpendicular to the given line that passes through the point. Then, you find the intersection of these two lines. Finally, use the Distance Formula to calculate the distance between the point and the intersection point. This method ensures that you are measuring the shortest distance, which is always along the perpendicular segment.

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