GEOMETRY APPLICATIONS

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1 GEOMETRY APPLICATIONS Chapter 3: Parallel & Perpendicular Lines Name: Teacher: Pd: 0

2 Table of Contents DAY 1: (Ch. 3-1 & 3-2) SWBAT: Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal. Pgs: 2-5 DAY 2: (Ch. 3-2) Calculate for missing angles when parallel lines are cut by a transversal Pgs: 6-10 DAY 3: Full Period Quiz: Day 1 to DAY 2 DAY 4: (Ch. 3-5) SWBAT: Calculate the slope of a line using the slope formula. Pgs: DAY 5: SWBAT: Use slopes to identify parallel and perpendicular lines Pgs:16-19 Take Home Quiz: Day 4 to DAY 5 DAY 6: SWBAT: Graph and Write Equations of Lines given a Slope and Point Pgs: DAY 7: SWBAT: Write the equation of a line given two points on the line Pgs: DAY 8: SWBAT: Graph Lines in Slope Intercept and Point Slope Form Pgs: DAY 9: SWBAT: Graph and Write Equations of Parallel & Perpendicular Lines given a Slope and Point Pgs: DAY 10: Full Period Quiz: Day 6 to DAY 9 DAY 11: SWBAT: Graph the Solutions to Quadratic Linear Systems Pgs: DAY 12: SWBAT: Graph the Solutions to Quadratic Linear Systems Pgs: DAY 13: Chapter 3 Practice Test DAY 14: Chapter 3 Test 1

3 Day & 3-2: Lines and Angles SWBAT: Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal. Warm Up: Matching Column supplementary angles point coplanar points linear pair points that lie in the same plane two angles whose sum is 180 the intersection of two distinct intersecting lines a pair of adjacent angles whose non-common sides are opposite rays Example 1: Lines Term Description Example 1 Example(s) are coplanar do not intersect intersect at 90 angles are not coplanar are not parallel do not intersect planes that do not intersect 2

4 Practice: Identify each of the following: a. A pair of parallel segments b. A pair of skew segments c. A pair of perpendicular segments d. A pair of parallel planes Example 2: Angles A is a line that intersects two coplanar lines at two different points. Term Description Example 1 Example(s) Lie on: the same side of the transversal t on the same sides of lines r and s Nonadjacent angles that lie on: opposite sides of the transversal t between lines r and s Lie on: opposite sides of the transversal t outside lines r and s Lie on: the same side of the transversal t between lines r and s 3

Practice Identify each of the following: a. A pair of alternate interior angles b. A pair of corresponding angles c. A pair of alternate exterior angles d.

5 Practice Identify each of the following: a. A pair of alternate interior angles b. A pair of corresponding angles c. A pair of alternate exterior angles d. A pair of same-side interior angles Example 3: Line l and Line m are parallel. Find each missing angle. Practice Line l and Line m are parallel. Find each missing angle. 4

6 Homework: In the diagram, parallel lines AB and CD are intersected by a transversal EF at points X and Y, m FYD = 123. Find AXY. 5

7 Day 2 - Chapter 3 2 (Parallel Lines and Related Angles) SWBAT: Calculate for missing angles when parallel lines are cut by a transversal Warm Up Classify each pair of angles as alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles, or vertical angles ) 2) 3) 2 1 4) 5) 6) State the angle relationship that justifies each statement. 7) m 3 + m 4 = 180 8) 1 5 9) ) ) m 4 + m 5 = 180 Find the m 1 and explain the angle relationship

8 Proving Lines Parallel Perpendicular Lines 18. Find the measure of b. 19. Find x and measure of b. b Algebra Related Questions In the accompanying diagram, m ABC = (4x + 22) and m DCE = (5x). Part a: Which relationship describes ABC and DCE? Part b: What is the value of x and what is m DCE? 7

9 Homework 1) In the accompanying diagram, l ll m and m 1 = (3x + 40) and m 2 = (5x 30). Part a: Which relationship describes 1 and 2? 1 l Part b: What is the value of x and what is m 1? 2 m 2) In the accompanying diagram, l ll m and m 1 = (9x - 8) and m 2 = (x + 72). Part a: Which relationship describes 1 and 2? 1 l Part b: What is the value of x and what is m 2? 2 m 3) In the accompanying diagram, p ll q. Part a: Which relationship describes the given angles? (x + 12) 5(x - 4) p Part b: What is the value of x? q 8

10 4) In the accompanying diagram, p ll q. If m 1 = (4x + 1) and m 2 = (5x 10) p 2 1 q Part a: Which relationship describes 1 and 2? Part b: What is the value of x? Part c: What is the m 2? 5) In the accompanying diagram, l ll m. If m 1 = (3x + 16) and m 2 = (x + 12) Part a: Which relationship describes 1 and 2? 1 l Part b: What is the value of x? 2 m Part c: What is the m 1 and m 2? 9

11 6) Find the m 6. 7) Find the measure of 3, 4, and 5. m 3 = m 4 = m 5= 8) m 1 = m 2 = m 3 = m 4 = m 6 = 10

12 Day 4 - Chapter 3-5 Slope of a Line SWBAT: Calculate the slope of a line using the slope formula. Warm Up Solve for x. The Slope m of a line passing through points (x 1, y 1 ) and (x 2, y 2 ) is the ratio of the difference in the y-coordinates to the corresponding difference in the x-coordinates. y rise run (x 1, y 1 ) Symbols: m = (x 2, y 2 ) x 11

13 Example 1: Find the slope of (3,3) and (8,7). Example 2: Find the slope of (2,3) and (-7,8). Example 3: Find the slope of (-5,3) and (2,3). 12

14 Finding Slope From Graphs and Tables. The graph or table shows a linear relationship. Find the slope. 4) 5) 6) 7) Finding Slope from an Equation 8) Find the slope of the line described by 4x 2y = 16. 9) Find the slope of the line described by 2x + 3y =

15 HOMEWORK: 1) Find the slope of (2, 5) and (8, 1). 2) Find the slope of (5, 7) and (6, 4). 14

16 Finding Slope from an Equation 14. Find the slope of the line described by 6x 3y = Find the slope of the line described by 3x + 4y =

Day 5 - Chapter 3-6: Slopes of Parallel and Perpendicular Lines SWBAT: Use slopes to identify parallel and perpendicular lines Use the slope formula to determine the slope of each line.

17 Day 5 - Chapter 3-6: Slopes of Parallel and Perpendicular Lines SWBAT: Use slopes to identify parallel and perpendicular lines Use the slope formula to determine the slope of each line. Pairs of Lines Parallel Lines Y = 5x + 8 Perpendicular Lines Y = 2x + 6 Neither Y = 3x 5 Coinciding Lines Y = 2x 4 Y = 5x - 4 Same Slope different y- intercept Y = -½x - 4 Slopes are Negative Reciprocals Y = 5x + 2 Different Slopes Y = 2x - 4 Same slope, Same y-intercept 16

18 Example 1 Find the slope of a line parallel to the graph of each equation. a) y = x 1 b) y = 4x - 1 c) 2x - 3y = 2 slope = slope = slope = Independent Practice Find the slope of a line parallel to the graph of each equation. a) y = - 5 x 1 b) y = -3x - 1 c) 4x - 2y = 2 3 slope = slope = slope = Example 2 Find the slope of a line perpendicular to the graph of each equation a) y = 2x + 1 b) y = 7 2 x - 4 c) 4x 2y = 9 slope = slope = slope = Independent Practice Find the slope of a line perpendicular to the graph of each equation a) y = -4x + 1 b) y = 3 x - 4 c) 6x 3y = 9 5 slope = slope = slope = 17

19 Example 3 Determine whether the lines are parallel, perpendicular, coincide, or neither. 3x + 5y = 2 and 3x + 6 = -5y Determine whether the lines are parallel, perpendicular, coincide, or neither. a) y 5 = 2x + 6 and y 3 = ½x b) 2y = 4x + 12 and 4x 2y = 8 c) 2y 4x = 16 and y 10 = 2x - 2 d) y + 3 = ¾x + 16 and 3y = -4x - 9 Regents Question Shanaya graphed the line represented by the equation y = 2x 6. A. Write an equation for a line that is parallel to the given line. B. Write an equation for a line that is perpendicular to the given line. C. Write an equation for a line that is identical to the given line but has different coefficients. Challenge: Determine whether the lines are parallel, perpendicular, coincide, or neither. y (-3) = ¾(x + 16), 3y = -4x

20 Homework: Find the slope of a line parallel to the graph of each equation. a) y = - 8 x 1 b) y = -9x - 1 c) 10x - 2y = 2 3 slope = slope = slope = Find the slope of a line perpendicular to the graph of each equation a) y = -6x + 1 b) y = 4 x - 4 c) 12x 3y = 9 5 slope = slope = slope = Determine whether the lines are parallel, perpendicular, coincide, or neither. 19

21 Day 6 - Chapter 3-6: Equations of Lines Given Slope and Point SWBAT: Graph and Write Equations of Lines given a Slope and Point Warm Up 1. Use the slope formula to determine the slope of the line that passes through A(3, 7) and B(-3, 1). 2. Graph the lines and use the slopes to determine whether they are parallel, perpendicular, or neither. and for A(-2,5) and B(-3, 1), X(0, -2) and Y(1, 2) 20

22 Writing Equations of Lines Example 1 1) Write an equation of a line that passes through the given point with the given slope: ( 1, 2) ; m = 2 Example 2 2) Write an equation of a line that passes through the given point with the given slope: (5, -2) ; m = 21

23 Practice 1) Write an equation of a line that passes through the given point with the given slope: (2, -5) ; m = -2 2) Write an equation of a line that passes through the given point with the given slope: (0, 3) ; m = 1 22

24 3) Write an equation of a line that passes through the given point with the given slope: (1, 2) ; m = -3 4) Write an equation of a line that passes through the given point with the given slope: (-1, 5) ; m = 23

25 Homework Write an equation of a line that passes through the given point with the given slope: 1) (3, 0) ; m = 2) (2, 6) ; m = 3) (3, -1) ; m = 24

26 Day 7 - Chapter 3-6: Equations of Lines Given Two Points SWBAT: Write the equation of a line given two points on the line Warm Up Find the slope of the line passing through the points (6,4) and (-2,-6). Writing Equations of Lines Example 1 Write the equation of the line through the two points (1,1) and (2,3). Example 2 Write the equation of the line through the two points (5,0) and (3,2) 25

27 Practice 1. Write the equation of the line through the two points (8,5) and (9,6) 2. Write the equation of the line through the two points (0,0) and (-3,4) 3. Write the equation of the line through the two points (-3,-4) and (-5,-6) 26

28 Homework Write the equation of the line through the two points. 1. (3,1) and (6,2) 2. (-2,6) and (-4,5) 3. (1,-4) and (-2,8) 4. (-3,4) and (0,6) 27

29 Day 8 - Chapter 3-6: Equations of Lines in Slope Intercept Form and Point Slope Form SWBAT: Graph Lines in Slope Intercept and Point Slope Form Warm Up 1) Write an equation of a line that passes through the point (4,-2) with slope 1. 2) Write an equation of a line that passes through the points ( 1, 0) and (1, 2). 28

30 Linear Equations written in the form y = mx + b are called the slope-intercept form. When an equation is written in this form, m is the and b is the. Find the slope and the y-intercept, then graph. a. y = x 4 b. y = 5 1 x + 2 slope = y - intercept = slope = y- intercept = c. y = 4x + 1 d. y = -2x slope = y - intercept = slope = y- intercept = 29

31 Linear Equations written in the form y y 1 = m(x x 1) are called the point-slope form. Find the slope and the y-intercept, then graph. a. y + 3 = -2(x 1) b. y - 3 = -2(x +4) slope = y - intercept = slope = y- intercept = c. y + 4 = 4(x +2) d. y - 1 = 3 2 (x + 3) slope = y - intercept = slope = y- intercept = 30

32 Write an equation of each line below. a. d. b. e. c. f. 31

33 Find the slope and the y-intercept, then graph. HOMEWORK a. y = -3x + 4 b. y - 5 = 2(x +6) slope = y - intercept = slope = y- intercept = c. x = 5 d. y + 4 = 3 2 (x - 6) slope = y - intercept = slope = y- intercept = 32

34 Find the slope and the y-intercept, then graph. e. y -7 = x + 4 f. y = 2 slope = y - intercept = slope = y- intercept = g. y x = -3 h. y = x + 1 slope = y - intercept = slope = y- intercept = 33

35 Day 9 Chapter 3-6: Equations of Parallel & Perpendicular Lines SWBAT: Graph and Write Equations of Parallel & Perpendicular Lines given a Slope and Point Warm Up Example 1 Writing Equations of Lines Practice: A) (-2, 2), y = 4x

36 B) (4, -2), y = -2x + 3 Example 2 (4, 2), y = 1 2 x + 1 Practice: C) (-8, -7), y = -x

37 D) (6, -2), y = -3x - 6 Challenge Problem (6, 4), y = 7x + 1 Wrap Up List 3 things you learned today; 2 key terms you learned; and 1 question you have about today s lesson

38 Homework 1) 2) 3) 4) 37

39 Day 11 - Chapter 3 6: Quadratic Linear Systems SWBAT: Graph the Solutions to Quadratic Linear Systems Warm Up 1. Write an equation of the line that passes through the given point and is parallel to the graph of the equation below. 2. Write an equation of the line that passes through the given point and is perpendicular to the graph of the equation below. 38

40 39

41 Example 2: Regents Questions 40

42 Practice 3: 41

43 Name: Date: Ms. Williams Homework SWBAT: Solve Quadratic-Linear Systems What is the equation of a line that is perpendicular to -3y = 7x 2 and passes through the point (0, -8)? 42

44 5. 43

45 Graph the lines and find the points of intersection. 1. Day 12 Quadratic Linear Systems

46 4. 5. y = x 2-9 y = y = x 2 2x 3 x = 1 45

2 and 6 4 and 8 1 and 5 3 and 7

2 and 6 4 and 8 1 and 5 3 and 7 Geo Ch 3 Angles formed by Lines Parallel lines are two coplanar lines that do not intersect. Skew lines are that are not coplanar and do not intersect. Transversal is a line that two or more lines at different