3 Parallel and Perpendicular Lines - Big Ideas Learning

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

Bike Path (p. 161)

Kiteboarding (p. 143) SEE the Big Idea Tree House (p. 130)

Crosswalk (p. 154) Gymnastics (p. 130)

Maintaining Mathematical Proficiency

Finding the Slope of a Line

Example 1 Find the slope of the line shown.

Let ( x1, y1 ) = (-2, -2) and ( x2, y2 ) = (1, 0).

slope = -- xy22 -- yx11 = -- 10 -- ((--22))

Write formula for slope. Substitute.

= --32

Simplify.

Find the slope of the line.

1.

y

3

(-1, 2) 1

-3 -1 1

x

(3, -1)

-3

2.

4y

(-2, 2) 2

-4 -2 -2

(-3, -1)

2 4x

y 4

2 (1, 0)

-4 -2

2 4x

2

3

(-2, -2)

3.

4y

2

-4 -2

2 4x

(-3, -2) (1, -2)

-4

Writing Equations of Lines

Example 2 Write an equation of the line that passes through the point (-4, 5) and has a slope of --34.

y = mx + b 5 = --34 (-4) + b 5 = -3 + b

Write the slope-intercept form. Substitute --34 for m, -4 for x, and 5 for y. Simplify.

8 = b

Solve for b.

So, an equation is y = --34 x + 8.

Write an equation of the line that passes through the given point and has the given slope.

4. (6, 1); m = -3 7. (2, -4); m = --12

5. (-3, 8); m = -2 8. (-8, -5); m = ---14

6. (-1, 5); m = 4 9. (0, 9); m = --23

10. ABSTRACT REASONING Why does a horizontal line have a slope of 0, but a vertical line has an undefined slope?

Dynamic Solutions available at 123

MPraatchteicmeastical

Mathematically proficient students use technological tools to explore concepts.

Characteristics of Lines in a Coordinate Plane

Core Concept

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.

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 x - y = 4 or perpendicular. They intersect at (2, -2).

4

b. 2x + y = 2 These lines are coincident 4x + 2y = 4 because their equations are equivalent.

4

-6

6

-6

6

-4

c. 2x + y = 2 These lines are parallel. 2x + y = 4 Each line has a slope of m = -2.

4

-4

d. 2x + y = 2 x - 2y = 4

These lines are perpendicular.

They have slopes and m2 = --12.

of

m1

=

-2

4

-6

6

-6

6

-4

-4

Monitoring 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 2x - y = 4

2. x + 2y = 2 2x + 4y = 4

3. x + 2y = 2 x + 2y = -2

4. x + 2y = 2 x - y = -4

124 Chapter 3 Parallel and Perpendicular Lines

3.1

Pairs of Lines and Angles

Essential Question What does it mean when two lines are parallel,

intersecting, coincident, or skew?

Points of Intersection

Work with a partner. Write the number of points of intersection of each pair of coplanar lines.

a. parallel lines

b. intersecting lines

c. coincident lines

CONSTRUCTING VIABLE ARGUMENTS

To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.

Classifying Pairs of Lines

Work with a partner. The figure shows a right rectangular prism. All its angles are right angles. Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers. (Two lines are skew lines when they do not intersect and are not coplanar.)

B A

F

E

I

C D

G H

Pair of Lines

a. AB and BC b. AD and BC c. EI and IH d. BF and EH e. EF and CG f. AB and GH

Classification

Reason

Identifying Pairs of Angles

Work with a partner. In the figure, two parallel lines are intersected by a third line called a transversal.

a. Identify all the pairs of vertical angles. Explain your reasoning.

b. Identify all the linear pairs of angles. Explain your reasoning.

12 43

5

6 8

7

Communicate Your Answer

4. What does it mean when two lines are parallel, intersecting, coincident, or skew?

5. In Exploration 2, find three more pairs of lines that are different from those given. Classify the pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers.

Section 3.1 Pairs of Lines and Angles 125

3.1 Lesson

Core Vocabulary

parallel lines, p. 126 skew lines, p. 126 parallel planes, p. 126 transversal, p. 128 corresponding angles,

p. 128 alternate interior angles,

p. 128 alternate exterior angles,

p. 128 consecutive interior angles,

p. 128 Previous perpendicular lines

REMEMBER

Recall that if two lines intersect to form a right angle, then they are perpendicular lines.

What You Will Learn

Identify lines and planes. Identify parallel and perpendicular lines. Identify pairs of angles formed by transversals.

Identifying Lines and Planes

Core Concept

Parallel Lines, Skew Lines, and Parallel Planes

Two lines that do not intersect are either parallel lines or skew lines. Two lines are parallel lines when they do not intersect and are coplanar. Two lines are skew lines when they do not intersect and are not coplanar. Also, two planes that do not intersect are parallel planes.

k

Lines m and n are parallel lines (m n).

m

Lines m and k are skew lines.

T

n

Planes T and U are parallel planes (T U ).

Lines k and n are intersecting lines, and there

U

is a plane (not shown) containing them.

Small directed arrows, as shown in red on lines m and n above, are used to show that lines are parallel. The symbol means "is parallel to," as in m n.

Segments and rays are parallel when they lie in parallel lines. A line is parallel to a plane when the line is in a plane parallel to the given plane. In the diagram above, line n is parallel to plane U.

Identifying Lines and Planes

Think of each segment in the figure as part of a line. Which line(s) or plane(s) appear to fit the description?

a. line(s) parallel to CD and containing point A b. line(s) skew to CD and containing point A c. line(s) perpendicular to CD and containing point A

d. plane(s) parallel to plane EFG and containing point A

C D

F E

B A

G H

SOLUTION

a. AB, HG, and EF all appear parallel to CD, but only AB contains point A. b. Both AG and AH appear skew to CD and contain point A. c. BC, AD, DE, and FC all appear perpendicular to CD, but only AD contains point A.

d. Plane ABC appears parallel to plane EFG and contains point A.

Monitoring Progress

Help in English and Spanish at

1. Look at the diagram in Example 1. Name the line(s) through point F that appear

skew to EH.

126 Chapter 3 Parallel and Perpendicular Lines

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