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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)