CHAPTER 4Parallels

Parallel Lines and Planes

Section 4-1

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.

Parallel Lines and Transversals

Section 4-2

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.

Transversals and Corresponding Angles

Section 4-3

Corresponding Angles

Have different verticesLie 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.

Proving Lines Parallel

Section 4-4

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.

Slope

Section 4-5

SlopeThe 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.

Equations of Lines

Section 4-6

Linear EquationAn equation whose graph is a straight line.

Y-InterceptThe y-value of the point where the lines crosses the y-axis.

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)