Parallel & Perpendicular Lines

Chapter 3

Parallel Lines & Transversals

Section 3.1

Vocabulary Parallel lines Parallel planes Skew lines Transversal Consecutive interior angles Alternate interior angles Alternate exterior angles Corresponding angles

Example 1

Example 2Identify the sets of lines to which each line is a

transversal.

Angle Relationships

Example 3Identify each pair of angles as alternate interior,

alternate exterior, corresponding, or consecutive interior angles.

Angles & Parallel Lines

Section 3.2

Postulates & Theorems3.1 – Corresponding Angles Postulate - If 2

parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

Example 1

Example 1

Theorems3.1 – Alternate Interior Angles – If two parallel

lines are cut by a transversal then each pair of alternate interior angles is congruent.

3.2 – Consecutive Interior Angles – If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary.

3.3 – Alternate Exterior Angles – If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

3.4 – Perpendicular Transversal Theroem – In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Example 3

Example 3

Example 2 – Using an auxiliary line

Example 2 – Using an auxiliary line

Slopes of Lines

Section 3.3

Example 1

Example 1

Example 1

Example 1

Example 1Find the slope of the line containing (-6, -2) and

(3, -5).

Example 1Find the slope of the line containing (8, -3) and

(-6, -2).

Example 2Between 2000 and 2003, annual sales of

exercise equipment increased by an average rate of $314.3 million per year. In 2003, the total sales were $4553 million. If sales of fitness equipment increase at the same rate, what will the total sales be in 2010?

Example 2In 2004, 200 million songs were legally

downloaded from the Internet. In 2003, 20 million songs were legally downloaded. If this increases at the same rate, how many songs will be legally downloaded in 2008?

Postulates3.2 Parallel Lines - Two nonvertical lines have

the same slope if and only if they are parallel. 3.3 Perpendicular Lines – Two nonvertical lines

have are perpendicular if and only if the product of their slopes is -1. *Remember opposite reciprocals*

Example 3Determine whether line AB and line CD are

parallel, perpendicular or neither. A(-2, -5), B(4, 7), C(0, 2), D(8, -2)

Example 3Determine whether line AB and line CD are

parallel, perpendicular or neither. A(-8, -7), B(4, -4), C(-2, -5), D(1, 7)

Example 3Determine whether line AB and line CD are

parallel, perpendicular or neither. A(14, 13), B(-11, 0), C(-3, 7), D(-4, -5)

Example 3Determine whether line AB and line CD are

parallel, perpendicular or neither. A(3, 6), B(-9, 2), C(-12, -6), D(15, 3)

Example 4Graph the line that contains P(-2, 1) and is perpendicular to line JK with J(-5, -4) and K(0, -2).

Example 4Graph the line that contains P(0, 1) and is perpendicular to line QR with Q(-6, -2) and R(0, -6).

Equations of Lines

Section 3.4

Example 1Write an equation in slope-intercept form of the

line with slope of -4 and y-intercept of 1.

Example 2Write an equation in point-slope form of the line

with a slope of -1/2 that contains (3, -7).

Example 3Write an equation in slope-intercept form for

line l.

Example 3Write an equation in slope-intercept form for the

line that contains (-2, 4) and (8, 10).

Example 4Write an equation in slope-intercept form for a

line containing (2, 0) that is perpendicular to the line with equation y = -x + 5.

Example 4Write an equation in slope-intercept form for a

line containing (-3, 6) that is parallel to the line with equation y = -3/4x + 3.

Example 5Gracia’s current wireless phone plan charges

$39.95 per month for unlimited calls and $0.05 per text message. Write an equation to represent the total monthly cost C for t text messages.

Proving Lines Parallel

Lesson 3.5

Postulates3.4 - If two lines are cut by a transversal so that

corresponding angles are congruent, then the lines are parallel.

3.5 – Parallel Postulate – If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

Theorems3.5 – If two lines are cut by a transversal so that

a pair of alternate exterior angles is congruent, then the two lines are parallel.

3.6 – If two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel.

3.7 – If two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.

3.8 – If two lines are perpendicular to the same line, then they are parallel.

Example 1

Example 1

Example 2

Example 2

Example 3 – PROVING Lines ParallelGiven: r ∥ s; ∡5≅∡6Prove: l ∥ m

Example 4Determine whether g ∥ f.

Example 4Line e contains points at (-5, 3) and (0, 4). Line

m contains points at (2, -2/3) and (12, 1). Determine whether the lines are parallel.

Perpendiculars & Distance

Section 3.6

Distance between a point & a line

Example 1Draw the segment that represents the distance

from P to line AB.

Theorem 3.9 – If two lines are equidistant from a third

line, then the two lines are parallel to each other.

Example 3Fine the distance between the parallel lines l

and n with equations y = -1/3x-3 and y = -1/3x + 1/3 respectively.

Example 3Fine the distance between the parallel lines a

and b with equations x + 3y = 6 and x + 3y = -14 respectively.