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**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

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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

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**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

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**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

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**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

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**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

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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

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**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

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**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

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**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

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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

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**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

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**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

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**Let’s Work This Together:**

Write the equation of a line that is parallel to 2x + y = -2 and passes through (2, -7)

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**Let’s Work This Together:**

Write the equation of a line perpendicular to 3y + x = 3 and passes through (6, -1)

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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)

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