3.6 PARALLEL LINES IN THE COORDINATE PLANE 1 m = GOAL SLOPE OF PARALLEL LINES y2 - y1 m = x2 - x1 EXAMPLE 1

Extra Example 1 The Cog Railway covers about 3.1 miles and gains about 3600 feet of altitude. What is the average slope of the track?

m = m = = When you use the formula for the slope, y2 - y1 m = x2 - x1 m = run change in x rise change in y = y2 - y1 x2 - x1 Subtraction order is the same the numerator and denominator must use the same subtraction order. CORRECT x1 - x2 y2 - y1 Subtraction order is different INCORRECT The order of subtraction is important. You can label either point as (x1, y1) and the other point as (x2, y2). However, both the numerator and denominator must use the same order. numerator y2 - y1 denominator x2 - x1 EXAMPLE 2

Extra Example 2 Find the slope of a line that passes through the points (–3, 0) and (4, 7).

Vertical lines are parallel. POSTULATE In the coordinate plane, nonvertical lines are parallel if and only if they have the same slope. Vertical lines are parallel. EXAMPLE 3

Extra Example 3 Find the slope of each line. EXAMPLE 4

Extra Example 4 Line p1 passes through (0, –3) and (1, –2). Line p2 passes through (5, 4) and (–4, –4). Line p3 passes through (–6, –1) and (3, 7). Find the slope of each line. Which lines are parallel?

Checkpoint Line k1 passes through (8, –1) and (–5, –9). Line k2 passes through (–6, –5) and (7, 3). Line k3 passes through (10, –4) and (–3, –4). Find the slope of each line. Which lines are parallel?

We will write equations in slope-intercept form: 3.6 PARALLEL LINES IN THE COORDINATE PLANE GOAL 2 WRITING EQUATIONS OF PARALLEL LINES We will write equations in slope-intercept form: EXAMPLE 5

Extra Example 5 Write an equation of the line through the point (4, 9) that has a slope of –2.

Checkpoint Write an equation of the line through the point (20, 5) that has a slope of EXAMPLE 6

Extra Example 6 Line k1 has the equation Line k2 is parallel to k1 and passes through the point (–5, 0). Write an equation of k2.

Checkpoint Line m1 has the equation y = 3x – 7. Line m2 is parallel to m1 and passes through the point (–2, 1). Write an equation of m2.

QUESTION: What are the six methods we have available to prove two lines are parallel? ANSWER: 1-3: Show alternate interior angles, alternate exterior angles, or corresponding angles are congruent. 4: Show consecutive interior angles are supplementary. 5: Show that the lines are perpendicular to the same line. 6: Show that the lines are parallel to the same line.