Parallel & Perpendicular Lines
Chapter 3
Parallel Lines & Transversals
Section 3.1
Vocabulary
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Parallel lines
Parallel planes
Skew lines
Transversal
Consecutive interior angles
Alternate interior angles
Alternate exterior angles
Corresponding angles
Example 1
Example 2
Identify the sets of lines to which each line is a transversal.
Angle Relationships
Example 3
Identify each pair of angles as alternate interior, alternate
exterior, corresponding, or consecutive interior angles.
Angles & Parallel Lines
Section 3.2
Postulates & Theorems
3.1 – Corresponding Angles Postulate - If 2 parallel lines are
cut by a transversal, then each pair of corresponding
angles is congruent.
Example 1
Example 1
Theorems
3.1 – 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 1
Find the slope of the line containing (-6, -2) and (3, -5).
Example 1
Find the slope of the line containing (8, -3) and (-6, -2).
Example 2
Between 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 2
In 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?
Postulates
3.2 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 3
Determine whether line AB and line CD are parallel,
perpendicular or neither.
A(-2, -5), B(4, 7), C(0, 2), D(8, -2)
Example 3
Determine whether line AB and line CD are parallel,
perpendicular or neither.
A(-8, -7), B(4, -4), C(-2, -5), D(1, 7)
Example 3
Determine whether line AB and line CD are parallel,
perpendicular or neither.
A(14, 13), B(-11, 0), C(-3, 7), D(-4, -5)
Example 3
Determine whether line AB and line CD are parallel,
perpendicular or neither.
A(3, 6), B(-9, 2), C(-12, -6), D(15, 3)
Example 4
Graph the line that
contains P(-2, 1) and is
perpendicular to
line JK with J(-5, -4)
and K(0, -2).
Example 4
Graph 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 1
Write an equation in slope-intercept form of the line with
slope of -4 and y-intercept of 1.
Example 2
Write an equation in point-slope form of the line with a
slope of -1/2 that contains (3, -7).
Example 3
Write an equation in slope-intercept form for line l.
Example 3
Write an equation in slope-intercept form for the line that
contains (-2, 4) and (8, 10).
Example 4
Write an equation in slope-intercept form for a line
containing (2, 0) that is perpendicular to the line with
equation y = -x + 5.
Example 4
Write 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 5
Gracia’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
Postulates
3.4 - 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.
Theorems
3.5 – 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 Parallel
Given: r ∥ s; ∡5≅∡6
Prove: l ∥ m
Example 4
Determine whether g ∥ f.
Example 4
Line 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 1
Draw 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 3
Fine the distance between the parallel lines l and n with
equations y = -1/3x-3 and y = -1/3x + 1/3 respectively.
Example 3
Fine the distance between the parallel lines a and b with
equations x + 3y = 6 and x + 3y = -14 respectively.