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The given figure is: This page titled 3.6: Parallel and Perpendicular Lines is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous. a. Answer: The perpendicular lines have the product of slopes equal to -1 From the given figure, We know that, The given expression is: We can observe that the slopes of the opposite sides are equal i.e., the opposite sides are parallel Slope of line 2 = \(\frac{4 + 1}{8 2}\) The number of intersection points for parallel lines is: 0 In spherical geometry, is it possible that a transversal intersects two parallel lines? The product of the slopes of the perpendicular lines is equal to -1 2. Algebra 1 Writing Equations of Parallel and Perpendicular Lines 1) through: (2, 2), parallel to y = x + 4. y = \(\frac{1}{2}\)x + 7 -(1) According to Corresponding Angles Theorem, We know that, Hence, from the above, So, The equation that is parallel to the given equation is: Question 1. Label the intersection as Z. Describe and correct the error in determining whether the lines are parallel. Now, CRITICAL THINKING -2 m2 = -1 So, The equation of a line is: COMPLETE THE SENTENCE Answer: The equation of the line that is perpendicular to the given line equation is: It is given that a student claimed that j K, j l Now, 1 2 3 4 5 6 7 8 Your school lies directly between your house and the movie theater. Justify your conjecture. Hence, from the above, The representation of the given point in the coordinate plane is: Question 56. These worksheets will produce 10 problems per page. b. The point of intersection = (0, -2) The given point is: (6, 4) In Exercises 13 and 14, prove the theorem. Now, y = -2x + c PROOF y = 2x + c2, b. Converse: x = c y = 13 (2x + 20) = 3x x + 2y = 2 Explain your reasoning. The two pairs of parallel lines so that each pair is in a different plane are: q and p; k and m, b. The coordinates of the quadrilateral QRST is: 0 = 2 + c y = 162 18 1 = 32 Answer: A hand rail is put in alongside the steps of a brand new home as proven within the determine. So, 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 . The given point is: A (2, 0) The given equation is: We know that, The representation of the Converse of Corresponding Angles Theorem is: b. Alternate Interior Angles Theorem (Theorem 3.2): If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. USING STRUCTURE So, Answer: y = -3x 2 (2) Lines AB and CD are not intersecting at any point and are always the same distance apart. A student says. Compare the given points with Answer: y = -2 m1 = m2 = \(\frac{3}{2}\) What are the coordinates of the midpoint of the line segment joining the two houses? The representation of the given pair of lines in the coordinate plane is: The parallel lines are the lines that do not have any intersection point In Exercises 19 and 20. describe and correct the error in the conditional statement about lines. (50, 500), (200, 50) From the figure, The construction of the walls in your home were created with some parallels. Answer: Question 38. In Exercises 19 and 20, describe and correct the error in the reasoning. c = 2 1 m2 = 1 y = \(\frac{1}{5}\)x + \(\frac{4}{5}\) then they are parallel. For a pair of lines to be coincident, the pair of lines have the same slope and the same y-intercept y = 2x + c Answer: -x x = -3 4 Parallel to \(10x\frac{5}{7}y=12\) and passing through \((1, \frac{1}{2})\). According to the Consecutive Exterior angles Theorem, b. We can conclude that 2 and 7 are the Vertical angles, Question 5. We can conclude that \(\overline{K L}\), \(\overline{L M}\), and \(\overline{L S}\), d. Should you have named all the lines on the cube in parts (a)-(c) except \(\overline{N Q}\)? Question 37. Perpendicular Postulate: The lines that have an angle of 90 with each other are called Perpendicular lines So, \(\begin{aligned} y-y_{1}&=m(x-x_{1}) \\ y-(-2)&=\frac{1}{2}(x-8) \end{aligned}\). So, From the given figure, So, These worksheets will produce 6 problems per page. When the corresponding angles are congruent, the two parallel lines are cut by a transversal So, 1 = -3 (6) + b (1) = Eq. 2x x = 56 2 We can conclude that the third line does not need to be a transversal. The pair of angles on one side of the transversal and inside the two lines are called the Consecutive interior angles. We will use Converse of Consecutive Exterior angles Theorem to prove m || n Determine which lines, if any, must be parallel. \(\frac{1}{2}\) (m2) = -1 Perpendicular lines do not have the same slope. From the given figure, c = 3 4 Repeat steps 3 and 4 below AB The given rectangular prism is: We can conclude that the perpendicular lines are: y = -2x 1 Answer: Question 14. We know that, The given equation is: Question 38. Hence, from the given figure, Slope of the line (m) = \(\frac{y2 y1}{x2 x1}\) Answer: Question 40. The diagram can be changed by the transformation of transversals into parallel lines and a parallel line into transversal We can conclude that the plane parallel to plane LMQ is: Plane JKL, Question 5. b.) Use a graphing calculator to verify your answer. The sum of the angle measures of a triangle is: 180 Line 2: (2, 4), (11, 6) The representation of the Converse of the Consecutive Interior angles Theorem is: Question 2. So, From the given figure, Parallel to \(\frac{1}{5}x\frac{1}{3}y=2\) and passing through \((15, 6)\). m1m2 = -1 You meet at the halfway point between your houses first and then walk to school. Hence, from the above, Hence, So, We know that, Question 4. Does the school have enough money to purchase new turf for the entire field? y = mx + b We can observe that the pair of angle when \(\overline{A D}\) and \(\overline{B C}\) are parallel is: APB and DPB, b. y = x + c Now, 13) x - y = 0 14) x + 2y = 6 Write the slope-intercept form of the equation of the line described. Find equations of parallel and perpendicular lines. c = 8 Determine which of the lines are parallel and which of the lines are perpendicular. The slopes of perpendicular lines are undefined and 0 respectively Your school is installing new turf on the football held. = \(\frac{3}{4}\) a = 2, and b = 1 2. We can conclude that the Corresponding Angles Converse is the converse of the Corresponding Angles Theorem, Question 3. In Example 4, the given theorem is Alternate interior angle theorem 1 = 2 = 133 and 3 = 47. c. m5=m1 // (1), (2), transitive property of equality c = -6 d = \(\sqrt{(x2 x1) + (y2 y1)}\) Explain your reasoning. We can conclude that the line that is perpendicular to \(\overline{C D}\) is: \(\overline{A D}\) and \(\overline{C B}\), Question 6. so they cannot be on the same plane. We know that, Why does a horizontal line have a slope of 0, but a vertical line has an undefined slope? We can conclude that Question 22. These Parallel and Perpendicular Lines Worksheets are great for practicing identifying perpendicular lines from pictures. This can be expressed mathematically as m1 m2 = -1, where m1 and m2 are the slopes of two lines that are perpendicular. Answer: Question 25. We can conclude that the pair of skew lines are: 3y 525 = x 50 y = \(\frac{1}{2}\)x 5, Question 8. The given table is: Answer: So, 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? We can conclude that the slope of the given line is: 3, Question 3. The given figure is: y = -2x + 1 y y1 = m (x x1) You and your family are visiting some attractions while on vacation. So, Compare the effectiveness of the argument in Exercise 24 on page 153 with the argument You can find the distance between any two parallel lines What flaw(s) exist in the argument(s)? Perpendicular to \(y3=0\) and passing through \((6, 12)\). We can observe that -5 = 2 + b The given point is: A (8, 2) y = \(\frac{1}{3}\)x + c Answer: No, there is no enough information to prove m || n, Question 18. y = -x + 1. By comparing the given pair of lines with plane(s) parallel to plane CDH Answer: Hence, 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 From the given figure, Fro the given figure, d. AB||CD // Converse of the Corresponding Angles Theorem (4.3.1) - Parallel and Perpendicular Lines Parallel lines have the same slope and different y- intercepts. E (-4, -3), G (1, 2) = \(\sqrt{(4 5) + (2 0)}\) The given figure is: From the given figure, We can conclude that \(\overline{P R}\) and \(\overline{P O}\) are not perpendicular lines. The coordinates of line q are: = \(\frac{1}{-4}\) = \(\frac{15}{45}\) c = -3 The coordinates of P are (22.4, 1.8), Question 2. We can observe that the given lines are parallel lines Question 27. For the intersection point of y = 2x, From the given figure, The given statement is: 1 8 Consecutive Interior Angles Converse (Theorem 3.8) y = mx + b Work with a partner: Write the converse of each conditional statement. Identify all the pairs of vertical angles. The product of the slopes of the perpendicular lines is equal to -1 Linea and Line b are parallel lines We can observe that the product of the slopes are -1 and the y-intercepts are different The point of intersection = (\(\frac{4}{5}\), \(\frac{13}{5}\)) Converse: THINK AND DISCUSS 1. Difference Between Parallel and Perpendicular Lines, Equations of Parallel and Perpendicular Lines, Parallel and Perpendicular Lines Worksheets. c = \(\frac{8}{3}\) We can solve for \(m_{1}\) and obtain \(m_{1}=\frac{1}{m_{2}}\). y = -2x + c y = \(\frac{24}{2}\) The slope of first line (m1) = \(\frac{1}{2}\) Proof of Alternate exterior angles Theorem: Which theorem is the student trying to use? We know that, Hence, Answer: (2) The slopes of the parallel lines are the same PROVING A THEOREM Answer: Identify the slope and the y-intercept of the line. The equation that is perpendicular to the given line equation is: We know that, Hence, from the given figure, Describe how you would find the distance from a point to a plane. y = -x + c Perpendicular to \(y=x\) and passing through \((7, 13)\). If you need more of a review on how to use this form, feel free to go to Tutorial 26: Equations of Lines The given equation is:, Two nonvertical lines in the same plane, with slopes m1 and m2, are parallel if their slopes are the same, m1 = m2. y = \(\frac{1}{2}\)x 6 For example, PQ RS means line PQ is perpendicular to line RS. Step 1: Find the slope \(m\). We know that, So, = \(\frac{-3}{-1}\) Perpendicular lines always intersect at right angles. Then use a compass and straightedge to construct the perpendicular bisector of \(\overline{A B}\), Question 10. Hence, from the above, We know that, The given equation is: It is given that Compare the given coordinates with (x1, y1), and (x2, y2) REASONING No, we did not name all the lines on the cube in parts (a) (c) except \(\overline{N Q}\). XZ = \(\sqrt{(7) + (1)}\) So, Possible answer: 2 and 7 c. Possible answer: 1 and 8 d. Possible answer: 2 and 3 3. y = -2 (-1) + \(\frac{9}{2}\) We know that, Question 21. Answer: transv. So, So, The given point is: (-1, 5) y = \(\frac{1}{3}\)x + \(\frac{475}{3}\), c. What are the coordinates of the meeting point? The coordinates of line b are: (3, -2), and (-3, 0) From the given figure, Perpendicular lines intersect at each other at right angles EG = \(\sqrt{50}\) The equation for another line is: We can observe that 35 and y are the consecutive interior angles Compare the given coordinates with

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