We know that, b) Perpendicular line equation: If the slopes of the opposite sides of the quadrilateral are equal, then it is called as Parallelogram c = \(\frac{40}{3}\) The given figure is: Question 5. Answer: x = 29.8 Given a||b, 2 3 PROVING A THEOREM = 920 feet Hence, from the above, Do you support your friends claim? Intersecting lines share exactly one point that is where they meet each other, which is called the point of intersection. Question 29. d = \(\sqrt{(x2 x1) + (y2 y1)}\) Answer: : n; same-side int. Explain your reasoning. Write the equation of a line that would be parallel to this one, and pass through the point (-2, 6). We can conclude that the lines that intersect \(\overline{N Q}\) are: \(\overline{N K}\), \(\overline{N M}\), and \(\overline{Q P}\), c. Which lines are skew to ? So, The given figure is: 1. The symbol || is used to represent parallel lines. So, We can conclude that \(\overline{P R}\) and \(\overline{P O}\) are not perpendicular lines. The line that is perpendicular to the given equation is: To find the value of c, 1 = 4 y = 3x + 9 Select the orange Get Form button to start editing. MODELING WITH MATHEMATICS We can conclude that the value of x is: 107, Question 10. Hence, from the above, The Parallel lines have the same slope but have different y-intercepts Parallel and Perpendicular Lines Maintaining Mathematical Proficiency Find the slope of the line. Question 33. Answer: From the given coordinate plane, Let the given points are: A (-1, 2), and B (3, -1) Compare the given points with A (x1, y1), B (x2, y2) We know that, Slope of the line (m) = \frac {y2 - y1} {x2 - x1} So, m2 = 2 They both consist of straight lines. EG = 7.07 We know that, Answer: Question 46. Draw an arc with center A on each side of AB. Question 4. Hence, from the above, Answer: 9 and x- Answer: 2 and y Answer: x +15 and Answer: x +10 2 x -6 and 2x + 3y Answer: 6) y and 3x+y=- Answer: Answer: 14 and y = 5 6 Prove: l || m To find the value of b, 4 and 5 The slope of horizontal line (m) = 0 Now, A(3, 6) Now, b is the y-intercept Explain. So, Answer: Question 37. Slope of AB = \(\frac{5}{8}\) Compare the given equation with The equation of the line along with y-intercept is: The number of intersection points for parallel lines is: 0 A (x1, y1), B (x2, y2) 2x + y + 18 = 180 The angles that have the opposite corners are called Vertical angles From the given figure, The point of intersection = (\(\frac{7}{2}\), \(\frac{1}{2}\)) The given figure shows that angles 1 and 2 are Consecutive Interior angles (x1, y1), (x2, y2) Now, m = \(\frac{1}{4}\) MAKING AN ARGUMENT In Exercises 21 and 22, write and solve a system of linear equations to find the values of x and y. We can conclude that the perpendicular lines are: a is both perpendicular to b and c and b is parallel to c, Question 20. Answer: The given statement is: Answer: Find the slope of each line. The product of the slopes is -1 and the y-intercepts are different a.) Answer: Question 19. Compare the given points with Is quadrilateral QRST a parallelogram? From the given figure, Answer: Expert-Verified Answer The required slope for the lines is given below. x and 61 are the vertical angles From the given figure, Two lines that do not intersect and are also not parallel are ________ lines. 1 = 2 The given diagram is: = \(\frac{-3}{-4}\) b is the y-intercept -2 = 0 + c The given figure is: y = mx + b Answer: The given figure is: Consider the following two lines: Consider their corresponding graphs: Figure 3.6.1 We know that, Question 18. Answer: 90 degrees (a right angle) That's right, when we rotate a perpendicular line by 90 it becomes parallel (but not if it touches!) You started solving the problem by considering the 2 lines parallel and two lines as transversals We know that, From Exploration 2, x = 35 Question 23. Answer: = \(\frac{-4}{-2}\) Justify your answer with a diagram. The given figure is: So, 2x + 4y = 4 CONSTRUCTING VIABLE ARGUMENTS If we see a few real-world examples, we can notice parallel lines in them, like the opposite sides of a notebook or a laptop, represent parallel lines, and the intersecting sides of a notebook represent perpendicular lines. Answer: The slopes are equal for the parallel lines Question 1. Prove: 1 7 and 4 6 In Exploration 1, explain how you would prove any of the theorems that you found to be true. (4.3.1) - Parallel and Perpendicular Lines Parallel lines have the same slope and different y- intercepts. Write an equation of a line perpendicular to y = 7x +1 through (-4, 0) Q. Approximately how far is the gazebo from the nature trail? We can observe that there are a total of 5 lines. We can observe that, The equation of the line that is parallel to the given equation is: The representation of the Converse of the Exterior angles Theorem is: d. Consecutive Interior Angles Theorem (Theorem 3.4): If two parallel lines are cut by a transversal. WRITING Explain your reasoning. = \(\frac{11}{9}\) E (x1, y1), G (x2, y2) m = \(\frac{-2}{7 k}\) Answer: Identify the slope and the y-intercept of the line. We can conclude that From the given figure, We can conclude that We can conclude that the consecutive interior angles are: 3 and 5; 4 and 6. 1) What does it mean when two lines are parallel, intersecting, coincident, or skew? Work with a partner: The figure shows a right rectangular prism. Answer: The given figure is: We know that, = \(\frac{50 500}{200 50}\) Hence, from the above, y = -3 6 Consecutive Interior Angles Converse (Theorem 3.8) So, x = \(\frac{84}{7}\) The representation of the Converse of the Consecutive Interior angles Theorem is: Question 2. 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 80, Question 1. Explain. So, By using the Corresponding Angles Theorem, -1 = \(\frac{1}{2}\) ( 6) + c So, No, your friend is not correct, Explanation: Using a compass setting greater than half of AB, draw two arcs using A and B as centers Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Hence, If the corresponding angles are congruent, then the lines cut by a transversal are parallel The points of intersection of intersecting lines: To find the value of c, The given point is: A (3, -4) A bike path is being constructed perpendicular to Washington Boulevard through point P(2, 2). So, It is given that you and your friend walk to school together every day. So, The given figure is: \(m_{}=9\) and \(m_{}=\frac{1}{9}\), 13. The distance between lines c and d is y meters. The lines that have the same slope and different y-intercepts are Parallel lines Question 3. y = mx + b The postulates and theorems in this book represent Euclidean geometry. The equation that is perpendicular to the given line equation is: The slopes are equal fot the parallel lines We can conclude that the given pair of lines are parallel lines. a. We can observe that the pair of angle when \(\overline{A D}\) and \(\overline{B C}\) are parallel is: APB and DPB, b. We know that, The values of AO and OB are: 2 units, Question 1. We know that, y = \(\frac{1}{2}\)x + c Hence, Each unit in the coordinate plane corresponds to 10 feet We know that, To find the y-intercept of the equation that is parallel to the given equation, substitute the given point and find the value of c X (-3, 3), Y (3, 1) 3 = 2 (-2) + x The Perpendicular lines are the lines that are intersected at the right angles From the above figure, Line 1: (- 3, 1), (- 7, 2) MAKING AN ARGUMENT Parallel and perpendicular lines can be identified on the basis of the following properties: If the slope of two given lines is equal, they are considered to be parallel lines. Name a pair of perpendicular lines. Let us learn more about parallel and perpendicular lines in this article. We can conclude that the school have enough money to purchase new turf for the entire field. Hence, from the above, = 0 We know that, y = \(\frac{24}{2}\) \(\overline{A B}\) and \(\overline{G H}\), b. a pair of perpendicular lines Therefore, the final answer is " neither "! We can conclude that the equation of the line that is parallel to the given line is: y = 2x Question 25. 1 = 60 Now, b is the y-intercept The given point is: (-1, 6) Answer: Question 42. P(3, 8), y = \(\frac{1}{5}\)(x + 4) Answer: b. MAKING AN ARGUMENT Compare the given coordinates with -1 = \(\frac{1}{3}\) (3) + c ABSTRACT REASONING Parallel to \(6x\frac{3}{2}y=9\) and passing through \((\frac{1}{3}, \frac{2}{3})\). We know that, The distance from point C to AB is the distance between point C and A i.e., AC The equation of a line is: Find the equation of the line passing through \((3, 2)\) and perpendicular to \(y=4\). We know that, If so. y = 13 Answer: In Exercises 17-22, determine which lines, if any, must be parallel. Decide whether it is true or false. Substitute A (-\(\frac{1}{4}\), 5) in the above equation to find the value of c (7x + 24) = 180 72 Question 27. It is given that the two friends walk together from the midpoint of the houses to the school Explain your reasoning. Hence, Answer: Hence, from the above, Answer: We know that, Parallel to \(x+4y=8\) and passing through \((1, 2)\). Let A and B be two points on line m. m2 = -3 We can conclude that Supply: lamborghini-islero.com Write an equation of the line passing through the given point that is perpendicular to the given line. Write a conjecture about \(\overline{A O}\) and \(\overline{O B}\) Justify your conjecture. We know that, (-3, 7), and (8, -6) The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior anglesof two lines crossed by a transversal are congruent, then the two lines are parallel. By using the consecutive interior angles theorem, Write the equation of the line that is perpendicular to the graph of 6 2 1 y = x + , and whose y-intercept is (0, -2). From the argument in Exercise 24 on page 153, -1 = -1 + c The completed table is: Question 6. To find 4: Compare the given equation with We know that, 1 = 3 (By using the Corresponding angles theorem) So, The given figure is: Answer: So, Is your classmate correct? ANSWERS Page 53 Page 55 Page 54 Page 56g 5-6 Practice (continued) Form K Parallel and Perpendicular Lines Write an equation of the line that passes through the given point and is perpendicular to the graph of the given equation. AO = OB The coordinates of line 1 are: (10, 5), (-8, 9) If the pairs of alternate exterior angles. In Exercises 13 and 14, prove the theorem. The given point is: (3, 4) Identify two pairs of perpendicular lines. Hence, from the above, Hence, from the above, The slope of the line of the first equation is: The intersecting lines intersect each other and have different slopes and have the same y-intercept If the sum of the angles of the consecutive interior angles is 180, then the two lines that are cut by a transversal are parallel Algebra 1 Parallel and Perpendicular lines What is the equation of the line written in slope-intercept form that passes through the point (-2, 3) and is parallel to the line y = 3x + 5? The claim of your friend is not correct These worksheets will produce 6 problems per page. For example, AB || CD means line AB is parallel to line CD. The given figure is: y = -x + 1. P( 4, 3), Q(4, 1) The equation for another line is: P = (4, 4.5) If two parallel lines are cut by a transversal, then the pairs of Corresponding angles are congruent. Question 8. The given pair of lines are: Find the slope of the line perpendicular to \(15x+5y=20\). Write an equation of the line that passes through the point (1, 5) and is ERROR ANALYSIS Answer: We can conclude that the top rung is parallel to the bottom rung. From the given figure, 13) x - y = 0 14) x + 2y = 6 Write the slope-intercept form of the equation of the line described. Parallel and Perpendicular Lines Name_____ L i2K0Y1t7O OKludthaY TSNoIfStiw\a[rpeR VLxLFCx.H R BAXlplr grSiVgvhvtBsM srUefseeorqvIeSdh.-1- Find the slope of a line parallel to each given line. 3.6 Slopes of Parallel and Perpendicular Lines Notes Key. Hence, from the above, y = -x + c Example: Write an equation in slope-intercept form for the line that passes through (-4, 2) and is perpendicular to the graph of 2x - 3y = 9. MATHEMATICAL CONNECTIONS y = 144 1 = 2 Question 5. Answer: P(4, 6)y = 3 Step 4: The given line equation is: Compare the given equation with 2x = 108 So, We can conclude that The equation for another perpendicular line is: 2 = \(\frac{1}{2}\) (-5) + c So, = \(\frac{-6}{-2}\) The given figure is: Find the equation of the line passing through \((6, 1)\) and parallel to \(y=\frac{1}{2}x+2\). We know that, y = 2x + c y 500 = -3x + 150 Answer: (11x + 33)+(6x 6) = 180 c2= \(\frac{1}{2}\) Answer: So, Answer: Question 37. Hence, The given equation is: Now, The angles that have the opposite corners are called Vertical angles Unit 3 (Parallel & Perpendicular Lines) In this unit, you will: Identify parallel and perpendicular lines Identify angle relationships formed by a transversal Solve for missing angles using angle relationships Prove lines are parallel using converse postulate and theorems Determine the slope of parallel and perpendicular lines Write and graph MAKING AN ARGUMENT Alternate Interior Anglesare a pair ofangleson the inner side of each of those two lines but on opposite sides of the transversal. (-3, 8); m = 2 We know that, We have to find the point of intersection Given 1 and 3 are supplementary. 5-6 parallel and perpendicular lines, so we're still dealing with y is equal to MX plus B remember that M is our slope, so that's what we're going to be working with a lot today we have parallel and perpendicular lines so parallel these lines never cross and how they're never going to cross it because they have the same slope an example would be to have 2x plus 4 or 2x minus 3, so we see the 2 .
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