concepts, and examples Lesson Objectives: I will be able to … Identify and graph parallel and perpendicular lines Write equations to describe lines parallel or perpendicular to a given line Language Objective: I will be able to … Read, write, and listen about vocabulary, key concepts, and examples

Page 29

Example 1: Identifying Parallel Lines Page 30 Example 1: Identifying Parallel Lines Identify which lines are parallel. The lines described by and both have slope . These lines are parallel. The lines described by y = x and y = x + 1 both have slope 1. These lines are parallel.

Example 2: Identifying Parallel Lines Page 31 Example 2: Identifying Parallel Lines Identify which lines are parallel. Step 1: Write each equation in slope-intercept (y = mx + b) form. 1. y = 2x – 3 2. 3. 2x + 3y = 8 4. y + 1 = 3(x – 3) √ slope = 2 slope = √ slope = 3y = -2x + 8 √ y + 1 = 3x – 9 y = 3x – 10 √ y=2x–3 y+1=3(x–3) slope = 3 Step 2: Find the slope of each line. Step 3: Determine whether the lines are parallel. and 2x + 3y = 8 are parallel because both have a slope of .

Identify which lines are parallel. Your Turn 2 Page 31 Identify which lines are parallel. Step 1: Write each equation in slope-intercept (y = mx + b) form. slope = 1. 2. -3x + 4y = 32 3. y = 3x 4. y – 1 = 3(x + 2) √ slope = 4y = 3x + 32 √ √ slope = 3 √ slope = 3 y – 1 = 3x + 6 y = 3x + 7 Step 2: Find the slope of each line. y–1=3(x+2) –3x+4y=32 y=3x Step 3: Determine whether the lines are parallel. y = 3x and y – 1 = 3(x + 2) are parallel because both have a slope of 3. (Note that and -3x + 4y = 32 are not parallel; they are the same line!)

Example 3: Geometry Application Page 32 Example 3: Geometry Application Show that JKLM is a parallelogram. Use the ordered pairs and the slope formula to find the slopes of MJ and KL. MJ is parallel to KL because they have the same slope. JK is parallel to ML because they are both horizontal. Since opposite sides are parallel, JKLM is a parallelogram.

AD is parallel to BC because they have the same slope. Your Turn 3 Page 32 Show that the points A(0, 2), B(4, 2), C(1, –3), D(–3, –3) are the vertices of a parallelogram. Use the ordered pairs and slope formula to find the slopes of AD and BC. • A(0, 2) B(4, 2) D(–3, –3) C(1, –3) AD is parallel to BC because they have the same slope. AB is parallel to DC because they are both horizontal. Since opposite sides are parallel, ABCD is a parallelogram.

Page 30

Page 30 Helpful Hint If you know the slope of a line, the slope of a perpendicular line will be the "opposite reciprocal.”

Example 4: Identifying Perpendicular Lines Page 33 Identify which lines are perpendicular: y = 3; x = –2; y = 3x; . y = 3 is a horizontal line, and x = –2 is a vertical line. These lines are perpendicular. x = –2 y = 3 The slope of the line described by y = 3x is 3. The slope of the line described by is . y =3x 3 and are opposite reciprocals, so these lines are perpendicular.

Your Turn 4 Identify which lines are perpendicular: Page 33 Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = x = 3 x = 3 is a vertical line, and y = –4 is a horizontal line. These lines are perpendicular. The slope of the line described by y – 6 = 5(x + 4) is 5. The slope of the line described by y = is 5 and are opposite reciprocals, so these lines are perpendicular. y = –4 y – 6 = 5(x + 4)

Example 5: Geometry Application Page 34 Example 5: Geometry Application Show that ABC is a right triangle. If ABC is a right triangle, AB will be perpendicular to AC. slope of slope of AB is perpendicular to AC because Therefore, ABC is a right triangle because it contains a right angle.

Example 6: Writing Equations of Parallel and Perpendicular Lines Page 34 Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8. Step 1 Find the slope of the line. Step 2 Write the equation in point-slope form. y = 3x + 8 The slope is 3. y – y1 = m(x – x1) Use the point-slope form. The parallel line also has a slope of 3. y – 10 = 3(x – 4) Step 3 Write the equation in slope-intercept form. y – 10 = 3(x – 4) Distribute 3 on the right side. y – 10 = 3x – 12 Add 10 to both sides. y = 3x – 2

Example 7: Writing Equations of Parallel and Perpendicular Lines Page 35 Example 7: Writing Equations of Parallel and Perpendicular Lines Write an equation in slope-intercept form for the line that passes through (2, –1) and is perpendicular to the line described by y = 2x – 5. Step 1 Find the slope of the line. Step 2 Write the equation in point-slope form. y = 2x – 5 The slope is 2. The perpendicular line has a slope of . y – y1 = m(x – x1) Step 3 Write the equation in slope-intercept form.

Your Turn 7 Page 36 Write an equation in slope-intercept form for the line that passes through (–5, 3) and is perpendicular to the line described by y = 5x. Step 1 Find the slope of the line. Step 2 Write the equation in point-slope form. y = 5x The slope is 5. y – y1 = m(x – x1) The perpendicular line has a slope of . Step 3 Write in slope-intercept form.

Chapter 5 Test in two classes! Homework Assignment #19.5 Holt 5-8 #9-15 odd, 22-24, 26, 34-36 Chapter 5 Test in two classes!