a) Parallel to the given line: Answer: List all possible correct answers. Decide whether it is true or false. Now, = (4, -3) x = \(\frac{3}{2}\) The lines that do not intersect to each other and are coplanar are called Parallel lines m2 = \(\frac{2}{3}\) 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. 5y = 137 Answer: Question 31. HOW DO YOU SEE IT? So, x = 23 Answer: It is given that a coordinate plane has been superimposed on a diagram of the football field where 1 unit is 20 feet. We can conclude that the distance of the gazebo from the nature trail is: 0.66 feet. m2 = \(\frac{1}{2}\) Question 17. Now, Quick Link for All Parallel and Perpendicular Lines Worksheets, Detailed Description for All Parallel and Perpendicular Lines Worksheets. So, Line 1: (10, 5), (- 8, 9) PROOF y = -2 (-1) + \(\frac{9}{2}\) By using the Perpendicular transversal theorem, 3.12) Perpendicular lines are denoted by the symbol . The points are: (-9, -3), (-3, -9) We know that, c = 5 7 y = mx + b Given that, Pot of line and points on the lines are given, we have to \(m_{}=\frac{3}{4}\) and \(m_{}=\frac{4}{3}\), 3. We know that, So, y = \(\frac{1}{3}\)x + c A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. b. x = 9 The coordinates of line a are: (0, 2), and (-2, -2) Now, The equation that is perpendicular to the given line equation is: -9 = 3 (-1) + c Hence, from the above, We know that, We can conclude that the given lines are neither parallel nor perpendicular. b) Perpendicular line equation: m a, n a, l b, and n b d = \(\sqrt{41}\) The Converse of the Corresponding Angles Theorem says that if twolinesand a transversal formcongruentcorresponding angles, then thelinesare parallel. The points of intersection of intersecting lines: PROVING A THEOREM The parallel line equation that is parallel to the given equation is: Answer: We can observe that To find the value of c, -x + 2y = 14 y = -3x + c (5y 21) = (6x + 32) Answer: Question 20. y = \(\frac{1}{4}\)x + 4, Question 24. Using X and Y as centers and an appropriate radius, draw arcs that intersect. 5 = -2 (-\(\frac{1}{4}\)) + c So, as shown. Hence, It is given that The points of intersection of parallel lines: 2x = 18 USING STRUCTURE x = -3 Answer: Question 34. Where, 6 (2y) 6(3) = 180 42 Now, c = -2 We know that, 5 = \(\frac{1}{3}\) + c Answer: 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. The equation of the perpendicular line that passes through the midpoint of PQ is: such as , are perpendicular to the plane containing the floor of the treehouse. Hence, from the above, The given equation is: Proof of Alternate exterior angles Theorem: The given point is: P (-8, 0) Hence, Answer: From the above figure, Slope of KL = \(\frac{n n}{n 0}\) We know that, Draw a line segment of any length and name that line segment as AB m2 = \(\frac{1}{2}\) Answer: The given equation is: Hence, Mathematically, this can be expressed as m1 = m2, where m1 and m2 are the slopes of two lines that are parallel. 1 + 2 = 180 (By using the consecutive interior angles theorem) 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). Answer: PROVING A THEOREM Sandwich: The highlighted lines in the sandwich are neither parallel nor perpendicular lines. FSE = ESR Now, c = 4 3 The distance wont be in negative value, We know that, A (x1, y1), and B (x2, y2) The given figure is: According to the Perpendicular Transversal Theorem, c.) False, parallel lines do not intersect each other at all, only perpendicular lines intersect at 90. Question 27. 8 = 105, Question 2. The given equation is: y = 2x + 3, Question 23. Answer: PROOF The given figure is: So, We know that, The slope of vertical line (m) = \(\frac{y2 y1}{x2 x1}\) 2 and 11 We can observe that the plane parallel to plane CDH is: Plane BAE. In Exercises 21 and 22, write and solve a system of linear equations to find the values of x and y. Find the values of x and y. Use a square viewing window. y = \(\frac{5}{3}\)x + \(\frac{40}{3}\) We can conclude that m || n, Question 15. So, 3 + 133 = 180 (By using the Consecutive Interior angles theorem) From the given figure, Hence, from the above, Compare the given points with y = (5x 17) what Given and Prove statements would you use? We can solve it by using the "point-slope" equation of a line: y y1 = 2 (x x1) And then put in the point (5,4): y 4 = 2 (x 5) That is an answer! The symbol || is used to represent parallel lines. The given figure is: If the angle measure of the angles is a supplementary angle, then the lines cut by a transversal are parallel Answer: \(\frac{6 (-4)}{8 3}\) Substitute A (-9, -3) in the above equation to find the value of c From the given figure, We know that, Answer: Find the distance from the point (6, 4) to the line y = x + 4. y = -2x + 1 Converse: 8x = 96 The slopes of the parallel lines are the same The equation for another line is: Algebra 1 Writing Equations of Parallel and Perpendicular Lines 1) through: (2, 2), parallel to y = x + 4. P = (4 + (4 / 5) 7, 1 + (4 / 5) 1) We can say that all the angle measures are equal in Exploration 1 Explain why the top step is parallel t0 the ground. The given figure is: x and 61 are the vertical angles So, An equation of the line representing the nature trail is y = \(\frac{1}{3}\)x 4. Perpendicular to \(5x+y=1\) and passing through \((4, 0)\). The given equation in the slope-intercept form is: The given point is: A (3, -1) Now, The given coordinates are: A (-2, 1), and B (4, 5) Question 5. Hence, from the above, Which angle pairs must be congruent for the lines to be parallel? Hence, Apply slope formula, find whether the lines are parallel or perpendicular. = 1 Hence, from the above, In Exercises 19 and 20. describe and correct the error in the conditional statement about lines. We can conclude that the given pair of lines are perpendicular lines, Question 2. From the given figure, 180 = x + x Prove: c || d 15) through: (4, -1), parallel to y = - 3 4 x16) through: (4, 5), parallel to y = 1 4 x - 4 17) through: (-2, -5), parallel to y = x + 318) through: (4, -4), parallel to y = 3 19) through . Indulging in rote learning, you are likely to forget concepts. The given figure is: Hence, Which lines intersect ? The given figure is: k = -2 + 7 Here the given line has slope \(m=\frac{1}{2}\), and the slope of a line parallel is \(m_{}=\frac{1}{2}\). Answer: So, Draw the portion of the diagram that you used to answer Exercise 26 on page 130. Difference Between Parallel and Perpendicular Lines, Equations of Parallel and Perpendicular Lines, Parallel and Perpendicular Lines Worksheets. So, c = -3 P = (7.8, 5) These worksheets will produce 10 problems per page. We know that, In a plane, if twolinesareperpendicularto the sameline, then they are parallel to each other. Yes, your classmate is correct, Explanation: Observe the horizontal lines in E and Z and the vertical lines in H, M and N to notice the parallel lines. It is given that Hence, from the above, 1 = 40 and 2 = 140. The given points are: P (-5, -5), Q (3, 3) We can conclude that the number of points of intersection of intersecting lines is: 1, c. The points of intersection of coincident lines: Answer: y = 3x + 2, (b) perpendicular to the line y = 3x 5. Answer: We can conclude that Answer: Hence, from the above, We can conclude that The slope of the line of the first equation is: We can conclude that So, Now, The slope of the equation that is parallel t the given equation is: \(\frac{1}{3}\) y = \(\frac{1}{2}\)x 2 line(s) parallel to So, \(m\cdot m_{\perp}=-\frac{5}{8}\cdot\frac{8}{5}=-\frac{40}{40}=-1\quad\color{Cerulean}{\checkmark}\). y = mx + b The given equation is: Answer: AC is not parallel to DF. The equation for another line is: x y + 4 = 0 The lines containing the railings of the staircase, such as , are skew to all lines in the plane containing the ground. The given figure is: Answer: Question 8. Alternate Interior Angles are a pair of angleson the inner side of each of those two lines but on opposite sides of the transversal. A(8, 0), B(3, 2); 1 to 4 m = 3 Perpendicular to \(x=\frac{1}{5}\) and passing through \((5, 3)\). Geometrically, we see that the line \(y=4x1\), shown dashed below, passes through \((1, 5)\) and is perpendicular to the given line. Proof: Hence, from the above, If you even interchange the second and third statements, you could still prove the theorem as the second line before interchange is not necessary Definition of Parallel and Perpendicular Parallel lines are lines in the same plane that never intersect. To find the value of c, 1 unit either in the x-plane or y-plane = 10 feet An equation of the line representing Washington Boulevard is y = \(\frac{2}{3}\)x. = \(\frac{8}{8}\) Perpendicular transversal theorem: Hence, from the above, y = \(\frac{1}{2}\)x + 2 1 + 57 = 180 In the equation form of a line y = mx +b lines that are parallel will have the same value for m. Perpendicular lines will have an m value that is the negative reciprocal of the . = \(\frac{6 0}{0 + 2}\) But, In spherical geometry, even though there is some resemblance between circles and lines, there is no possibility to form parallel lines as the lines will intersect at least at 1 point on the circle which is called a tangent From the given figure, Is it possible for consecutive interior angles to be congruent? The equation of the line that is perpendicular to the given line equation is: x = 133 Start by finding the parallels, work on some equations, and end up right where you started. Answer: So, Hence, from the above, Each step is parallel to the step immediately above it. Use the numbers and symbols to create the equation of a line in slope-intercept form Given a b c = 6 0 alternate interior, alternate exterior, or consecutive interior angles. 2 = \(\frac{1}{2}\) (-5) + c = 2 (460) (x1, y1), (x2, y2) By using the dynamic geometry, 1 = 2 = 3 = 4 = 5 = 6 = 7 = 53.7, Work with a partner. \(\begin{aligned} y-y_{1}&=m(x-x_{1}) \\ y-(-2)&=\frac{1}{2}(x-8) \end{aligned}\). m1m2 = -1 It is given that m || n Compare the given coordinates with y = -2x + c The lines that do not have any intersection points are called Parallel lines Answer: y = 27.4 The given point is: A (3, -4) To find the distance from point A to \(\overline{X Z}\), Answer: y = 162 18 So, From the above definition, Justify your answer with a diagram. So, When we compare the converses we obtained from the given statement and the actual converse, The given figure is: We can observe that the given lines are parallel lines Hence, from the above, Are the numbered streets parallel to one another? So, 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. y = 162 2 (9) Substitute (4, 0) in the above equation From the given figure, We have to find the point of intersection Now, The slope of the given line is: m = \(\frac{2}{3}\) Slope of line 1 = \(\frac{9 5}{-8 10}\) Step 1: Find the slope \(m\). We can conclude that 1 and 5 are the adjacent angles, Question 4. Answer: Substitute (6, 4) in the above equation We know that, The lines that do not intersect or not parallel and non-coplanar are called Skew lines a.) x = 0 For the intersection point, Name a pair of perpendicular lines. According to the Consecutive Exterior angles Theorem, y = 3x + 9 -(1) So, 2 and 7 are vertical angles Now, Parallel lines are those that never intersect and are always the same distance apart. Find the slope \(m\) by solving for \(y\). d = \(\sqrt{(x2 x1) + (y2 y1)}\) Save my name, email, and website in this browser for the next time I comment. Answer: From the given figure, We can conclude that the tallest bar is parallel to the shortest bar, b. It is given that the given angles are the alternate exterior angles Since it must pass through \((3, 2)\), we conclude that \(x=3\) is the equation. (- 5, 2), y = 2x 3 2x y = 4 1 5 If we try to find the slope of a perpendicular line by finding the opposite reciprocal, we run into a problem: \(m_{}=\frac{1}{0}\), which is undefined. 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. Hence, What point on the graph represents your school? (1) So, The standard form of the equation is: Any fraction that contains 0 in the denominator has its value undefined Slope of AB = \(\frac{5}{8}\) (6, 1); m = 3 So, We can conclude that the value of x is: 90, Question 8. Answer: MODELING WITH MATHEMATICS Which pair of angle measures does not belong with the other three? Slope of ST = \(\frac{2}{-4}\) y = \(\frac{1}{6}\)x 8 The intersection point of y = 2x is: (2, 4) The given points are: 3 + 4 = c y = \(\frac{3}{2}\)x 1 Explain your reasoning. So, Slope (m) = \(\frac{y2 y1}{x2 x1}\) Answer: Question 30. We know that, = 2 The given equation is: If you use the diagram below to prove the Alternate Exterior Angles Converse. 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. MAKING AN ARGUMENT 8 = \(\frac{1}{5}\) (3) + c Now, We get -x = x 3 = \(\frac{8 + 3}{7 + 2}\) From the given figure, (2x + 12) + (y + 6) = 180 Now, Hence, Hence, from the above, Compare the given coordinates with (x1, y1), and (x2, y2) The given figure is: Hence, from the above, Homework Sheets. Parallel Lines - Lines that move in their specific direction without ever intersecting or meeting each other at a point are known as the parallel lines. Hence, So, The claim of your friend is not correct Question 1. According to the Converse of the Corresponding Angles Theorem, m || n is true only when the corresponding angles are congruent y = \(\frac{2}{3}\) Answer: a. m1 + m8 = 180 //From the given statement From the above figure, So, (7x + 24) = 108 y = -x + c The sum of the angle measures are not supplementary, according to the Consecutive Exterior Angles Converse, The coordinates of line 2 are: (2, -1), (8, 4) Answer: Given: m5 + m4 = 180 From the given figure, 1 = 180 57 Answer: Question 50. Determine if the lines are parallel, perpendicular, or neither. Answer: 1. d = | 2x + y | / \(\sqrt{5}\)} = 6.26 Parallel and perpendicular lines worksheet answers key geometry - Note: This worksheet is supported by a flash presentation, under Mausmi's Math Q2: Determine. Q. The representation of the given point in the coordinate plane is: Question 54. Now, We can conclude that the top step is also parallel to the ground since they do not intersect each other at any point, Question 6. Substitute (0, 1) in the above equation The given points are: b. (x1, y1), (x2, y2) Intersecting lines can intersect at any . We can observe that It is important to have a geometric understanding of this question. m = \(\frac{1}{4}\) The coordinates of line 1 are: (-3, 1), (-7, -2) We know that, The product of the slopes of the perpendicular lines is equal to -1 Parallel to \(x+y=4\) and passing through \((9, 7)\). Hence, from the above, Answer: The equation of the line along with y-intercept is: The given figure is: how many right angles are formed by two perpendicular lines? c = 0 The mathematical notation \(m_{}\) reads \(m\) parallel.. Parallel & Perpendicular Lines Practice Answer Key Parallel and Perpendicular Lines Key *Note:If Google Docs displays "Sorry, we were unable to retrieve the document for viewing," refresh your browser. w y and z x The given point is: A (-\(\frac{1}{4}\), 5) We can observe that 141 and 39 are the consecutive interior angles So, We can conclude that The given statement is: 1 8 To find the value of c, Because j K, j l What missing information is the student assuming from the diagram? The equation that is parallel to the given equation is: Answer: We know that, The given figure is: \(\overline{C D}\) and \(\overline{A E}\) 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 So, Use the diagram. -3 = -4 + c Slope (m) = \(\frac{y2 y1}{x2 x1}\) These Parallel and Perpendicular Lines Worksheets will give the slope of a line and ask the student to determine the slope for any line that is parallel and the slope that is perpendicular to the given line. To find the coordinates of P, add slope to AP and PB c = -3 Slope of TQ = 3 (b) perpendicular to the given line. The coordinates of the line of the second equation are: (1, 0), and (0, -2) (y + 7) = (3y 17) y = -3x + 19, Question 5. The sum of the angle measure between 2 consecutive interior angles is: 180 We know that, a. m5 + m4 = 180 //From the given statement The slopes of perpendicular lines are undefined and 0 respectively When we compare the given equation with the obtained equation, We can conclude that the value of y when r || s is: 12, c. Can r be parallel to s and can p, be parallel to q at the same time? The point of intersection = (\(\frac{4}{5}\), \(\frac{13}{5}\)) When two lines are cut by a transversal, the pair ofangles on one side of the transversal and inside the two lines are called the Consecutive interior angles So, The given coordinates are: A (-2, -4), and B (6, 1) (5y 21) = 116 We can conclude that FCA and JCB are alternate exterior angles. The equation of the line that is perpendicular to the given line equation is: The given figure is: The two pairs of perpendicular lines are l and n, c. Identify two pairs of skew line The given point is: (-1, 6) In the same way, when we observe the floor from any step, \(m_{}=\frac{4}{3}\) and \(m_{}=\frac{3}{4}\), 15. XY = \(\sqrt{(3 + 1.5) + (3 2)}\) The equation that is perpendicular to the given line equation is: Answer: So, y = 2x In other words, if \(m=\frac{a}{b}\), then \(m_{}=\frac{b}{a}\). 2x x = 56 2 Converse: Parallel lines do not intersect each other 0 = \(\frac{1}{2}\) (4) + c (A) Corresponding Angles Converse (Thm 3.5) c = \(\frac{40}{3}\) If the slopes of the opposite sides of the quadrilateral are equal, then it is called as Parallelogram From the given bars, We can conclude that the perpendicular lines are: Compare the given points with (x1, y1), and (x2, y2) The angles that have the common side are called Adjacent angles Substitute (-5, 2) in the given equation So, So, Hence, from the given figure, Answer: x = \(\frac{153}{17}\) Show your steps. Proof: We know that, Parallel and Perpendicular Lines Name_____ L i2K0Y1t7O OKludthaY TSNoIfStiw\a[rpeR VLxLFCx.H R BAXlplr grSiVgvhvtBsM srUefseeorqvIeSdh.-1- Find the slope of a line parallel to each given line. So, = 44,800 square feet 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 In Exercises 9 and 10, trace \(\overline{A B}\). = \(\frac{4}{-18}\) 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 Parallel to \(7x5y=35\) and passing through \((2, 3)\). Hence, For example, if the equations of two lines are given as: y = 1/4x + 3 and y = - 4x + 2, we can see that the slope of one line is the negative reciprocal of the other. y = -x -(1) P = (2 + (2 / 8) 8, 6 + (2 / 8) (-6)) Find m1 and m2. Prove c||d Explain. y = -2x + c1 y = \(\frac{1}{2}\)x 3, b. Answer: USING STRUCTURE Prove 1, 2, 3, and 4 are right angles. Substitute the given point in eq. Students must unlock 5 locks by: 1: determining if two given slopes are parallel, perpendicular or neither. y = -2x + 8 = 2.12 The given coordinates are: A (1, 3), and B (8, 4) Question 39. Slope (m) = \(\frac{y2 y1}{x2 x1}\) We can conclude that the line that is parallel to the given line equation is: y = -3 6 Answer: Question 44. x = 20 We can conclude that the distance from point A to the given line is: 2.12, Question 26.

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