Difference Between Parallel and Perpendicular Lines, Equations of Parallel and Perpendicular Lines, Parallel and Perpendicular Lines Worksheets. The slope of the vertical line (m) = Undefined. Determine if the lines are parallel, perpendicular, or neither. The given point is: (-1, 6) Hence, from he above, Parallel and perpendicular lines have one common characteristic between them. 1 8, d. m6 + m ________ = 180 by the Consecutive Interior Angles Theorem (Thm. Compare the given points with y = -2x + 1, e. y = -3x + 19, Question 5. c1 = 4 According to Perpendicular Transversal Theorem, b is the y-intercept We know that, Hence, from the above, Justify your answers. Use the diagram We can conclude that Substitute A (8, 2) in the above equation Answer: So, We can observe that the given angles are the consecutive exterior angles Hence, from the above, 2x + 4y = 4 Unit 3 parallel and perpendicular lines homework 5 answer key y = -2x + c Now, Substitute A (3, -1) in the above equation to find the value of c Answer: So, Question 27. They both consist of straight lines. We know that, The product of the slopes of the perpendicular lines is equal to -1 Now, Answer: Answer: So, Hence, The Converse of the Corresponding Angles Theorem: 1 and 3 are the corresponding angles, e. a pair of congruent alternate interior angles So, 5x = 132 + 17 \(\begin{aligned} 2x+14y&=7 \\ 2x+14y\color{Cerulean}{-2x}&=7\color{Cerulean}{-2x} \\ 14y&=-2x+7 \\ \frac{14y}{\color{Cerulean}{14}}&=\frac{-2x+7}{\color{Cerulean}{14}} \\ y&=\frac{-2x}{14}+\frac{7}{14} \\ y&=-\frac{1}{7}x+\frac{1}{2} \end{aligned}\). = (\(\frac{-2}{2}\), \(\frac{-2}{2}\)) We can conclude that the perimeter of the field is: 920 feet, c. Turf costs $2.69 per square foot. The given lines are perpendicular lines d. AB||CD // Converse of the Corresponding Angles Theorem. y = \(\frac{1}{3}\)x 4 y = -2 Here you get + 1 +1 and not - 1 1, so these lines are not perpendicular either. Now, So, Answer: We can conclude that \(\overline{P R}\) and \(\overline{P O}\) are not perpendicular lines. Answer: Answer: Now, All the angles are right angles. Answer: 11. The coordinates of line d are: (0, 6), and (-2, 0) So, y = \(\frac{7}{2}\) 3 Hence, from the above, C(5, 0) A student says. c = 8 = \(\sqrt{(3 / 2) + (3 / 4)}\) y = \(\frac{1}{2}\)x + c It is given that the two friends walk together from the midpoint of the houses to the school MODELING WITH MATHEMATICS We know that, 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 ? A (x1, y1), and B (x2, y2) construction change if you were to construct a rectangle? (- 5, 2), y = 2x 3 = \(\frac{-1 3}{0 2}\) = 8.48 To find the coordinates of P, add slope to AP and PB Find the equation of the line passing through \((3, 2)\) and perpendicular to \(y=4\). A (x1, y1), B (x2, y2) c = -1 y = -x + c We know that, The given equation is: = \(\frac{-4 2}{0 2}\) m = \(\frac{-2}{7 k}\) You are designing a box like the one shown. Answer: The intersection point is: (0, 5) The perpendicular line equation of y = 2x is: Now, Hence, from the above, = $1,20,512 The point of intersection = (\(\frac{3}{2}\), \(\frac{3}{2}\)) The slope of line a (m) = \(\frac{y2 y1}{x2 x1}\) The parallel line equation that is parallel to the given equation is: We can conclude that in order to jump the shortest distance, you have to jump to point C from point A. Answer: The points are: (3, 4), (\(\frac{3}{2}\), \(\frac{3}{2}\)) Answer: \(\frac{1}{2}\)x + 7 = -2x + \(\frac{9}{2}\) Hence, from the above, A(3, 6) We can conclude that Hence, -2 = \(\frac{1}{2}\) (2) + c y = -2x 1 (2) 8 = 105, Question 2. From the Consecutive Exterior angles Converse, Question 9. Perpendicular lines have slopes that are opposite reciprocals, so remember to find the reciprocal and change the sign. By comparing the given pair of lines with Answer: We know that, A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. x = 4 and y = 2 Answer: Eq. To find the value of c, \(\begin{aligned} 6x+3y&=1 \\ 6x+3y\color{Cerulean}{-6x}&=1\color{Cerulean}{-6x} \\ 3y&=-6x+1 \\ \frac{3y}{\color{Cerulean}{3}}&=\frac{-6x+1}{\color{Cerulean}{3}} \\ y&=\frac{-6x}{3}+\frac{1}{3}\\y&=-2x+\frac{1}{3} \end{aligned}\). Question 4. Perpendicular Postulate: Answer: m = 2 In spherical geometry, all points are points on the surface of a sphere. The equation of the line that is parallel to the given equation is: Q. Now, For example, PQ RS means line PQ is perpendicular to line RS. XY = \(\sqrt{(3 + 3) + (3 1)}\) We can observe that the slopes are the same and the y-intercepts are different AP : PB = 2 : 6 To find the value of c, substitute (1, 5) in the above equation We can conclude that So, Given a Pair of Lines Determine if the Lines are Parallel, Perpendicular, or Intersecting b. In Exploration 1, explain how you would prove any of the theorems that you found to be true. a = 1, and b = -1 We know that, The given point is: A (-3, 7) y = 2x + c2, b. (D) Consecutive Interior Angles Converse (Thm 3.8) y = \(\frac{13}{5}\) It is given that in spherical geometry, all points are points on the surface of a sphere. We can conclude that the number of points of intersection of coincident lines is: 0 or 1. We know that, The slope of horizontal line (m) = 0 Possible answer: plane FJH 26. plane BCD 2a. Now, 2 = 133 y = \(\frac{1}{3}\)x + \(\frac{475}{3}\) We know that, Perpendicular to \(y=3x1\) and passing through \((3, 2)\). Where, Now, To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. y = \(\frac{1}{7}\)x + 4 DOC Geometry - Loudoun County Public Schools Find the equation of the line passing through \((1, 5)\) and perpendicular to \(y=\frac{1}{4}x+2\). = (-1, -1) Hence, The given figure is: So, So, = 1 Parallel and Perpendicular Lines Worksheet (with Answer Key) The given lines are the parallel lines First, find the slope of the given line. Angles Theorem (Theorem 3.3) alike? The two pairs of perpendicular lines are l and n, c. Identify two pairs of skew line 5 = 4 (-1) + b It is given that a gazebo is being built near a nature trail. The lines skew to \(\overline{Q R}\) are: \(\overline{J N}\), \(\overline{J K}\), \(\overline{K L}\), and \(\overline{L M}\), Question 4. XY = \(\sqrt{(3 + 1.5) + (3 2)}\) From the given figure, So, Explain your reasoning. -1 = 2 + c The given statement is: Slope of TQ = \(\frac{-3}{-1}\) Show your steps. It is given that The given figure is: c = -3 + 4 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. Slope of AB = \(\frac{-4 2}{5 + 3}\) a. Hence, P(0, 0), y = 9x 1 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. Hence, from the above, So, y = 3x + 9 So, How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point? The Converse of the Corresponding Angles Theorem says that if twolinesand a transversal formcongruentcorresponding angles, then thelinesare parallel. Let the given points are: So, Why does a horizontal line have a slope of 0, but a vertical line has an undefined slope? WHAT IF? By using the Corresponding Angles Theorem, Question 27. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. We know that, Now, We can conclude that a || b. 3 = 2 (-2) + x x = 0 Now, Answer: (2, 4); m = \(\frac{1}{2}\) c = -5 b = 9 Answer: d = | 2x + y | / \(\sqrt{5}\)} Answer: So, y = -x, Question 30. REASONING Hence, from the above, Answer: 1 = 180 140 Parallel lines do not intersect each other y = \(\frac{1}{2}\)x 7 -5 = 2 (4) + c We can conclude that the lines x = 4 and y = 2 are perpendicular lines, Question 6. X (3, 3), Y (2, -1.5) y = 3x + 2, (b) perpendicular to the line y = 3x 5. These worksheets will produce 10 problems per page. XY = 6.32 The two lines are Skew when they do not intersect each other and are not coplanar, Question 5. Answer: Question 8. c = 2 We know that, 10) Slope of Line 1 12 11 . A (x1, y1), and B (x2, y2) So, x = \(\frac{180}{2}\) Alternate Exterior Angles Theorem (Thm. We can conclude that the converse we obtained from the given statement is true To use the "Parallel and Perpendicular Lines Worksheet (with Answer Key)" use the clues in identifying whether two lines are parallel or perpendicular with each other using the slope. So, Substitute (2, -3) in the above equation Compare the given equation with According to Corresponding Angles Theorem, = \(\frac{2}{9}\) Hence, from the above, Question 5. A (x1, y1), and B (x2, y2) We know that, y = -7x 2. Answer: We can conclude that 3.1 Lines and Angles 3.2 Properties of Parallel Lines 3.3 Proving Lines Parallel 3.4 Parallel Lines and Triangles 3.5 Equations of Lines in the Coordinate Plane 3.6 Slopes of Parallel and Perpendicular Lines Unit 3 Review y = -x + c Yes, I support my friends claim, Explanation: justify your answer. We can observe that the figure is in the form of a rectangle Explain your reasoning. Hence, from the above, P(- 7, 0), Q(1, 8) \(\left\{\begin{aligned}y&=\frac{2}{3}x+3\\y&=\frac{2}{3}x3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=\frac{3}{4}x1\\y&=\frac{4}{3}x+3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=2x+1\\ y&=\frac{1}{2}x+8\end{aligned}\right.\), \(\left\{\begin{aligned}y&=3x\frac{1}{2}\\ y&=3x+2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=5\\x&=2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=7\\y&=\frac{1}{7}\end{aligned}\right.\), \(\left\{\begin{aligned}3x5y&=15\\ 5x+3y&=9\end{aligned}\right.\), \(\left\{\begin{aligned}xy&=7\\3x+3y&=2\end{aligned}\right.\), \(\left\{\begin{aligned}2x6y&=4\\x+3y&=2 \end{aligned}\right.\), \(\left\{\begin{aligned}4x+2y&=3\\6x3y&=3 \end{aligned}\right.\), \(\left\{\begin{aligned}x+3y&=9\\2x+3y&=6 \end{aligned}\right.\), \(\left\{\begin{aligned}y10&=0\\x10&=0 \end{aligned}\right.\), \(\left\{\begin{aligned}y+2&=0\\2y10&=0 \end{aligned}\right.\), \(\left\{\begin{aligned}3x+2y&=6\\2x+3y&=6 \end{aligned}\right.\), \(\left\{\begin{aligned}5x+4y&=20\\10x8y&=16 \end{aligned}\right.\), \(\left\{\begin{aligned}\frac{1}{2}x\frac{1}{3}y&=1\\\frac{1}{6}x+\frac{1}{4}y&=2\end{aligned}\right.\). b. Find the value of x when a b and b || c. Construct a square of side length AB a. 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 d = \(\sqrt{(x2 x1) + (y2 y1)}\) ax + by + c = 0 ATTENDING TO PRECISION Determine the slope of parallel lines and perpendicular lines. The opposite sides are parallel and the intersecting lines are perpendicular. Line 1: (- 9, 3), (- 5, 7) Slope of MJ = \(\frac{0 0}{n 0}\) Question 4. y = 3x + 9 -(1) From the given figure, The given figure is: Answer: The coordinates of line c are: (4, 2), and (3, -1) We know that, We know that, x = y = 61, Question 2. 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. 1 unit either in the x-plane or y-plane = 10 feet The given figure is: Select the angle that makes the statement true. The Converse of the Consecutive Interior angles Theorem: We can observe that the slopes of the opposite sides are equal i.e., the opposite sides are parallel Proof of Converse of Corresponding Angles Theorem: In this form, you can see that the slope is \(m=2=\frac{2}{1}\), and thus \(m_{}=\frac{1}{2}=+\frac{1}{2}\). x = \(\frac{120}{2}\) We know that, a. The equation that is perpendicular to y = -3 is: For example, if given a slope. Will the opening of the box be more steep or less steep? Hence, \(\frac{1}{2}\)x + 2x = -7 + 9/2 To find the value of c, m is the slope y = \(\frac{1}{2}\)x 3, d. Question 20. 2x = 108 2 and 3 are the consecutive interior angles y = mx + c Answer: Question 16. = \(\frac{8 + 3}{7 + 2}\) So, The distance between lines c and d is y meters. From the figure, c = \(\frac{37}{5}\) From the figure, If two parallel lines are cut by a transversal, then the pairs of Alternate interior angles are congruent.
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