ATTENDING TO PRECISION 2 = \(\frac{1}{4}\) (8) + c = 2 In geometry, there are three different types of lines, namely, parallel lines, perpendicular lines, and intersecting lines. We can observe that the angle between b and c is 90 If two lines are intersected by a third line, is the third line necessarily a transversal? We can conclude that the value of x is: 20, Question 12. We know that, We can conclude that the equation of the line that is perpendicular bisector is: Compare the given points with We know that, Sandwich: The highlighted lines in the sandwich are neither parallel nor perpendicular lines. We can conclude that the alternate interior angles are: 4 and 5; 3 and 6, Question 14. Question 16. Answer: (x1, y1), (x2, y2) Hence, Hence, it can be said that if the slope of two lines is the same, they are identified as parallel lines, whereas, if the slope of two given lines are negative reciprocals of each other, they are identified as perpendicular lines. To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. You and your friend walk to school together every day. According to Perpendicular Transversal Theorem, By using the consecutive interior angles theorem, Are the numbered streets parallel to one another? It also shows that a and b are cut by a transversal and they have the same length c = -1 3 = 53.7 and 4 = 53.7 d = 17.02 (7x + 24) = 108 y = 3x + 9 The given figure is: Fro the given figure, y = 2x + c Line 1: (1, 0), (7, 4) Question 8. 8x = (4x + 24) = \(\frac{325 175}{500 50}\) Answer: Answer/Step-by-step Explanation: To determine if segment AB and CD are parallel, perpendicular, or neither, calculate the slope of each. The equation of the perpendicular line that passes through the midpoint of PQ is: (x1, y1), (x2, y2) Hence, from the above, So, Answer: The equation that is perpendicular to the given line equation is: PDF Solving Equations Involving Parallel and Perpendicular Lines Examples m = 2 a is perpendicular to d and b isperpendicular to c, Question 22. We know that, To find 4: 7x = 108 24 Now, Geometrically, we see that the line \(y=4x1\), shown dashed below, passes through \((1, 5)\) and is perpendicular to the given line. \(\frac{3}{2}\) . Answer: Question 28. The parallel lines do not have any intersecting points For the intersection point, Answer: consecutive interior So, \(\frac{6 (-4)}{8 3}\) The product of the slopes of the perpendicular lines is equal to -1 We know that, Substitute this slope and the given point into point-slope form. The given figure is: Answer: ANALYZING RELATIONSHIPS = (\(\frac{-2}{2}\), \(\frac{-2}{2}\)) 2 = 133 2 = 180 3 Here is a quick review of the point/slope form of a line. Hence, from the above, The y-intercept is: -8, Writing Equations of Parallel and Perpendicular Lines, Work with a partner: Write an equation of the line that is parallel or perpendicular to the given line and passes through the given point. -5 = \(\frac{1}{2}\) (4) + c y = mx + c Parallel and perpendicular lines worksheet answers key geometry Let the given points are: x = 97, Question 7. 1 8, d. m6 + m ________ = 180 by the Consecutive Interior Angles Theorem (Thm. The given table is: Hence, from the given figure, 6 (2y) 6(3) = 180 42 The slope of the given line is: m = \(\frac{1}{2}\) XY = \(\sqrt{(3 + 3) + (3 1)}\) Finding Parallel and Perpendicular Lines - mathsisfun.com So, The equation for another line is: A (-2, 2), and B (-3, -1) According to the Perpendicular Transversal theorem, Question 43. line(s) perpendicular to . The given point is:A (6, -1) Hence, from the above, From the given figure, Sketch what the segments in the photo would look like if they were perpendicular to the crosswalk. We know that, The given equation is: x = 147 14 (1) The two lines are Coincident when they lie on each other and are coplanar (B) If line E is parallel to line F and line F is parallel to line G, then line E is parallel to line G. Question 49. We know that, By using the Perpendicular transversal theorem, x = \(\frac{84}{7}\) Label its intersection with \(\overline{A B}\) as O. The equation for another perpendicular line is: The Converse of the alternate exterior angles Theorem: Here the given line has slope \(m=\frac{1}{2}\), and the slope of a line parallel is \(m_{}=\frac{1}{2}\). a) Parallel line equation: The equation of the parallel line that passes through (1, 5) is: So, Answer: If the slopes of two distinct nonvertical lines are equal, the lines are parallel. Answer: Question 24. Hence, from the above, 3 = 76 and 4 = 104 c = -2 y = \(\frac{1}{3}\)x 4 4 and 5 are adjacent angles a. \(\overline{C D}\) and \(\overline{A E}\) are Skew lines because they are not intersecting and are non coplanar 2 = 180 1 Given m1 = 115, m2 = 65 Hence, Write the equation of the line that is perpendicular to the graph of 9y = 4x , and whose y-intercept is (0, 3). Question 27. The slope of the equation that is parallel t the given equation is: 3 PROVING A THEOREM Substitute (1, -2) in the above equation From the given figure, Given 1 2, 3 4 The sum of the angle measure between 2 consecutive interior angles is: 180 So, The slopes are equal fot the parallel lines Substitute (-2, 3) in the above equation Perpendicular lines are lines in the same plane that intersect at right angles (\(90\) degrees). According to the Alternate Interior Angles theorem, the alternate interior angles are congruent Now, Now, The product of the slopes of the perpendicular lines is equal to -1 REASONING We know that, So, -9 = 3 (-1) + c Explain. The slope that is perpendicular to the given line is: Now, Answer: We can observe that The equation of the line that is perpendicular to the given line equation is: We can observe that, Now, So, Explain your reasoning. parallel Answer: Explanation: In the above image we can observe two parallel lines. as shown. Which of the following is true when are skew? 132 = (5x 17) Answer: Quick Link for All Parallel and Perpendicular Lines Worksheets, Detailed Description for All Parallel and Perpendicular Lines Worksheets. Use these steps to prove the Transitive Property of Parallel Lines Theorem We can conclude that y = \(\frac{1}{2}\)x 2 Answer: m = -1 [ Since we know that m1m2 = -1] XZ = \(\sqrt{(4 + 3) + (3 4)}\) The vertical angles are: 1 and 3; 2 and 4 For example, PQ RS means line PQ is perpendicular to line RS. Parallel to \(\frac{1}{5}x\frac{1}{3}y=2\) and passing through \((15, 6)\). Each unit in the coordinate plane corresponds to 10 feet Answer: According to the Corresponding Angles Theorem, the corresponding angles are congruent We can observe that 35 and y are the consecutive interior angles Now, The slope of the horizontal line (m) = \(\frac{y2 y2}{x2 x1}\) We know that, Question 4. Hence, The line that passes through point F that appear skew to \(\overline{E H}\) is: \(\overline{F C}\), Question 2. \(\frac{1}{2}\)x + 7 = -2x + \(\frac{9}{2}\) The given lines are: a. k = -2 + 7 Now, So, ERROR ANALYSIS 2. If you use the diagram below to prove the Alternate Exterior Angles Converse. From the given figure, X (-3, 3), Z (4, 4) a. The coordinates of the school = (400, 300) Parallel and Perpendicular Lines Maintaining Mathematical Proficiency Find the slope of the line. 3 + 133 = 180 (By using the Consecutive Interior angles theorem) When finding an equation of a line perpendicular to a horizontal or vertical line, it is best to consider the geometric interpretation. Answer: From the given figure, Let A and B be two points on line m. Question 5. intersecting Answer: Explanation: We can conclude that the parallel lines are: Answer: Question 18. Now, 2x + 72 = 180 The lines that have the same slope and different y-intercepts are Parallel lines We can conclude that the distance between the given 2 points is: 6.40. Perpendicular to \(\frac{1}{2}x\frac{1}{3}y=1\) and passing through \((10, 3)\). From the given figure, So, Hence, from the above, Justify your conjecture. (8x + 6) = 118 (By using the Vertical Angles theorem) Yes, there is enough information in the diagram to conclude m || n. Explanation: Quiz: Parallel and Perpendicular Lines - Quizizz -x + 2y = 12 The given figure shows that angles 1 and 2 are Consecutive Interior angles We know that, So, corresponding Answer: For a pair of lines to be perpendicular, the product of the slopes i.e., the product of the slope of the first line and the slope of the second line will be equal to -1 y = \(\frac{1}{2}\)x + 7 -(1) So, Now, y = \(\frac{1}{2}\)x 2 We can conclude that the pair of skew lines are: Grade: Date: Parallel and Perpendicular Lines. = 2, The slope of line c (m) = \(\frac{y2 y1}{x2 x1}\) So, We can conclude that the alternate interior angles are: 3 and 6; 4 and 5, Question 7. Answer: Question 4. (50, 500), (200, 50) 200), d. What is the distance from the meeting point to the subway? Answer: Question 20. 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, y = \(\frac{1}{5}\)x + \(\frac{4}{5}\) Geometrically, we note that if a line has a positive slope, then any perpendicular line will have a negative slope. y = 2x + 3, Question 23. Question 3. Hence, Which angle pair does not belong with the other three? We get So, Explain. We can observe that -x = x 3 We can conclude that the value of x is: 60, Question 6. From the above figure, Slope of RS = 3, Slope of ST = \(\frac{3 1}{1 5}\) 1 = 2 = 133 and 3 = 47. In spherical geometry, is it possible that a transversal intersects two parallel lines? Now, Two lines are termed as parallel if they lie in the same plane, are the same distance apart, and never meet each other. Parallel lines are always equidistant from each other. We can observe that the given pairs of angles are consecutive interior angles Answer: b.) We know that, = 2 (2) Eq. Name them. Question 3. y = 2x + c A(- \(\frac{1}{4}\), 5), x + 2y = 14 We know that, The given figure is: By using the Corresponding angles Theorem, = \(\sqrt{(9 3) + (9 3)}\) We know that, For perpendicular lines, In Exercises 13 16. write an equation of the line passing through point P that s parallel to the given line. x = \(\frac{108}{2}\) We can conclude that option D) is correct because parallel and perpendicular lines have to be lie in the same plane, Question 8. So, what Given and Prove statements would you use? Each unit in the coordinate plane corresponds to 50 yards. We can conclude that y = -3x + 150 + 500 We know that, Answer: x = 29.8 and y = 132, Question 7. Answer: F if two coplanar strains are perpendicular to the identical line then the 2 strains are. Answer: 5x = 149 Find the measures of the eight angles that are formed. = \(\frac{3 2}{-2 2}\) The given figure is: The given coplanar lines are: Now, The slope of first line (m1) = \(\frac{1}{2}\) Alternate Exterior Angles Theorem (Thm. Slope of AB = \(\frac{1 + 4}{6 + 2}\) The equation of the line along with y-intercept is: MODELING WITH MATHEMATICS Work with a partner: Write the equations of the parallel or perpendicular lines. Write an equation for a line parallel to y = 1/3x - 3 through (4, 4) Q. x = 20 Hence, from the above, Answer: THOUGHT-PROVOKING The given figure is: We know that, Question 4. The given figure is: 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 equation is: b = -5 The two lines are Parallel when they do not intersect each other and are coplanar So, Hence, from the above, Parallel to \(y=\frac{1}{4}x5\) and passing through \((2, 1)\). From the given figure, For example, the letter H, in which the vertical lines are parallel and the horizontal line is perpendicular to both the vertical lines. y = -3x + 19, Question 5. So, The coordinates of a quadrilateral are: So, So, Answer: Question 14. The given point is: A(3, 6) x1 = x2 = x3 . You started solving the problem by considering the 2 lines parallel and two lines as transversals A (-1, 2), and B (3, -1) The given figure is: 4 = 105, To find 5: 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. Explain your reasoning. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. Hence, XY = \(\sqrt{(x2 x1) + (y2 y1)}\) The given equation is: We can conclude that the slope of the given line is: 3, Question 3. -x + 4 = x 3 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. These Parallel and Perpendicular Lines Worksheets are great for practicing identifying perpendicular lines from pictures. m2 = -1 We can observe that d = \(\sqrt{41}\) We can observe that the given lines are parallel lines Hence, from the above, Hence, from the above, m2 = -3 Question 41. FCA and __________ are alternate exterior angles. So, Likewise, parallel lines become perpendicular when one line is rotated 90. Perpendicular to \(4x5y=1\) and passing through \((1, 1)\). -x + 2y = 14 Compare the given points with x = 6 Answer: Question 32. DRAWING CONCLUSIONS b is the y-intercept Equations parallel and perpendicular lines answer key b) Perpendicular line equation: c = 2 + 2 = 2, The slope of line b (m) = \(\frac{y2 y1}{x2 x1}\) So, We can observe that 8x = 118 6 These worksheets will produce 6 problems per page. If the slopes of the opposite sides of the quadrilateral are equal, then it is called as Parallelogram We can conclude that quadrilateral JKLM is a square. Answer: Answer: y = \(\frac{3}{2}\) + 4 and y = \(\frac{3}{2}\)x \(\frac{1}{2}\) We can conclude that the distance from point A to the given line is: 1.67. According to the Transitive Property of parallel lines, The equation of a line is: A(- 2, 3), y = \(\frac{1}{2}\)x + 1 -2 = \(\frac{1}{3}\) (-2) + c By using the linear pair theorem, Now, P(4, 6)y = 3 3 = -2 (-2) + c We know that, The equation that is parallel to the given equation is: Perpendicular to \(y3=0\) and passing through \((6, 12)\). Answer: So, Compare the given equation with Hence, from the above, Now, (2) (1) \(m_{}=\frac{4}{3}\) and \(m_{}=\frac{3}{4}\), 15. The slopes are the same but the y-intercepts are different Hence, from the above figure, (b) perpendicular to the given line.
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