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Parallel & Perpendicular Lines Chapter 3 Parallel Lines & Transversals Section 3.1 Vocabulary 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.