1. Section 6.6 What we are Learning:
To determine if two lines are parallel or perpendicular by their slopes
To write equations of lines that pass through a given point, parallel or perpendicular to the graph of a given equation

2. Parallel Lines:
Lines that are in the same plane and that never intersect
Lines in a coordinate plane that have the same slope
All vertical lines in a coordinate plane are parallel
We use the symbol ∥ to say two lines are parallel

3. Parallelogram A quadrilateral in which opposite sides are parallel
We can use slope to determine if a quadrilateral is a parallelogram
If we are given the vertices:
First, graph the points of the vertices and connect
Second, find the slope of each side
Third, determine if opposite sides have the same slope

4. Parallelogram Example:
Determine whether quadrilateral ABCD is a parallelogram. Its vertices are A(-5, 4), B(-3, 8), C(4, 1), D(2, -3) B
First, graph the vertices A C D

5. Parallelogram Example (cont):
Second, use slope to find the slopes of the sides
AB is opposite CD, their slopes are the same
BC is opposite AD, their slopes are the same
Quadrilateral ABCD is a parallelogram
AB m =
4 - 8
BC m =
8 – 1
CD m =
1 – (-3)
AD m =
4 – (-3)
-5 – (-3)
-3 - 4
4 – 2
-5 - 2 -4
= 2 7
= -1 4 -2 -7 2

6. Finding an Equation for a Parallel Line Given an Equation and a Point:
First, rewrite the given equation in Slope-Intercept form to find the slope of the given equation
Second, use Point-Slope Form or Slope-Intercept Form to find the equation of the parallel line using the given point

7. Example:
Write the equation of the line that is parallel to 2x + 3y = -1 and passes through (-5, -4)
First, rewrite the equation is slope-intercept form
m = -2/3
2x + 3y = -1
-2x x
3y = -2x – 1 3
y = -2/3x – 1/3

8. Example (cont): point (-5, -4), m = -2/3
Point-Slope Form
Slope-Intercept Form
y – y1 = m(x – x1)
y – (-4) = -2/3(x – (-5))
y + 4 = -2/3(x + 5)
y + 4 = -2/3x – 10/3
y + 4 – 4 = -2/3x – 10/3 – 4
y = -2/3x – 10/3 – 12/3
y = -2/3x – 22/3
y = mx + b
-4 = -2/3(-5) + b
-4 = 10/3 + b
-4 – 10/3 = 10/3 – 10/3 + b
-12/3 – 10/3 = b
-22/3 = b
y = -2/3x – 22/3

9. Perpendicular Lines: Lines that intersect at right angles
In a coordinate plane the slopes of perpendicular lines are negative reciprocals of each other
Flip the slope over and change the sign
The product of the slopes of perpendicular lines is -1
Horizontal and vertical lines are perpendicular

10. Finding an Equation for a Perpendicular Line Given an Equation and a Point:
First, rewrite the given equation in Slope-Intercept form to find the slope of the given equation
Second, use Point-Slope Form or Slope-Intercept Form to find the equation of the parallel line using the given point

11. Example:
Write an equation for a line that is perpendicular to 5x – 7 = 3y and passes through (8, -2).
First rewrite equation in Slope-Intercept Form
m = 5/3
5x – 7 = 3y
5/3x – 7/3 = y

12. Example (cont): point (8, -2), m = 5/3
Point-Slope Form
Slope-Intercept Form
y – y1 = m(x – x1)
y – (-2) = 5/3(x – 8)
y + 2 = 5/3x – 40/3
y + 2 – 2 = 5/3x – 40/3 – 2
y = 5/3x – 40/3 – 6/3
y = 5/3x – 46/3
y = mx + b
-2 = 5/3(8) + b
-2 = 40/3 + b
-2 – 40/3 = 40/3 – 40/3 + b
-6/3 – 40/3 = b
-46/3 = b
y = 5/3x – 46/3

13. Let’s Work These Together:
Determine if the equations are parallel, perpendicular, or neither
y = - 2x + 11
y + 2x = 23
y = -5x
y = 5x + 18
y = -2/3x – 3
-3x + 2y = 10

14. Let’s Work This Together:
Write the equation of a line that is parallel to 2x + y = -2 and passes through (2, -7)

15. Let’s Work This Together:
Write the equation of a line perpendicular to 3y + x = 3 and passes through (6, -1)

16. Homework:
Page 367:
19 to 41 odd
Determine if quadrilateral ABCD is a parallelogram if its vertices are A(-7, 1), B(-2, -1), C(7, 4), D(2, 6)