CHAPTER 4 Parallels

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 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.

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

Slope  The slope m of a line containing two points with coordinates (x 1, y 1 ) and (x 2, y 2 ) is given by the formula m =y 2 – y 1 x 2 – x 1

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 Equation  An equation whose graph is a straight line.

Y-Intercept  The 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)