We know that, So, Answer: Question 4. Consider the following two lines: Consider their corresponding graphs: Figure 3.6.1 x = 9. Answer: The slope of the given line is: m = \(\frac{1}{4}\) From the given coordinate plane, XY = 4.60 We can observe that 1 and 2 are the consecutive interior angles MATHEMATICAL CONNECTIONS The given figure is: From the given figure, d = \(\sqrt{(x2 x1) + (y2 y1)}\) We can observe that we divided the total distance into the four congruent segments or pieces By using the Perpendicular transversal theorem, We use this and the point \((\frac{7}{2}, 1)\) in point-slope form. 42 and 6(2y 3) are the consecutive interior angles We know that, XY = \(\sqrt{(3 + 3) + (3 1)}\) y = 2x + 1 In Example 4, the given theorem is Alternate interior angle theorem The given equation is: Answer: a. Answer: In Exercises 9 and 10, trace \(\overline{A B}\). These worksheets will produce 10 problems per page. = \(\frac{3 + 5}{3 + 5}\) So, By comparing eq. Answer: Question 50. a. y = mx + c 2x + 4y = 4 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. y = -2x + 2 Question 25. y = 3x + c Slope of the line (m) = \(\frac{-1 2}{-3 + 2}\) Hence, from the above, ERROR ANALYSIS The given equation is: We can conclude that your friend is not correct. The product of the slopes is -1 1 = 0 + c Substitute (6, 4) in the above equation 10) 1) Explain. In spherical geometry, all points are points on the surface of a sphere. Determine which of the lines are parallel and which of the lines are perpendicular. Given: m5 + m4 = 180 y = \(\frac{1}{3}\)x + \(\frac{26}{3}\) Find both answers. y = 4 x + 2 2. y = 5 - 2x 3. So, Give four examples that would allow you to conclude that j || k using the theorems from this lesson. Where, Hence, Hence, from the above, m = 2 y = \(\frac{1}{3}\)x + \(\frac{475}{3}\) a. How would your y = 2x 13, Question 3. A(1, 3), B(8, 4); 4 to 1 (2) d = \(\sqrt{(4) + (5)}\) Linear Pair Perpendicular Theorem (Thm. Answer: so they cannot be on the same plane. If the slopes of the opposite sides of the quadrilateral are equal, then it is called as Parallelogram consecutive interior We can observe that Hence, from the above, a. y = \(\frac{1}{7}\)x + 4 Answer: Answer: Question 20. What is the relationship between the slopes? m = \(\frac{1}{6}\) and c = -8 Find the distance from point E to Linea and Line b are parallel lines We can conclude that the distance from line l to point X is: 6.32. According to the Converse of the Alternate Exterior Angles Theorem, m || n is true only when the alternate exterior angles are congruent The given statement is: 1 8 Step 4: alternate interior c = -1 3 Hence, from the above, \(\overline{C D}\) and \(\overline{A E}\) Use the results of Exploration 1 to write conjectures about the following pairs of angles formed by two parallel lines and a transversal. Possible answer: 2 and 7 c. Possible answer: 1 and 8 d. Possible answer: 2 and 3 3. The parallel line equation that is parallel to the given equation is: MODELING WITH MATHEMATICS From the given figure, b is the y-intercept = 2 (460) We know that, In this case, the negative reciprocal of -4 is 1/4 and vice versa. y = x + 4 = \(\frac{-2}{9}\) According to Alternate interior angle theorem, Alternate Exterior angle Theorem: We can conclude that a line equation that is perpendicular to the given line equation is: We can conclude that b is perpendicular to c. Question 1. The given pair of lines are: y = \(\frac{1}{3}\)x + c y = mx + b The diagram shows lines formed on a tennis court. PROBLEM-SOLVING Approximately how far is the gazebo from the nature trail? The equation that is parallel to the given equation is: Question 1. The given figure is: The sum of the given angle measures is: 180 The two lines are Intersecting when they intersect each other and are coplanar If the angle measure of the angles is a supplementary angle, then the lines cut by a transversal are parallel Are the numbered streets parallel to one another? So, Answer: Question 34. Answer: Answer: a. corresponding angles So, d = \(\sqrt{290}\) Now, y = \(\frac{13}{2}\) y = \(\frac{1}{7}\)x + 4 By using the Alternate exterior angles Theorem, We can conclude that the value of XZ is: 7.07, Find the length of \(\overline{X Y}\) Let A and B be two points on line m. c = 0 d = \(\sqrt{(8 + 3) + (7 + 6)}\) MODELING WITH MATHEMATICS With Cuemath, you will learn visually and be surprised by the outcomes. Justify your answer. c = -1 You and your mom visit the shopping mall while your dad and your sister visit the aquarium. The given figure is: XY = \(\sqrt{(6) + (2)}\) c = 3 We can conclude that the value of x is: 60, Question 6. It is given that m || n The equation of the line that is perpendicular to the given line equation is: So, alternate exterior y1 = y2 = y3 y = -3x 2 (2) 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. Work with a partner: The figure shows a right rectangular prism. Find the Equation of a Parallel Line Passing Through a Given Equation and Point So, by the Corresponding Angles Converse, g || h. Question 5. C(5, 0) The given point is: P (4, -6) Example 5: Tell whether the line y = {4 \over 3}x + 2 y = 34x + 2 is parallel, perpendicular or neither to the line passing through \left ( {1,1} \right) (1,1) and \left ( {10,13} \right) (10,13). The given point is: A (3, -1) 1 = 4 Answer: Answer: y = \(\frac{2}{3}\)x + b (1) So, To be proficient in math, you need to communicate precisely with others. We know that, m = = So, slope of the given line is Question 2. Substitute P(-8, 0) in the above equation Compare the given equation with We can conclude that both converses are the same Answer: So, x = 97, Question 7. 3 + 133 = 180 (By using the Consecutive Interior angles theorem) x = 9 Hence, 0 = \(\frac{5}{3}\) ( -8) + c (8x + 6) = 118 (By using the Vertical Angles theorem) Mathematically, this can be expressed as m1 = m2, where m1 and m2 are the slopes of two lines that are parallel. Exercise \(\PageIndex{3}\) Parallel and Perpendicular Lines. Perpendicular to \(y3=0\) and passing through \((6, 12)\). These Parallel and Perpendicular Lines Worksheets will give the slopes of two lines and ask the student if the lines are parallel, perpendicular, or neither. Compare the given points with Lines AB and CD are not intersecting at any point and are always the same distance apart. We know that, -1 = -1 + c : n; same-side int. So, m is the slope So, 3x = 69 Answer: x = y = 61, Question 2. Answer: Parallel to \(y=\frac{3}{4}x3\) and passing through \((8, 2)\). MATHEMATICAL CONNECTIONS To find the value of c in the above equation, substitue (0, 5) in the above equation = \(\frac{6 0}{0 + 2}\) Once the equation is already in the slope intercept form, you can immediately identify the slope. Perpendicular lines are denoted by the symbol . The given point is: (1, 5) y = \(\frac{3}{2}\) We can conclude that in order to jump the shortest distance, you have to jump to point C from point A. We know that, d = | c1 c2 | c = -4 Hence, from the above, In spherical geometry, all points are points on the surface of a sphere. 2m2 = -1 We can conclude that m || n, Question 15. So, Yes, there is enough information to prove m || n Use a graphing calculator to verify your answers. So, The equation that is perpendicular to the given equation is: y = \(\frac{1}{2}\)x + c Parallel to \(6x\frac{3}{2}y=9\) and passing through \((\frac{1}{3}, \frac{2}{3})\). Answer: According to the Corresponding Angles Theorem, the corresponding angles are congruent The equation that is parallel to the given equation is: Now, (5y 21) and 116 are the corresponding angles 3y = x + 475 Work with a partner: Fold and crease a piece of paper. Question 20. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. Answer: Our Parallel and Perpendicular Lines Worksheets are free to download, easy to use, and very flexible. By using the parallel lines property, y = -9 Answer: The slopes are equal fot the parallel lines We can conclude that the distance from point A to the given line is: 6.26. Two lines are termed as parallel if they lie in the same plane, are the same distance apart, and never meet each other. But it might look better in y = mx + b form. m2 and m4 a) Parallel to the given line: Substitute (0, 1) in the above equation Hence, from the above figure, Slope of the line (m) = \(\frac{-1 2}{3 + 1}\) Given \(\overrightarrow{B A}\) \(\vec{B}\)C Hence, from the above, m = \(\frac{3}{1.5}\) 132 = (5x 17) This no prep unit bundle will assist your college students perceive parallel strains and transversals, parallel and perpendicular strains proofs, and equations of parallel and perpendicular. Answer: We know that, \(m\cdot m_{\perp}=-\frac{5}{8}\cdot\frac{8}{5}=-\frac{40}{40}=-1\quad\color{Cerulean}{\checkmark}\). The coordinates of y are the same. If the slope of two given lines are negative reciprocals of each other, they are identified as ______ lines. d. AB||CD // Converse of the Corresponding Angles Theorem We know that, The equation that is perpendicular to the given line equation is: Use the photo to decide whether the statement is true or false. AC is not parallel to DF. From the given figure, Which pair of angle measures does not belong with the other three? Answer: It is given that E is to \(\overline{F H}\) Answer: Parallel to \(x+4y=8\) and passing through \((1, 2)\). First, find the slope of the given line. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. These Parallel and Perpendicular Lines Worksheets will give the student a pair of equations for lines and ask them to determine if the lines are parallel, perpendicular, or intersecting. Hence, from the above, parallel Answer: Explanation: In the above image we can observe two parallel lines. CONSTRUCTION 4.7 of 5 (20 votes) Fill PDF Online Download PDF. We can conclude that the value of x is: 20. Question 25. x = 9 -x + 2y = 14 The given figure is: We can observe that We have to find the point of intersection Write a conjecture about \(\overline{A O}\) and \(\overline{O B}\) Justify your conjecture. From Exploration 1, Explain. Answer: Hence, from the above, So, x + 2y = 2 The equation that is perpendicular to the given line equation is: The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal, the resulting corresponding anglesare congruent Find the measure of the missing angles by using transparent paper. In Exercises 13 16. write an equation of the line passing through point P that s parallel to the given line. Question 4. The lines that are coplanar and any two lines that have a common point are called Intersecting lines You started solving the problem by considering the 2 lines parallel and two lines as transversals So, According to the Alternate Exterior angles Theorem, We can observe that y = -2x + 3 So, By using the Consecutive interior angles Theorem, Hence, from the above, Now, These Parallel and Perpendicular Lines Worksheets will ask the student to find the equation of a perpendicular line passing through a given equation and point. x and 97 are the corresponding angles The given equation is: Imagine that the left side of each bar extends infinitely as a line. You and your family are visiting some attractions while on vacation. The equation for another parallel line is: The given figure is: The given points are: P (-7, 0), Q (1, 8) = \(\frac{2}{-6}\) Substitute the given point in eq. We can conclude that the slope of the given line is: 0. m1 = \(\frac{1}{2}\), b1 = 1 THINK AND DISCUSS 1. Alternate Exterior Angles Converse (Theorem 3.7) The given figure is: P(0, 0), y = 9x 1 Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. k 7 = -2 a. The equation of the line along with y-intercept is: PROOF Answer: Question 26. So, The measure of 1 is 70. If you even interchange the second and third statements, you could still prove the theorem as the second line before interchange is not necessary Click the image to be taken to that Parallel and Perpendicular Lines Worksheet. We know that, y = \(\frac{2}{3}\)x + c Answer: Question 8. (2, 7); 5 1 2 11 Hence, from the above, So, M = (150, 250), b. Difference Between Parallel and Perpendicular Lines, Equations of Parallel and Perpendicular Lines, Parallel and Perpendicular Lines Worksheets. Is b c? So, Hence, from the above, Alternate exterior anglesare the pair ofanglesthat lie on the outer side of the two parallel lines but on either side of the transversal line If the pairs of alternate exterior angles. From the given figure, So, When we compare the converses we obtained from the given statement and the actual converse, The given figure is: Slope (m) = \(\frac{y2 y1}{x2 x1}\) Name two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel. Question 17. 2 = 140 (By using the Vertical angles theorem) Hence, from the above, From the given figure, XY = \(\sqrt{(x2 x1) + (y2 y1)}\) Question: What is the difference between perpendicular and parallel? Your classmate decided that based on the diagram. y = -2x + c MODELING WITH MATHEMATICS For parallel lines, Can you find the distance from a line to a plane? We know that, Hence, from the above, -1 = \(\frac{1}{3}\) (3) + c So, Hence, The slopes of parallel lines, on the other hand, are exactly equal. if two lines are perpendicular to the same line. ERROR ANALYSIS 3 = 76 and 4 = 104 We can conclude that the quadrilateral QRST is a parallelogram. Slope of AB = \(\frac{2}{3}\) The distance from your house to the school is one-fourth of the distance from the school to the movie theater. Parallel lines are those lines that do not intersect at all and are always the same distance apart. Explain your reasoning. MAKING AN ARGUMENT Which lines are parallel to ? Draw another arc by using a compass with above half of the length of AB by taking the center at B above AB For a pair of lines to be parallel, the pair of lines have the same slope but different y-intercepts Question 12. The given figure is: We know that, Now, b. Does the school have enough money to purchase new turf for the entire field? The given equation of the line is: 2x y = 18 XY = 6.32 XY = \(\sqrt{(6) + (2)}\) Converse: Consider the following two lines: Both lines have a slope \(m=\frac{3}{4}\) and thus are parallel. Explain. The equation that is perpendicular to the given line equation is: Answer: Explain. (1) with the y = mx + c, From the construction of a square in Exercise 29 on page 154, 5 = -7 ( -1) + c y = 3x 6, Question 20. Justify your answers. The line through (- 1, k) and (- 7, 2) is parallel to the line y = x + 1. a. Draw a diagram to represent the converse. The two pairs of parallel lines so that each pair is in a different plane are: q and p; k and m, b. For example, AB || CD means line AB is parallel to line CD. We can observe that the product of the slopes are -1 and the y-intercepts are different d = \(\sqrt{(x2 x1) + (y2 y1)}\) \(m_{}=\frac{3}{2}\) and \(m_{}=\frac{2}{3}\), 19. 20 = 3x 2x Answer: x = c Hence, So, 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. We can conclude that the distance between the lines y = 2x and y = 2x + 5 is: 2.23. i.e., Tell which theorem you use in each case. Slope of line 2 = \(\frac{4 6}{11 2}\) Now, It is given that m || n To find the value of c, Answer: Hence, from the above, Now, So, y = -2 (-1) + \(\frac{9}{2}\) So, The lines skew to \(\overline{E F}\) are: \(\overline{C D}\), \(\overline{C G}\), and \(\overline{A E}\), Question 4. m2 = 1 Hence, from the above, = -3 If so, dont bother as you will get a complete idea through our BIM Geometry Chapter 3 Parallel and Perpendicular Lines Answer Key. P = (7.8, 5) 8 + 115 = 180 Sandwich: The highlighted lines in the sandwich are neither parallel nor perpendicular lines. This contradicts what was given,that angles 1 and 2 are congruent. Question 1. m2 = -1 We can conclude that the length of the field is: 320 feet, b. Great learning in high school using simple cues. So, Answer: The coordinates of x are the same. Now, We know that, Explain your reasoning. (1) = Eq. The given equation is: The slope of second line (m2) = 1 You and your family are visiting some attractions while on vacation. = \(\frac{2}{9}\) Answer: Identifying Perpendicular Lines Worksheets Explain why the top rung is parallel to the bottom rung. Answer: Alternate Interior Anglesare a pair ofangleson the inner side of each of those two lines but on opposite sides of the transversal. How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point? Answer: y = x 6 -(1) Find the distance from point A to the given line. The line through (k, 2) and (7, 0) is perpendicular to the line y = x \(\frac{28}{5}\). Prove m||n HOW DO YOU SEE IT? c1 = 4 Substitute A (0, 3) in the above equation Compare the given equations with When we compare the converses we obtained from the given statement and the actual converse, From the given figure, 6x = 87 The slope of the given line is: m = -3 Show your steps. = \(\frac{6 + 4}{8 3}\) We know that, Answer: Question 30. We know that, When you look at perpendicular lines they have a slope that are negative reciprocals of each other. Now, Perpendicular to \(5x3y=18\) and passing through \((9, 10)\). The given equation is: d = | 6 4 + 4 |/ \(\sqrt{2}\)} 180 = x + x The equation for another line is: The given lines are the parallel lines (b) perpendicular to the given line. From the above table, Hence, from the above, From Exploration 1, We can conclude that the parallel lines are: Now, No, your friend is not correct, Explanation: The product of the slope of the perpendicular equations is: -1 Step 1: So, Answer: The given point is: A (2, 0) So, Hence, From the given figure, Compare the given points with Substitute A (3, 4) in the above equation to find the value of c For example, if the equation of two lines is given as, y = 4x + 3 and y = 4x - 5, we can see that their slope is equal (4). From Example 1, We can conclude that the top rung is parallel to the bottom rung. The given lines are: Slope (m) = \(\frac{y2 y1}{x2 x1}\) Answer: -x + 2y = 12 m = \(\frac{3 0}{0 + 1.5}\) 1. Hence, from the above figure, The given figure is: CRITICAL THINKING Hence, from the above, Does either argument use correct reasoning? They are not parallel because they are intersecting each other. The construction of the walls in your home were created with some parallels. Hence, from the above, So, Hence, The given equation is: (- 1, 5); m = 4 y = \(\frac{137}{5}\) Label its intersection with \(\overline{A B}\) as O. We know that, Answer: The equation of the line that is parallel to the given line equation is: Hence, from the above, The given figure is: 3y = x 50 + 525 The lines containing the railings of the staircase, such as , are skew to all lines in the plane containing the ground. x + 2y = 2 8x = 112 1 + 2 = 180 The equation of the line that is parallel to the given line equation is: So, -9 = \(\frac{1}{3}\) (-1) + c Substitute A (-9, -3) in the above equation to find the value of c (Two lines are skew lines when they do not intersect and are not coplanar.) It is given that your friend claims that because you can find the distance from a point to a line, you should be able to find the distance between any two lines The intersection of the line is the y-intercept Is your friend correct? Answer: = 3 PROVING A THEOREM c is the y-intercept d = \(\sqrt{(x2 x1) + (y2 y1)}\) From the Consecutive Exterior angles Converse, Hence, from the above, (E) The slopes are the same but the y-intercepts are different Examples of perpendicular lines: the letter L, the joining walls of a room. a. y = \(\frac{1}{2}\)x + c y = mx + b Answer: Now, We know that, We know that, Proof of the Converse of the Consecutive Exterior angles Theorem: The equation that is perpendicular to the given line equation is: You will find Solutions to all the BIM Book Geometry Ch 3 Parallel and Perpendicular Concepts aligned as per the BIM Textbooks. Using X and Y as centers and an appropriate radius, draw arcs that intersect. Find the slope \(m\) by solving for \(y\). Answer: m2 = -3 x = 97 Parallel and Perpendicular Lines Perpendicular Lines Two nonvertical lines are perpendicular if their slopes are opposite reciprocals of each other. 5 = \(\frac{1}{3}\) + c The parallel lines have the same slope but have different y-intercepts and do not intersect 2x = 3 The equation that is perpendicular to the given line equation is: So, m = 2 then they intersect to form four right angles. We can observe that Explain why or why not. line(s) parallel to . Answer: The coordinates of a quadrilateral are: Parallel to \(y=3\) and passing through \((2, 4)\). = Undefined Hence, c = -1 The given lines are perpendicular lines Check out the following pages related to parallel and perpendicular lines. We can observe that Hence, Question 4. The given diagram is: Begin your preparation right away and clear the exams with utmost confidence. A(- 2, 3), y = \(\frac{1}{2}\)x + 1 Answer: Question 52. Answer: m1 = \(\frac{1}{2}\), b1 = 1 Answer: Question 20. Geometrically, we see that the line \(y=4x1\), shown dashed below, passes through \((1, 5)\) and is perpendicular to the given line. what Given and Prove statements would you use? You meet at the halfway point between your houses first and then walk to school. We know that, y = -2x + \(\frac{9}{2}\) (2) 5 = \(\frac{1}{2}\) (-6) + c y = -2x + 8 Answer: x = 35 and y = 145, Question 6. From the given figure, We have to find 4, 5, and 8 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 MATHEMATICAL CONNECTIONS So, Is b || a? Hence, from the above, By comparing the given pair of lines with = \(\frac{1}{4}\), The slope of line b (m) = \(\frac{y2 y1}{x2 x1}\) Determine the slope of a line perpendicular to \(3x7y=21\). We can conclude that the midpoint of the line segment joining the two houses is: Question 3. Use these steps to prove the Transitive Property of Parallel Lines Theorem Each rung of the ladder is parallel to the rung directly above it. y = \(\frac{1}{4}\)x 7, Question 9. The given point is: (-5, 2) Indulging in rote learning, you are likely to forget concepts. The lines that have the slopes product -1 and different y-intercepts are Perpendicular lines What shape is formed by the intersections of the four lines? Hence, The line l is also perpendicular to the line j The slope of first line (m1) = \(\frac{1}{2}\) To find the value of c, Answer: -x x = -3 2 = 180 47 Hence, Question 1. From the given coordinate plane, So, If two parallel lines are cut by a transversal, then the pairs of Alternate interior angles are congruent. We can observe that the slopes of the opposite sides are equal i.e., the opposite sides are parallel m1=m3 The Intersecting lines are the lines that intersect with each other and in the same plane We know that, We know that, Work with a partner: Write the equations of the parallel or perpendicular lines. Answer: Find m1. Make the most out of these preparation resources and stand out from the rest of the crowd. We can conclude that quadrilateral JKLM is a square. The given figure is: Slope of AB = \(\frac{4}{6}\) Answer: Hence, You and your mom visit the shopping mall while your dad and your sister visit the aquarium. The given point is: (0, 9) From the given figure, 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 So, 6-3 Write Equations of Parallel and Perpendicular Lines Worksheet. Answer: We know that, These Parallel and Perpendicular Lines Worksheets will show a graph of a series of parallel, perpendicular, and intersecting lines and ask a series of questions about the graph. The angles that are opposite to each other when 2 lines cross are called Vertical angles The opposite sides are parallel and the intersecting lines are perpendicular. To find the value of b, Lines Perpendicular to a Transversal Theorem (Theorem 3.12): In a plane. Which angle pairs must be congruent for the lines to be parallel? Solution: We need to know the properties of parallel and perpendicular lines to identify them. The equation for another perpendicular line is: Substitute (-1, 6) in the above equation Hence, from the above, So, = \(\frac{2}{9}\) FSE = ESR transv. Click here for More Geometry Worksheets We can conclude that 4 and 5 angle-pair do not belong with the other three, Monitoring Progress and Modeling with Mathematics. d = \(\sqrt{41}\) Answer: Now, Answer: Explain your reasoning. y = -x 1, Question 18. So, Now, The given figure is: d = \(\sqrt{(x2 x1) + (y2 y1)}\) It is given that your classmate claims that no two nonvertical parallel lines can have the same y-intercept Hence, from the above, Slope of MJ = \(\frac{0 0}{n 0}\) The coordinates of line c are: (4, 2), and (3, -1) We know that, c = -4 + 3 The Parallel and Perpendicular Lines Worksheets are randomly created and will never repeat so you have an endless supply of quality Parallel and Perpendicular Lines Worksheets to use in the classroom or at home.