In the parallel lines, For example, if the equations of two lines are given as, y = -3x + 6 and y = -3x - 4, we can see that the slope of both the lines is the same (-3). Answer: By comparing the given pair of lines with The representation of the given pair of lines in the coordinate plane is: Answer: So, Alternate exterior angles are the pair of anglesthat lie on the outer side of the two parallel lines but on either side of the transversal line. Answer: Use the diagram to find the measure of all the angles. (1) = Eq. Parallel to \(x+y=4\) and passing through \((9, 7)\). EG = \(\sqrt{(1 + 4) + (2 + 3)}\) 3y 525 = x 50 From the figure, c = \(\frac{8}{3}\) Label the point of intersection as Z. y = \(\frac{3}{2}\)x 1 Hence, from the above, We can conclude that 1 and 5 are the adjacent angles, Question 4. x 2y = 2 If the support makes a 32 angle with the floor, what must m1 so the top of the step will be parallel to the floor? The equation of the line along with y-intercept is: What are Parallel and Perpendicular Lines? m is the slope 13) x - y = 0 14) x + 2y = 6 Write the slope-intercept form of the equation of the line described. The slope of the line of the first equation is: The coordinates of line p are: b. Then, according to the parallel line axiom, there is a different line than L2 that passes through the intersection point of L2 and L3 (point A in the drawing), which is parallel to L1. Compare the given points with 9 0 = b From the given figure, The lines skew to \(\overline{Q R}\) are: \(\overline{J N}\), \(\overline{J K}\), \(\overline{K L}\), and \(\overline{L M}\), Question 4. The given point is: (0, 9) c = \(\frac{16}{3}\) We know that, A _________ line segment AB is a segment that represents moving from point A to point B. Use these steps to prove the Transitive Property of Parallel Lines Theorem Substitute (-1, -9) in the given equation The point of intersection = (\(\frac{4}{5}\), \(\frac{13}{5}\)) The slope of line a (m) = \(\frac{y2 y1}{x2 x1}\) The given point is: A (-1, 5) We can conclue that m1m2 = -1 \(\left\{\begin{aligned}y&=\frac{2}{3}x+3\\y&=\frac{2}{3}x3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=\frac{3}{4}x1\\y&=\frac{4}{3}x+3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=2x+1\\ y&=\frac{1}{2}x+8\end{aligned}\right.\), \(\left\{\begin{aligned}y&=3x\frac{1}{2}\\ y&=3x+2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=5\\x&=2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=7\\y&=\frac{1}{7}\end{aligned}\right.\), \(\left\{\begin{aligned}3x5y&=15\\ 5x+3y&=9\end{aligned}\right.\), \(\left\{\begin{aligned}xy&=7\\3x+3y&=2\end{aligned}\right.\), \(\left\{\begin{aligned}2x6y&=4\\x+3y&=2 \end{aligned}\right.\), \(\left\{\begin{aligned}4x+2y&=3\\6x3y&=3 \end{aligned}\right.\), \(\left\{\begin{aligned}x+3y&=9\\2x+3y&=6 \end{aligned}\right.\), \(\left\{\begin{aligned}y10&=0\\x10&=0 \end{aligned}\right.\), \(\left\{\begin{aligned}y+2&=0\\2y10&=0 \end{aligned}\right.\), \(\left\{\begin{aligned}3x+2y&=6\\2x+3y&=6 \end{aligned}\right.\), \(\left\{\begin{aligned}5x+4y&=20\\10x8y&=16 \end{aligned}\right.\), \(\left\{\begin{aligned}\frac{1}{2}x\frac{1}{3}y&=1\\\frac{1}{6}x+\frac{1}{4}y&=2\end{aligned}\right.\). y = \(\frac{13}{2}\) Check out the following pages related to parallel and perpendicular lines. We can conclude that there are not any parallel lines in the given figure, Question 15. So, Answer: Question 27. From the given figure, From the given bars, In Exercises 13-18. decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Question 25. So, Question 18. Cops the diagram with the Transitive Property of Parallel Lines Theorem on page 141. 6x = 87 = \(\frac{8}{8}\) Download Parallel and Perpendicular Lines Worksheet - Mausmi Jadhav. y = \(\frac{1}{2}\)x 3 Parallel and Perpendicular Lines Maintaining Mathematical Proficiency Page 123, Parallel and Perpendicular Lines Mathematical Practices Page 124, 3.1 Pairs of Lines and Angles Page(125-130), Lesson 3.1 Pairs of Lines and Angles Page(126-128), Exercise 3.1 Pairs of Lines and Angles Page(129-130), 3.2 Parallel Lines and Transversals Page(131-136), Lesson 3.2 Parallel Lines and Transversals Page(132-134), Exercise 3.2 Parallel Lines and Transversals Page(135-136), 3.3 Proofs with Parallel Lines Page(137-144), Lesson 3.3 Proofs with Parallel Lines Page(138-141), Exercise 3.3 Proofs with Parallel Lines Page(142-144), 3.1 3.3 Study Skills: Analyzing Your Errors Page 145, 3.4 Proofs with Perpendicular Lines Page(147-154), Lesson 3.4 Proofs with Perpendicular Lines Page(148-151), Exercise 3.4 Proofs with Perpendicular Lines Page(152-154), 3.5 Equations of Parallel and Perpendicular Lines Page(155-162), Lesson 3.5 Equations of Parallel and Perpendicular Lines Page(156-159), Exercise 3.5 Equations of Parallel and Perpendicular Lines Page(160-162), 3.4 3.5 Performance Task: Navajo Rugs Page 163, Parallel and Perpendicular Lines Chapter Review Page(164-166), Parallel and Perpendicular Lines Test Page 167, Parallel and Perpendicular Lines Cumulative Assessment Page(168-169), Big Ideas Math Answers Grade 2 Chapter 15 Identify and Partition Shapes, Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors, enVision Math Common Core Grade 7 Answer Key | enVision Math Common Core 7th Grade Answers, Envision Math Common Core Grade 5 Answer Key | Envision Math Common Core 5th Grade Answers, Envision Math Common Core Grade 4 Answer Key | Envision Math Common Core 4th Grade Answers, Envision Math Common Core Grade 3 Answer Key | Envision Math Common Core 3rd Grade Answers, enVision Math Common Core Grade 2 Answer Key | enVision Math Common Core 2nd Grade Answers, enVision Math Common Core Grade 1 Answer Key | enVision Math Common Core 1st Grade Answers, enVision Math Common Core Grade 8 Answer Key | enVision Math Common Core 8th Grade Answers, enVision Math Common Core Kindergarten Answer Key | enVision Math Common Core Grade K Answers, enVision Math Answer Key for Class 8, 7, 6, 5, 4, 3, 2, 1, and K | enVisionmath 2.0 Common Core Grades K-8, enVision Math Common Core Grade 6 Answer Key | enVision Math Common Core 6th Grade Answers, Go Math Grade 8 Answer Key PDF | Chapterwise Grade 8 HMH Go Math Solution Key. Then use a compass and straightedge to construct the perpendicular bisector of \(\overline{A B}\), Question 10. We know that, 4 = 5 35 + y = 180 In this case, the negative reciprocal of 1/5 is -5. are parallel, or are the same line. So, Find the distance between the lines with the equations y = \(\frac{3}{2}\) + 4 and 3x + 2y = 1. Answer: Identify the slope and the y-intercept of the line. These Parallel and Perpendicular Lines Worksheets are great for practicing identifying parallel, perpendicular, and intersecting lines from pictures. We know that, The pair of angles on one side of the transversal and inside the two lines are called the Consecutive interior angles. y = mx + c So, Hence, = \(\frac{-1 3}{0 2}\) m || n is true only when x and 73 are the consecutive interior angles according to the Converse of Consecutive Interior angles Theorem 8 = 65. m = -1 [ Since we know that m1m2 = -1] 2 = 2 (-5) + c y = \(\frac{1}{7}\)x + 4 We can conclude that the slope of the given line is: \(\frac{-3}{4}\), Question 2. The equation that is perpendicular to the given line equation is: Now, Question 3. From the given figure, If we draw the line perpendicular to the given horizontal line, the result is a vertical line. x = 4 We can observe that there are 2 perpendicular lines For example, if given a slope. Now, The given equation of the line is: The given points are: The slope of the given line is: m = \(\frac{1}{2}\) The points are: (-2, 3), (\(\frac{4}{5}\), \(\frac{13}{5}\)) For parallel lines, we cant say anything y = \(\frac{2}{3}\)x + 9, Question 10. Question 22. Hence, from the above, A(1, 3), B(8, 4); 4 to 1 line(s) perpendicular to Line 1: (- 3, 1), (- 7, 2) ax + by + c = 0 Two lines are termed as parallel if they lie in the same plane, are the same distance apart, and never meet each other. The opposite sides are parallel and the intersecting lines are perpendicular. Hence, c = 2 The Converse of the Consecutive Interior angles Theorem: The given coordinates are: A (-2, 1), and B (4, 5) We can conclude that -2 = 1 + c b.) Hence, from the given figure, The slope of second line (m2) = 1 Quick Link for All Parallel and Perpendicular Lines Worksheets, Detailed Description for All Parallel and Perpendicular Lines Worksheets. The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem, Question 16. So, We know that, (1) x = 54 5 = \(\frac{1}{2}\) (-6) + c We know that, If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Identify two pairs of parallel lines so that each pair is in a different plane. Compare the given equation with Answer: Question 36. The slopes are equal fot the parallel lines (C) Alternate Exterior Angles Converse (Thm 3.7) Hence, from the above, = (-1, -1) Now, 3 = 68 and 8 = (2x + 4) The equation of the line that is perpendicular to the given line equation is: = 0 y = -2x + c Hence, from the above, Observe the following figure and the properties of parallel and perpendicular lines to identify them and differentiate between them. construction change if you were to construct a rectangle? The slope of the vertical line (m) = Undefined. = 104 Answer: We can observe that Question 11. y = \(\frac{1}{5}\)x + \(\frac{37}{5}\) = 5.70 (D) Consecutive Interior Angles Converse (Thm 3.8) These worksheets will produce 6 problems per page. Explain your reasoning. Hene, from the given options, alternate interior The lines skew to \(\overline{E F}\) are: \(\overline{C D}\), \(\overline{C G}\), and \(\overline{A E}\), Question 4. The slope of first line (m1) = \(\frac{1}{2}\) x + 2y = 2 The given point is: A (-9, -3) y = \(\frac{1}{2}\)x 7 Hence, Hence, from the above, The equation of the line that is parallel to the given line equation is: 1 (m2) = -3 In which of the following diagrams is \(\overline{A C}\) || \(\overline{B D}\) and \(\overline{A C}\) \(\overline{C D}\)? alternate exterior Which rays are not parallel? We can conclude that the line that is parallel to the given line equation is: Justify your conjecture. Answer: (2) to get the values of x and y The given table is: 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. 2 and 3 are the congruent alternate interior angles, Question 1. Answer: Compare the given points with b. The intersection point of y = 2x is: (2, 4) \(\frac{1}{2}\)x + 7 = -2x + \(\frac{9}{2}\) We know that, m2 = \(\frac{1}{3}\) Hence, from the above, Now, From the given figure, For a square, The given equation is: c = -4 Hence, from the above, y = \(\frac{1}{2}\)x + b (1) y = 4 x + 2 2. y = 5 - 2x 3. S. Giveh the following information, determine which lines it any, are parallel. CRITICAL THINKING y = mx + c We can conclude that option D) is correct because parallel and perpendicular lines have to be lie in the same plane, Question 8. Slope (m) = \(\frac{y2 y1}{x2 x1}\) We can conclude that the number of points of intersection of intersecting lines is: 1, c. The points of intersection of coincident lines: So, Answer: So, Answer: Answer: Question 40. y = 2x 13, Question 3. c. m5=m1 // (1), (2), transitive property of equality What is the relationship between the slopes? The given point is: A (3, -4) y = \(\frac{1}{2}\)x + c Use a square viewing window. Sketch what the segments in the photo would look like if they were perpendicular to the crosswalk. a. a pair of skew lines Part 1: Determine the parallel line using the slope m = {2 \over 5} m = 52 and the point \left ( { - 1, - \,2} \right) (1,2). We can conclude that Whereas, if the slopes of two given lines are negative reciprocals of each other, they are considered to be perpendicular lines. x and 61 are the vertical angles Hence, from the given figure, The perpendicular bisector of a segment is the line that passes through the _______________ of the segment at a _______________ angle. Examples of parallel lines: Railway tracks, opposite sides of a whiteboard. Question 29. Answer: Compare the given equation with In this form, we can see that the slope of the given line is \(m=\frac{3}{7}\), and thus \(m_{}=\frac{7}{3}\). So, y = \(\frac{1}{3}\)x + \(\frac{16}{3}\), Question 5. Perpendicular to \(\frac{1}{2}x\frac{1}{3}y=1\) and passing through \((10, 3)\). In Exploration 1, explain how you would prove any of the theorems that you found to be true. Use a graphing calculator to graph the pair of lines. The representation of the perpendicular lines in the coordinate plane is: In Exercises 21 24, find the distance from point A to the given line. 8x = 96 So, Identifying Parallel, Perpendicular, and Intersecting Lines from a Graph Slope of the line (m) = \(\frac{y2 y1}{x2 x1}\) y = -2x 2, f. Construct a square of side length AB x = \(\frac{7}{2}\) We can observe that Label points on the two creases. y = x + c 17x = 180 27 Remember that horizontal lines are perpendicular to vertical lines. The product of the slopes of the perpendicular lines is equal to -1 We can conclude that quadrilateral JKLM is a square. Hence, from the above, \(\overline{A B}\) and \(\overline{G H}\), b. a pair of perpendicular lines The equation that is perpendicular to the given line equation is: The given equations are: y = -2x + 1 The slope of second line (m2) = 2 y = \(\frac{1}{3}\) (10) 4 c = -1 11y = 77 2-4 Additional Practice Parallel And Perpendicular Lines Answer Key November 7, 2022 admin 2-4 Extra Observe Parallel And Perpendicular Strains Reply Key. The given rectangular prism of Exploration 2 is: We can conclude that the perpendicular lines are: Answer: The slopes are equal fot the parallel lines Answer: Question 12. m = 2 A(2, 0), y = 3x 5 THOUGHT-PROVOKING We know that, -5 = 2 (4) + c Answer: If parallel lines are cut by a transversal line, thenconsecutive exterior anglesare supplementary. y = -3x + 150 + 500 So, The equation that is perpendicular to the given equation is: Question 39. The given figure is: Answer: We know that, Will the opening of the box be more steep or less steep? Answer: Given: m5 + m4 = 180 = \(\sqrt{(-2 7) + (0 + 3)}\) The representation of the perpendicular lines in the coordinate plane is: Question 19. The rungs are not intersecting at any point i.e., they have different points y = -2x + 8 The distance between the perpendicular points is the shortest m1m2 = -1 Parallel lines are always equidistant from each other. 0 = \(\frac{1}{2}\) (4) + c The lines that have an angle of 90 with each other are called Perpendicular lines Yes, there is enough information to prove m || n Part - A Part - B Sheet 1 5) 6) Identify the pair of parallel and perpendicular line segments in each shape. 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. Explain your reasoning. They are always the same distance apart and are equidistant lines. Answer: From the given diagram, The point of intersection = (-1, \(\frac{13}{2}\)) Given a Pair of Lines Determine if the Lines are Parallel, Perpendicular, or Intersecting 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? It is given that m || n How are they different? We can conclude that 75 and 75 are alternate interior angles, d. Slope of QR = \(\frac{-2}{4}\) A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. So, Name a pair of perpendicular lines. m2 = -3 Homework Sheets. Answer: Answer: We can conclude that the linear pair of angles is: Perpendicular lines have slopes that are opposite reciprocals. These worksheets will produce 6 problems per page. Hence, from the above, Answer: a.) c = 6 0 -2 = 0 + c 5 + 4 = b We know that, Question 33. If the line cut by a transversal is parallel, then the corresponding angles are congruent We know that, Proof of Alternate exterior angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent b is the y-intercept Find the distance from the point (- 1, 6) to the line y = 2x. When you look at perpendicular lines they have a slope that are negative reciprocals of each other. m1 = \(\frac{1}{2}\), b1 = 1 We know that, The equation that is perpendicular to the given equation is: XY = \(\sqrt{(x2 x1) + (y2 y1)}\) Perpendicular lines are those that always intersect each other at right angles. Hence, from the given figure, Answer: Now, Answer: Question 19. A(- 2, 4), B(6, 1); 3 to 2 Substitute (-2, 3) in the above equation In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. Use a graphing calculator to verify your answers. Ruler: The highlighted lines in the scale (ruler) do not intersect or meet each other directly, and are the same distance apart, therefore, they are parallel lines. The line x = 4 is a vertical line that has the right angle i.e., 90 So, = \(\frac{10}{5}\) 6-3 Write Equations of Parallel and Perpendicular Lines Worksheet. Now, Hence, from the above, Is it possible for all eight angles formed to have the same measure? Now, Slope (m) = \(\frac{y2 y1}{x2 x1}\) The equation of line q is: Compare the given equation with Write an equation of a line perpendicular to y = 7x +1 through (-4, 0) Q. If the slope of two given lines are negative reciprocals of each other, they are identified as ______ lines. Question 16. Now, Answer: (-3, 8); m = 2 The slopes of parallel lines, on the other hand, are exactly equal. So, The slopes of the parallel lines are the same We know that, Now, 42 + 6 (2y 3) = 180 In Euclidean geometry, the two perpendicular lines form 4 right angles whereas, In spherical geometry, the two perpendicular lines form 8 right angles according to the Parallel lines Postulate in spherical geometry. Identify two pairs of perpendicular lines. But it might look better in y = mx + b form. = 0 So, We can conclude that the distance of the gazebo from the nature trail is: 0.66 feet. We can observe that The given rectangular prism is: = \(\sqrt{(250 300) + (150 400)}\) c = 8 \(\frac{3}{5}\) Substitute P (4, 0) in the above equation to find the value of c The given figure is: Substitute (4, 0) in the above equation y = mx + b Answer: The coordinates of the subway are: (500, 300) 2m2 = -1 We can conclude that the perpendicular lines are: x = \(\frac{18}{2}\) The distance from the point (x, y) to the line ax + by + c = 0 is: Slope of line 2 = \(\frac{4 + 1}{8 2}\) 5 7 Explain your reasoning? y = -2x + 3 = \(\frac{6 + 4}{8 3}\) If not, what other information is needed? So, Question 20. Let us learn more about parallel and perpendicular lines in this article. m1 and m3 Question 5. Explain your reasoning. Answer: d = \(\frac{4}{5}\) We know that, a. Question 27. We know that, The Alternate Interior angles are congruent d = \(\sqrt{(x2 x1) + (y2 y1)}\) Do you support your friends claim? The given figure is: We know that, m = = So, slope of the given line is Question 2. AC is not parallel to DF. The Parallel lines have the same slope but have different y-intercepts Hence, from the above, So, Answer: How would your The product of the slopes of the perpendicular lines is equal to -1 How do you know that the lines x = 4 and y = 2 are perpendiculars? Hence, By using the parallel lines property, (5y 21) and 116 are the corresponding angles d = 17.02 Hence, from the above, We can observe that We can conclude that 2 and 11 are the Vertical angles. X (-3, 3), Y (3, 1) We can conclude that \(\overline{P R}\) and \(\overline{P O}\) are not perpendicular lines. PROBLEM-SOLVING Make a conjecture about how to find the coordinates of a point that lies beyond point B along \(\vec{A}\)B. ABSTRACT REASONING We can observe that the given lines are parallel lines In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. The coordinates of a quadrilateral are: We know that, It is given that, Question 1. Explain your reasoning. 4. First, find the slope of the given line. Question 1. The points are: (2, -1), (\(\frac{7}{2}\), \(\frac{1}{2}\)) Proof: Question 17. We can observe that 1 and 2 are the alternate exterior angles x = 29.8 and y = 132, Question 7. In Exercises 7 and 8, determine which of the lines are parallel and which of the lines are perpendicular. The given equation is: It is given that m || n Solve eq. The equation of the line along with y-intercept is: Draw \(\overline{P Z}\), CONSTRUCTION The vertical angles are congruent i.e., the angle measures of the vertical angles are equal Consider the following two lines: Both lines have a slope \(m=\frac{3}{4}\) and thus are parallel. Answer: Question 10. Hence, Write an equation of a line parallel to y = x + 3 through (5, 3) Q. The slope of the parallel line that passes through (1, 5) is: 3 MATHEMATICAL CONNECTIONS XZ = \(\sqrt{(4 + 3) + (3 4)}\) d = \(\sqrt{(x2 x1) + (y2 y1)}\) d = \(\sqrt{(x2 x1) + (y2 y1)}\) -3 = -4 + c 2x x = 56 2 (A) Corresponding Angles Converse (Thm 3.5) y = \(\frac{3}{5}\)x \(\frac{6}{5}\) 2 = 180 123 In Exercises 9 and 10, trace \(\overline{A B}\). It is given that the two friends walk together from the midpoint of the houses to the school y = -2x + b (1) \(\frac{1}{3}\)x 2 = -3x 2 y = 3x + c 3: write the equation of a line through a given coordinate point . A triangle has vertices L(0, 6), M(5, 8). . In Exercises 9 12, tell whether the lines through the given points are parallel, perpendicular, or neither. We know that, Hence, from the above, m1m2 = -1 x = \(\frac{87}{6}\) So, The equation for another line is: We know that, The pair of lines that are different from the given pair of lines in Exploration 2 are: m1m2 = -1 So, Now, It is given that in spherical geometry, all points are points on the surface of a sphere. -x x = -3 The slope of perpendicular lines is: -1 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. Proof of the Converse of the Consecutive Interior angles Theorem: Answer: Each bar is parallel to the bar directly next to it. By using the Consecutive Interior angles Converse, XY = \(\sqrt{(3 + 3) + (3 1)}\) Now, Given: a || b, 2 3 So, The given figure is: We know that, The given point is: A (-6, 5) y = mx + b The symbol || is used to represent parallel lines. When two lines are cut by a transversal, the pair ofangleson one side of the transversal and inside the two lines are called theconsecutive interior angles. Answer: Answer: Lines Perpendicular to a Transversal Theorem (Theorem 3.12): In a plane. The given line has slope \(m=\frac{1}{4}\), and thus \(m_{}=+\frac{4}{1}=4\). The diagram that represents the figure that it can not be proven that any lines are parallel is: y = mx + c y = -x 12 (2) c = -1 3 Now, Compare the given points with The distance from the point (x, y) to the line ax + by + c = 0 is: Cellular phones use bars like the ones shown to indicate how much signal strength a phone receives from the nearest service tower. From the given figure, From the given figure, Hence, 1 and 4; 2 and 3 are the pairs of corresponding angles 7 = -3 (-3) + c So, Use the diagram Verticle angle theorem: y = -2x + c1 So, Now, The lines that have the same slope and different y-intercepts are Parallel lines We can conclude that the claim of your friend can be supported, Question 7. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also.