CHAPTER 4
Parallels
Section 4-1
Parallel Lines and Planes
Parallel Lines

Two lines are parallel if and
only if they are in the same
plane and do not intersect.
Parallel Planes

Planes that do not intersect.
Skew Lines

Two lines that are not in the
same plane are skew if and
only if they do not intersect.
Section 4-2
Parallel Lines and Transversals
Transversal

In a plane, a line is a transversal
if and only if it intersects two or
more lines, each at a different
point.
Alternate Interior Angles

Interior angles that are on
opposite sides of the transversal
Consecutive Interior Angles
Interior angles that are on the
same side of the transversal.
 Also called, same-side interior
angles.

Alternate Exterior Angles

Exterior angles that are on
opposite sides of the
transversal.
Theorem 4-1

If two parallel lines are cut by a
transversal, then each pair of
alternate interior angles is
congruent.
Theorem 4-2

If two parallel lines are cut by a
transversal, then each pair of
consecutive interior angles is
supplementary.
Theorem 4-3

If two parallel lines are cut by a
transversal, then each pair of
alternate exterior angles is
congruent.
Section 4-3
Transversals and
Corresponding Angles
Corresponding Angles
Have different vertices
 Lie on the same side of the
transversal
 One angle is interior and one
angle is exterior

Postulate 4-1

If two parallel lines are cut by a
transversal, then each pair of
corresponding angles is
congruent.
Theorem 4-4

If a transversal is perpendicular
to one of two parallel lines, it is
perpendicular to the other.
Section 4-4
Proving Lines Parallel
Postulate 4-2

In a plane, if two lines are cut
by a transversal so that a pair of
corresponding angles is
congruent, then the lines are
parallel.
Theorem 4-5

In a plane, if two lines are cut
by a transversal so that a pair of
alternate interior angles is
congruent, then the two lines are
parallel.
Theorem 4-6

In a plane, if two lines are cut
by a transversal so that a pair of
alternate exterior angles is
congruent, then the two lines are
parallel.
Theorem 4-7

In a plane, if two lines are cut
by a transversal so that a pair of
consecutive interior angles is
supplementary, then the two
lines are parallel.
Theorem 4-8

In a plane, if two lines are
perpendicular to the same line,
then the two lines are parallel.
Section 4-5
Slope
Slope
The slope m of a line
containing two points with
coordinates (x1, y1) and
(x2, y2) is given by the formula
m =y2 – y1
x2 – x1

Vertical Line

The slope of a vertical line is
undefined.
Postulate 4-3

Two distinct non-vertical lines
are parallel if and only if they
have the same slope.
Postulate 4-4

Two non-vertical lines are
perpendicular if and only if the
product of their slopes is -1.
Section 4-6
Equations of Lines
Linear Equation
An equation whose graph is
a straight line.

Y-Intercept
The y-value of the point
where the lines crosses the yaxis.

Slope-Intercept Form
An equation of the line having
slope m and y-intercept b is
y = mx + b.

Examples
Name the slope and y-intercept
of each line

y = 1/2x + 5

y=3

x = -2

2x – 3y = 18
Examples


Graph each equation
2x + y = 3
-x + 3y = 9
Examples
Write an equation of each line
 Passes through ( 8, 6) and (-3, 3)
Parallel to y = 2x – 5 and
through the point (3, 7)
Perpendicular to y = 1/4x + 5
and through the point (-3, 8)