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We can conclude that y = -2 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. So, Question 13. In the proof in Example 4, if you use the third statement before the second statement. Fro the given figure, Hence, from the above, y = 2x + 1 True, the opposite sides of a rectangle are parallel lines. ERROR ANALYSIS d = \(\sqrt{(x2 x1) + (y2 y1)}\) Answer: We have to keep the lengths of the length of the rectangles the same and the widths of the rectangle also the same, Question 3. Answer: m2 = -1 The portion of the diagram that you used to answer Exercise 26 on page 130 is: Question 2. We know that, We can conclude that the alternate exterior angles are: 1 and 8; 7 and 2. The parallel line equation that is parallel to the given equation is: The angles that have the opposite corners are called Vertical angles Hence, from the above, Hence, from the above, So, P(3, 8), y = \(\frac{1}{5}\)(x + 4) Answer: = 9.48 Hence, from the above, Compare the given points with (x1, y1), and (x2, y2) \(\frac{13-4}{2-(-1)}\) y = \(\frac{1}{2}\)x \(\frac{1}{2}\), Question 10. Hence, from the above, We can conclude that Hence, from the above, Whereas, if the slopes of two given lines are negative reciprocals of each other, they are considered to be perpendicular lines. (8x + 6) = 118 (By using the Vertical Angles theorem) We can conclude that the value of k is: 5. y = \(\frac{1}{3}\)x + c Prove that horizontal lines are perpendicular to vertical lines. The given figure is: We can observe that 1 and 2 are the alternate exterior angles In spherical geometry, all points are points on the surface of a sphere. We can conclude that 1 and 3 pair does not belong with the other three. Explain your reasoning. 2y + 4x = 180 From the coordinate plane, We know that, Write an equation of the line that passes through the given point and has the given slope. They are always equidistant from each other. your friend claims to be able to make the shot Shown in the diagram by hitting the cue ball so that m1 = 25. Hence, from the above, 1 = 2 = 123, Question 11. k = -2 + 7 The sum of the angle measure between 2 consecutive interior angles is: 180 Answer: Answer: Question 28. We know that, -x = x 3 plane(s) parallel to plane LMQ You and your mom visit the shopping mall while your dad and your sister visit the aquarium. We can conclude that we can not find the distance between any two parallel lines if a point and a line is given to find the distance, Question 2. y = -2x + c We know that, Slope of the line (m) = \(\frac{-1 2}{3 + 1}\) Hence, from the above, Question 12. x = c According to the Corresponding Angles Theorem, the corresponding angles are congruent 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 y = 27.4 Work with a partner: Write the equations of the parallel or perpendicular lines. So, Hence, from the above, So, By comparing eq. We can conclude that both converses are the same Hence, from the above, y = \(\frac{1}{2}\)x 7 Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. The equation that is perpendicular to the given equation is: y = mx + b Hence, Draw \(\overline{P Z}\), Question 8. 3 = 60 (Since 4 5 and the triangle is not a right triangle) Answer: Explain our reasoning. CRITICAL THINKING 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 (x1, y1), (x2, y2) what Given and Prove statements would you use? Explain your reasoning. So, \(m_{}=9\) and \(m_{}=\frac{1}{9}\), 13. a.) The slopes are the same and the y-intercepts are different 1 5 x = 29.8 and y = 132, Question 7. Examine the given road map to identify parallel and perpendicular streets. By using the Perpendicular transversal theorem, y = -2x + 8 So, Now, x = 54 The points are: (-3, 7), (0, -2) Answer: Observe the horizontal lines in E and Z and the vertical lines in H, M and N to notice the parallel lines. Answer: Here is a quick review of the point/slope form of a line. Answer: Answer: EG = 7.07 From the figure, E (-4, -3), G (1, 2) So, So, 69 + 111 = 180 y = \(\frac{1}{3}\)x + c It is given that m || n To find the distance from point X to \(\overline{W Z}\), Hence, from the coordinate plane, So, From the given bars, We can conclude that the parallel lines are: Question 37. We can conclude that Use a graphing calculator to verify your answer. We have to find the distance between A and Y i.e., AY We know that, We have to find the distance between X and Y i.e., XY Now, We can observe that the given angles are consecutive exterior angles Verticle angle theorem: What is the distance that the two of you walk together? In Exercises 47 and 48, use the slopes of lines to write a paragraph proof of the theorem. Substitute (-1, 6) in the above equation Lines Perpendicular to a Transversal Theorem (Thm. Answer: Question 8. The distance wont be in negative value, Answer: We know that, The Converse of the consecutive Interior angles Theorem states that if the consecutive interior angles on the same side of a transversal line intersecting two lines are supplementary, then the two lines are parallel. Where, By using the parallel lines property, Hence, from the above, Answer: XY = 6.32 y = 2x + c The given figure is: y = \(\frac{1}{4}\)x + b (1) By comparing the slopes, The slope of one line is the negative reciprocal of the other line. These worksheets will produce 6 problems per page. The midpoint of PQ = (\(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\)) MODELING WITH MATHEMATICS d = \(\sqrt{(x2 x1) + (y2 y1)}\) Answer: Parallel to \(x=2\) and passing through (7, 3)\). We get The slope of second line (m2) = 1 In spherical geometry. We can observe that the given lines are perpendicular lines Hence, from the given figure, The conjectures about perpendicular lines are: 10. Hence, from the above, c = 8 11y = 77 If you will see a tiger, then you go to the zoo-> False. Are the two linear equations parallel, perpendicular, or neither? We can observe that By comparing the given pair of lines with y = \(\frac{1}{2}\)x + 1 -(1) Slope of ST = \(\frac{1}{2}\), Slope of TQ = \(\frac{3 6}{1 2}\) Answer: Question 24. The given figure is: -2 = \(\frac{1}{2}\) (2) + c From the given figure, So, A(- \(\frac{1}{4}\), 5), x + 2y = 14 Slope of line 1 = \(\frac{9 5}{-8 10}\) m = 2 3 (y 175) = x 50 Now, Hence, from the above, So, From the given figure, We can conclude that the number of points of intersection of intersecting lines is: 1, c. The points of intersection of coincident lines: b. Alternate Exterior angles Theorem Use the theorems from Section 3.2 and the converses of those theorems in this section to write three biconditional statements about parallel lines and transversals. Describe and correct the error in the students reasoning 4 = 105, To find 5: From the above figure, So, We can conclude that the given pair of lines are parallel lines. Question 30. The given figure is: Geometry chapter 3 parallel and perpendicular lines answer key Apps can be a great way to help learners with their math. Hence, Answer: Slope (m) = \(\frac{y2 y1}{x2 x1}\) The given equation is: We can observe that From the figure, Answer: Question 1. We can conclude that the given pair of lines are non-perpendicular lines, work with a partner: Write the number of points of intersection of each pair of coplanar lines. We know that, The given figure is: 2 6, c. 1 ________ by the Alternate Exterior Angles Theorem (Thm. The measure of 1 is 70. Line 2: (7, 0), (3, 6) = 104 Answer: We can conclude that So, The best editor is directly at your fingertips offering you a range of advantageous instruments for submitting a Algebra 1 Worksheet 3 6 Parallel And Perpendicular Lines. Answer: Question 50. 1 2 3 4 5 6 7 8 What are the coordinates of the midpoint of the line segment joining the two houses? ax + by + c = 0 So, Answer: We can conclude that 75 and 75 are alternate interior angles, d. x = 133 Use the results of Exploration 1 to write conjectures about the following pairs of angles formed by two parallel lines and a transversal. The product of the slopes of perpendicular lines is equal to -1 We can observe that there are 2 perpendicular lines These Parallel and Perpendicular Lines Worksheets are great for practicing identifying parallel lines from pictures. Question 35. Question 39. In which of the following diagrams is \(\overline{A C}\) || \(\overline{B D}\) and \(\overline{A C}\) \(\overline{C D}\)? 1 = 53.7 and 5 = 53.7 How are they different? We can observe that 35 and y are the consecutive interior angles So, y = mx + c We can conclude that the equation of the line that is perpendicular bisector is: Each unit in the coordinate plane corresponds to 10 feet 1 = 123 and 2 = 57. construction change if you were to construct a rectangle? Now, So, The equation that is perpendicular to the given equation is: 2x = 108 Answer Key Parallel and Perpendicular Lines : Shapes Write a relation between the line segments indicated by the arrows in each shape. By using the corresponding angles theorem, The standard form of the equation is: To find the value of c, \(\begin{aligned} y-y_{1}&=m(x-x_{1}) \\ y-(-2)&=\frac{1}{2}(x-8) \end{aligned}\). 3 + 8 = 180 In Exploration 2, It is given that Answer: b. Unfold the paper and examine the four angles formed by the two creases. The point of intersection = (-3, -9) The intersection point of y = 2x is: (2, 4) line(s) perpendicular to . y = \(\frac{2}{3}\)x + 9, Question 10. Hence, Now, 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. m is the slope Solution: We need to know the properties of parallel and perpendicular lines to identify them. 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 . m1m2 = -1 XZ = 7.07 (5y 21) and 116 are the corresponding angles The given equation is: We know that, x = 14.5 and y = 27.4, Question 9. So, AP : PB = 2 : 6 Think of each segment in the diagram as part of a line. The given points are: x + 2y = 2 From the given figure, c = -2 We know that, S. Giveh the following information, determine which lines it any, are parallel. Write the equation of the line that is perpendicular to the graph of 53x y = , and Hence, The slopes of the parallel lines are the same Hence, The equation of the line along with y-intercept is: Question 13. We can observe that Answer: The given figure is: Slope of AB = \(\frac{-4 2}{5 + 3}\) y = 3x + c We get, Now, We can conclude that the converse we obtained from the given statement is true Lets draw that line, and call it P. Lets also call the angle formed by the traversal line and this new line angle 3, and we see that if we add some other angle, call it angle 4, to it, it will be the same as angle 2. -3 = -2 (2) + c Slope (m) = \(\frac{y2 y1}{x2 x1}\) Hence, from the above, b is the y-intercept \(\frac{1}{3}\)m2 = -1 So, Justify your answers. answer choices Parallel Perpendicular Neither Tags: MGSE9-12.G.GPE.5 Question 7 300 seconds According to Alternate interior angle theorem, Exploration 2 comes from Exploration 1 Write a conjecture about \(\overline{A B}\) and \(\overline{C D}\). We can observe that the given lines are perpendicular lines y = 2x + c Step 5: We can conclude that option D) is correct because parallel and perpendicular lines have to be lie in the same plane, Question 8. Answer: Question 16. 4 and 5 Step 2: Let the congruent angle be P = \(\frac{3}{4}\) When we compare the given equation with the obtained equation, Now, We know that, m1 m2 = -1 Perpendicular lines have slopes that are opposite reciprocals, so remember to find the reciprocal and change the sign. According to the Perpendicular Transversal Theorem, 2x + y = 162(1) Write the equation of the line that is perpendicular to the graph of 9y = 4x , and whose y-intercept is (0, 3). Substitute (3, 4) in the above equation 8x and 96 are the alternate interior angles In the diagram below. Therefore, they are parallel lines. Two nonvertical lines in the same plane, with slopes \(m_{1}\) and \(m_{2}\), are perpendicular if the product of their slopes is \(1: m1m2=1\). d = \(\sqrt{(x2 x1) + (y2 y1)}\) Because j K, j l What missing information is the student assuming from the diagram? So, The coordinates of line 2 are: (2, -4), (11, -6) Name two pairs of congruent angles when \(\overline{A D}\) and \(\overline{B C}\) are parallel? So, Question 14. Answer: Question 25. DIFFERENT WORDS, SAME QUESTION By using the Alternate Exterior Angles Theorem, Answer: Compare the given points with (x1, y1), (x2, y2) XY = \(\sqrt{(x2 x1) + (y2 y1)}\) Draw another arc by using a compass with above half of the length of AB by taking the center at B above AB Explain your reasoning. . a.) Each step is parallel to the step immediately above it. The points are: (-2, 3), (\(\frac{4}{5}\), \(\frac{13}{5}\)) (5y 21) = 116 Alternate Interior Angles are a pair of angleson the inner side of each of those two lines but on opposite sides of the transversal. Compare the given coordinates with how many right angles are formed by two perpendicular lines? ATTENDING TO PRECISION Find the measure of the missing angles by using transparent paper. 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 80, Question 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. Find the distance from point A to the given line. y = \(\frac{137}{5}\) To find the distance from point A to \(\overline{X Z}\), (B) Students must unlock 5 locks by: 1: determining if two given slopes are parallel, perpendicular or neither. We can conclude that the distance from point A to \(\overline{X Z}\) is: 4.60. c = 3 Hence, from the above, Which of the following is true when are skew? So, Answer: For the intersection point of y = 2x, 1 = 4 Prove: t l We know that, We can observe that the slopes of the opposite sides are equal i.e., the opposite sides are parallel Hence, 1 = 2 m1m2 = -1 c = 2 We can observe that y = -x + 8 y = \(\frac{3}{2}\)x 1 Answer: = \(\frac{1}{-4}\) According to the Perpendicular Transversal Theorem, Answer: Question 2. Hence,f rom the above, You meet at the halfway point between your houses first and then walk to school. A(- 3, 7), y = \(\frac{1}{3}\)x 2 m2 and m4 We know that, So, Consider the following two lines: Both lines have a slope \(m=\frac{3}{4}\) and thus are parallel. Slope of QR = \(\frac{-2}{4}\) line(s) parallel to . 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 The given figure is: Find both answers. b = 2 Line 2: (2, 1), (8, 4) 7x = 84 According to Euclidean geometry, For example, if the equation of two lines is given as, y = 1/5x + 3 and y = - 5x + 2, we can see that the slope of one line is the negative reciprocal of the other. According to the Consecutive Exterior angles Theorem, To find the value of c, Answer: 5 = 8 (1) So, Hence, from the above, x = 20 Hence, from the above, So, P(0, 0), y = 9x 1 From the given figure, Slope of AB = \(\frac{4}{6}\) We know that, b is the y-intercept y y1 = m (x x1) y = 2x + 3, Question 23. XY = \(\sqrt{(3 + 3) + (3 1)}\) y = \(\frac{1}{4}\)x + c These worksheets will produce 6 problems per page. To find the value of c, substitute (1, 5) in the above equation a. We can conclude that A new road is being constructed parallel to the train tracks through points V. An equation of the line representing the train tracks is y = 2x. : n; same-side int. Hence, A (x1, y1), and B (x2, y2) Determine the slope of a line parallel to \(y=5x+3\). A(6, 1), y = 2x + 8 In spherical geometry, is it possible that a transversal intersects two parallel lines? We have to find the point of intersection Substitute A (-6, 5) in the above equation to find the value of c Prove: l || m The midpoint of PQ = (\(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\)) y = mx + b Two nonvertical lines in the same plane, with slopes m1 and m2, are parallel if their slopes are the same, m1 = m2. So, y = \(\frac{1}{6}\)x 8 = \(\frac{0 + 2}{-3 3}\) We can conclude that the value of x when p || q is: 54, b. a. m5 + m4 = 180 //From the given statement The given parallel line equations are: The product of the slopes of the perpendicular lines is equal to -1 P = (7.8, 5) ax + by + c = 0 The coordinates of the line of the first equation are: (-1.5, 0), and (0, 3) Parallel and Perpendicular Lines Perpendicular Lines Two nonvertical lines are perpendicular if their slopes are opposite reciprocals of each other. ERROR ANALYSIS Verify your answer. (- 5, 2), y = 2x 3 A(3, 4),y = x + 8 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. So, Answer: Any fraction that contains 0 in the numerator has its value equal to 0 So, Justify your answers. x = 6 We know that, Slope of JK = \(\frac{n 0}{0 0}\) To find the value of c, We can conclude that x = \(\frac{3}{2}\) We can conclude that Now, a. So, XY = 6.32 We can observe that We can observe that 48 and y are the consecutive interior angles and y and (5x 17) are the corresponding angles We can observe that when r || s, To find an equation of a line, first use the given information to determine the slope. y= \(\frac{1}{3}\)x + 4 Difference Between Parallel and Perpendicular Lines, Equations of Parallel and Perpendicular Lines, Parallel and Perpendicular Lines Worksheets. y = 3x 6, Question 20. m1m2 = -1 3.2). From the given figure, The number of intersection points for parallel lines is: 0 It is given that We know that, m is the slope alternate interior, alternate exterior, or consecutive interior angles. The consecutive interior angles are: 2 and 5; 3 and 8. 1 = 2 So, m2 = \(\frac{1}{3}\) Slope of AB = \(\frac{5 1}{4 + 2}\) The representation of the given pair of lines in the coordinate plane is: From the given coordinate plane, A triangle has vertices L(0, 6), M(5, 8). Substitute (-5, 2) in the above equation We can observe that the given angles are the consecutive exterior angles Hence, from the above, The point of intersection = (0, -2) Explain your reasoning? Hence, from the above, 3.12) From the given figure, Hence, from the above, We know that, Answer: ABSTRACT REASONING We know that, Each bar is parallel to the bar directly next to it. Question 5. Answer: x = \(\frac{40}{8}\) The length of the field = | 20 340 | Answer: The slope of vertical line (m) = \(\frac{y2 y1}{x2 x1}\) Explain your reasoning. 1 unit either in the x-plane or y-plane = 10 feet 2x = 135 15 From the given figure, Answer: Slope of the line (m) = \(\frac{y2 y1}{x2 x1}\) y = -3 6 Answer: It is given that a gazebo is being built near a nature trail. You decide to meet at the intersection of lines q and p. Each unit in the coordinate plane corresponds to 50 yards. Tell which theorem you use in each case. y 500 = -3x + 150 So, So, Therefore, these lines can be identified as perpendicular lines. Often you will be asked to find the equation of a line given some geometric relationshipfor instance, whether the line is parallel or perpendicular to another line. In exercises 25-28. copy and complete the statement. plane(s) parallel to plane ADE Now, Substitute (-1, -9) in the given equation y = 3x + 9 Now, So, Consider the following two lines: Consider their corresponding graphs: Figure 3.6.1 The equation that is perpendicular to the given line equation is: w v and w y 17x + 27 = 180 2 = 57 The slope that is perpendicular to the given line is: x = y = 61, Question 2. that passes through the point (4, 5) and is parallel to the given line. (x1, y1), (x2, y2) We can observe that 4.5 Equations of Parallel and Perpendicular Lines Solving word questions According to Perpendicular Transversal Theorem, (2) Work with a partner: The figure shows a right rectangular prism. It is given that m || n Now, y = -x + c According to Corresponding Angles Theorem, Parallel to \(x+4y=8\) and passing through \((1, 2)\). 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? We get Now, m2 = \(\frac{1}{2}\), b2 = -1 We know that, The given figure is: The pair of angles on one side of the transversal and inside the two lines are called the Consecutive interior angles. Step 1: Find the slope \(m\). The coordinates of P are (4, 4.5). Substitute A (-\(\frac{1}{4}\), 5) in the above equation to find the value of c 1 = 2 6x = 140 53 The given point is: P (4, 0) If we draw the line perpendicular to the given horizontal line, the result is a vertical line. m is the slope Answer: Now, We can conclude that the distance between the meeting point and the subway is: 364.5 yards, Question 13. d = 32 Hence, from the above, One answer is the line that is parallel to the reference line and passing through a given point. The slopes of parallel lines, on the other hand, are exactly equal. Question: What is the difference between perpendicular and parallel? When we compare the given equation with the obtained equation, So, Proof of the Converse of the Consecutive Exterior angles Theorem: 2 + 3 = 180 We know that, The equation of the line that is perpendicular to the given equation is: y = \(\frac{1}{3}\)x + \(\frac{475}{3}\) 1 + 57 = 180 x y + 4 = 0 0 = 3 (2) + c Given 1 2, 3 4 From the given figure, \(\frac{6 (-4)}{8 3}\) The slopes are equal fot the parallel lines So, So, The equation that is perpendicular to the given line equation is: Hence, from the above, We have to find the point of intersection The equation for another perpendicular line is: m2 = 2 What is the length of the field? The given figure is: The given figure is: The equation that is perpendicular to y = -3 is: