The given figure is: The standard form of the equation is: m2 = \(\frac{1}{2}\) The given lines are: 1 = 123 and 2 = 57. Write an equation for a line parallel to y = 1/3x - 3 through (4, 4) Q. MODELING WITH MATHEMATICS So, Hence, from the above, Slope (m) = \(\frac{y2 y1}{x2 x1}\) Hence, from the above, Find the distance front point A to the given line. c = \(\frac{16}{3}\) We can observe that The missing information the student assuming from the diagram is: P(3, 8), y = \(\frac{1}{5}\)(x + 4) Explain your reasoning. y = x 6 -(1) So, The coordinates of x are the same. Question 25. From the given figure, y = mx + c The given statement is: 1 8 Hence, from the above, 17x = 180 27 MODELING WITH MATHEMATICS Substitute A (8, 2) in the above equation Now, x = \(\frac{180}{2}\) 5y = 116 + 21 Parallel and perpendicular lines are an important part of geometry and they have distinct characteristics that help to identify them easily. Is b c? These worksheets will produce 6 problems per page. Here the given line has slope \(m=\frac{1}{2}\), and the slope of a line parallel is \(m_{}=\frac{1}{2}\). So, Substitute A (-9, -3) in the above equation to find the value of c Answer: The given line equation is: We can observe that the slopes of the opposite sides are equal i.e., the opposite sides are parallel In spherical geometry, is it possible that a transversal intersects two parallel lines? 1. y = -x + c We can conclude that the claim of your friend can be supported, Question 7. We know that, We can observe that Given 1 3 Slope of the line (m) = \(\frac{-1 2}{-3 + 2}\) From the figure, The equation of the line that is parallel to the given line equation is: Answer: The angle measures of the vertical angles are congruent So, REASONING Step 2: THOUGHT-PROVOKING If you multiply theslopesof twoperpendicular lines in the plane, you get 1 i.e., the slopes of perpendicular lines are opposite reciprocals. We know that, We know that, y = 2x + c We can conclude that We have to find the distance between X and Y i.e., XY Answer: The slopes are the same but the y-intercepts are different b.) The map shows part of Denser, Colorado, Use the markings on the map. The line that passes through point F that appear skew to \(\overline{E H}\) is: \(\overline{F C}\), Question 2. Hence, from the above, 5 = 8 y = \(\frac{2}{3}\)x + b (1) Now, y = \(\frac{1}{3}\)x + \(\frac{26}{3}\) 3.12) z x and w z The given figure is: We know that, Start by finding the parallels, work on some equations, and end up right where you started. Now, 1) In Example 5. yellow light leaves a drop at an angle of m2 = 41. Answer: Q1: Find the slope of the line passing through the pairs of points and describe the line as rising 745 Math Consultants 8 Years on market 51631+ Customers Get Homework Help (-3, 8); m = 2 Line b and Line c are perpendicular lines. So, x 2y = 2 d = \(\frac{4}{5}\) (1) 1 = 60 We can observe that the given angles are consecutive exterior angles We can observe that, It is given that 1 = 105 So, We can observe that b. Justify your answer. We can observe that, Hence, from the above, The given equation is: Explain. Compare the given points with (x1, y1), and (x2, y2) She says one is higher than the other. So, Now, y = \(\frac{1}{2}\)x + 2 From the given figure, Answer: Find the equation of the line passing through \((1, 5)\) and perpendicular to \(y=\frac{1}{4}x+2\). We know that, We can conclude that the value of x when p || q is: 54, b. So, by the Corresponding Angles Converse, g || h. Question 5. Write an equation of the line that passes through the given point and is parallel to the Get the best Homework key y = -2x + 2. It is given that From the above, The slope of the given line is: m = 4 The theorems involving parallel lines and transversals that the converse is true are: The equation of the line that is parallel to the given line equation is: Answer: The Intersecting lines have a common point to intersect The line parallel to \(\overline{Q R}\) is: \(\overline {L M}\), Question 3. Answer: Question 28. Question 39. (B) intersect The equation of the line along with y-intercept is: Question 5. -5 = \(\frac{1}{2}\) (4) + c From the given bars, If parallel lines are cut by a transversal line, thenconsecutive exterior anglesare supplementary. In Exercises 11 and 12. find m1, m2, and m3. we can conclude that the converse we obtained from the given statement is false, c. Alternate Exterior Angles Theorem (Theorem 3.3): If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. The given figure is: y = \(\frac{3}{2}\)x + c Step 6: \(\begin{aligned} 2x+14y&=7 \\ 2x+14y\color{Cerulean}{-2x}&=7\color{Cerulean}{-2x} \\ 14y&=-2x+7 \\ \frac{14y}{\color{Cerulean}{14}}&=\frac{-2x+7}{\color{Cerulean}{14}} \\ y&=\frac{-2x}{14}+\frac{7}{14} \\ y&=-\frac{1}{7}x+\frac{1}{2} \end{aligned}\). MATHEMATICAL CONNECTIONS b is the y-intercept 1 = 2 (By using the Vertical Angles theorem) Hence, from the above, We can conclude that the quadrilateral QRST is a parallelogram. Work with a partner: Fold a piece of pair in half twice. We can conclude that It is given that m || n From the given figure, We can conclude that 8 right angles are formed by two perpendicular lines in spherical geometry. The given expression is: Possible answer: plane FJH plane BCD 2a. XY = \(\sqrt{(6) + (2)}\) y = 145 m = 3 and c = 9 x + 73 = 180 We can conclude that the top rung is parallel to the bottom rung. x = \(\frac{7}{2}\) Explain. It is given that m || n y = \(\frac{1}{2}\)x 6 13) x - y = 0 14) x + 2y = 6 Write the slope-intercept form of the equation of the line described. Solve each system of equations algebraically. m = -7 Question 4. We can conclude that both converses are the same 1 5 In which of the following diagrams is \(\overline{A C}\) || \(\overline{B D}\) and \(\overline{A C}\) \(\overline{C D}\)? Hence, from the above, x = n The equation that is perpendicular to the given equation is: We know that, The angles that have the opposite corners are called Vertical angles We can conclude that the values of x and y are: 9 and 14 respectively. The given point is: A (0, 3) c = 2 It is given that m || n y = 162 18 The slopes are the same and the y-intercepts are different y = \(\frac{1}{2}\)x + b (1) b = 19 The given figure is: Compare the given points with (x1, y1), and (x2, y2) a. In Exercises 21 and 22, write and solve a system of linear equations to find the values of x and y. Substitute the given point in eq. Substitute (-1, -9) in the above equation Let the given points are: Justify your answer. Classify each pair of angles whose measurements are given. Answer: Compare the given coordinates with (x1, y1), and (x2, y2) We can conclude that the value of x is: 20, Question 12. We know that, 5 = 3 (1) + c Hence, -2 = \(\frac{1}{2}\) (2) + c Compare the given coordinates with We can conclude that construction change if you were to construct a rectangle? m2 = \(\frac{1}{3}\) Find the distance from point A to the given line. Question 21. The given points are: (k, 2), and (7, 0) 2y + 4x = 180 Question 18. Therefore, they are perpendicular lines. The given statement is: Homework 1 - State whether the given pair of lines are parallel. Geometrically, we see that the line \(y=4x1\), shown dashed below, passes through \((1, 5)\) and is perpendicular to the given line. So, Now, From the given figure, Answer: X (3, 3), Y (2, -1.5) (1) \(m_{}=9\) and \(m_{}=\frac{1}{9}\), 13. y = mx + c 6x = 87 Perpendicular lines are denoted by the symbol . CRITICAL THINKING We can conclude that the consecutive interior angles are: 3 and 5; 4 and 6. Another answer is the line perpendicular to it, and also passing through the same point. Question 14. Now, Question 31. We know that, 1 = 80 Answer: c = -4 + 3 Which point should you jump to in order to jump the shortest distance? Slope of QR = \(\frac{1}{2}\), Slope of RS = \(\frac{1 4}{5 6}\) From ESR, 5 = 105, To find 8: 1 (m2) = -3 The coordinates of the quadrilateral QRST is: Proof: The measure of 1 is 70. The equation that is perpendicular to the given equation is: You and your mom visit the shopping mall while your dad and your sister visit the aquarium. Hence, Explain. According to Euclidean geometry, Then write Supply: lamborghini-islero.com Answer: Use the diagram to find the measure of all the angles. To find the value of c, So, We can conclude that AC || DF, Question 24. We know that, State the converse that Answer: Question 4. The equation that is perpendicular to the given line equation is: So, = 3, The slope of line d (m) = \(\frac{y2 y1}{x2 x1}\) We know that, Determine whether quadrilateral JKLM is a square. The equation that is parallel to the given equation is: 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. If two parallel lines are cut by a transversal, then the pairs of Alternate interior angles are congruent. We can conclude that The parallel line equation that is parallel to the given equation is: Select all that apply. a = 1, and b = -1 PROVING A THEOREM To be proficient in math, you need to analyze relationships mathematically to draw conclusions. So, Hence, from the above, You and your family are visiting some attractions while on vacation. Find the equation of the line passing through \((6, 1)\) and parallel to \(y=\frac{1}{2}x+2\). = \(\frac{-1 2}{3 4}\) d = | x y + 4 | / \(\sqrt{1 + (-1)}\) Hence, from the above, We know that, (- 5, 2), y = 2x 3 A gazebo is being built near a nature trail. In Exploration 3. find AO and OB when AB = 4 units. (B) Explain your reasoning. Draw \(\overline{P Z}\), Question 8. We know that, According to the Perpendicular Transversal Theorem, Possible answer: 2 and 7 c. Possible answer: 1 and 8 d. Possible answer: 2 and 3 3. Two lines are cut by a transversal. Prove the Perpendicular Transversal Theorem using the diagram in Example 2 and the Alternate Exterior Angles Theorem (Theorem 3.3). c is the y-intercept Make a conjecture about how to find the coordinates of a point that lies beyond point B along \(\vec{A}\)B. So, \(\frac{13-4}{2-(-1)}\) We know that, c = -13 So, Use the diagram By the Vertical Angles Congruence Theorem (Theorem 2.6). 2 and 3 are vertical angles m2 = \(\frac{1}{2}\) The slopes are equal for the parallel lines Look at the diagram in Example 1. Question 1. y = -x + c A(- \(\frac{1}{4}\), 5), x + 2y = 14 2x = 180 Question 8. m = = So, slope of the given line is Question 2. The equation that is parallel to the given equation is: = \(\sqrt{(3 / 2) + (3 / 2)}\) They are not perpendicular because they are not intersecting at 90. We know that, Perpendicular to \(6x+3y=1\) and passing through \((8, 2)\). m1m2 = -1 c = 12 m1 m2 = -1 The parallel line equation that is parallel to the given equation is: We can observe that the product of the slopes are -1 and the y-intercepts are different A triangle has vertices L(0, 6), M(5, 8). Use a graphing calculator to graph the pair of lines. We know that, The product of the slopes of the perpendicular lines is equal to -1 HOW DO YOU SEE IT? Answer: We can observe that The perpendicular lines have the product of slopes equal to -1 Note: Parallel lines are distinguished by a matching set of arrows on the lines that are parallel. So, Answer: Question 36. Hence, Describe the point that divides the directed line segment YX so that the ratio of YP Lo PX is 5 to 3. 72 + (7x + 24) = 180 (By using the Consecutive interior angles theory) So, The are outside lines m and n, on . It is given that 1 = 58 Answer: The equation that is perpendicular to the given line equation is: From the given figure, 2x + 4y = 4 MATHEMATICAL CONNECTIONS The slope of the equation that is parallel t the given equation is: \(\frac{1}{3}\) Answer: Question 14. We can observe that The given figure is: Hence, from the above, Perpendicular to \(5x+y=1\) and passing through \((4, 0)\). m1m2 = -1 1 4. To find 4: Hence, from the given figure, The given equation is: We can conclude that 2 and 11 are the Vertical angles. transv. The given equation is: m1m2 = -1 2x = 135 15 a. m = 2 We know that, 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 given figure is: Hw Key Hw Part 2 key Updated 9/29/22 #15 - Perpendicular slope 3.6 (2017) #16 - Def'n of parallel 3.1 . m2 and m4 For example, AB || CD means line AB is parallel to line CD. The equation that is perpendicular to the given line equation is: The coordinates of line b are: (2, 3), and (0, -1) 3y = x 50 + 525 Now, a. corresponding angles Question 13. So, The slope of the line of the first equation is: So, Now, 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. y = \(\frac{1}{2}\)x 4, Question 22. So, Hence, The equation that is parallel to the given equation is: b. Compare the given coordinates with The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior anglesare congruent, then the lines are parallel a. m1 + m8 = 180 //From the given statement Each unit in the coordinate plane corresponds to 50 yards. Use these steps to prove the Transitive Property of Parallel Lines Theorem Explain your reasoning. 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 m1=m3 d = \(\sqrt{(x2 x1) + (y2 y1)}\) Question 1. We can conclude that the third line does not need to be a transversal. y 3y = -17 7 Answer: Question 28. a. consecutive interior We can conclude that the equation of the line that is perpendicular bisector is: Proof of the Converse of the Consecutive Interior angles Theorem: The given point is: (1, 5) Hence, from the above figure, We can observe that Mark your diagram so that it cannot be proven that any lines are parallel. (1) = 9.48 The converse of the given statement is: Answer: Hence, from the above, Now, We can conclude that PROVING A THEOREM We know that, an equation of the line that passes through the midpoint and is perpendicular to \(\overline{P Q}\). Answer: Prove c||d a.) Hence, from the above, Graph the equations of the lines to check that they are perpendicular. The slopes of perpendicular lines are undefined and 0 respectively In Exercises 11 and 12. prove the theorem. Remember that horizontal lines are perpendicular to vertical lines. Repeat steps 3 and 4 below AB Now, In exercises 25-28. copy and complete the statement. The coordinates of line 1 are: (-3, 1), (-7, -2) Q. Using P as the center, draw two arcs intersecting with line m. x = \(\frac{18}{2}\) We know that, = \(\frac{6 0}{0 + 2}\) Solving Equations Involving Parallel and Perpendicular Lines www.BeaconLC.org2001 September 22, 2001 9 Solving Equations Involving Parallel and Perpendicular Lines Worksheet Key Find the slope of a line that is parallel and the slope of a line that is perpendicular to each line whose equation is given. So, y = -3x + 150 + 500 In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. Hence, Question 4. \(m_{}=\frac{3}{4}\) and \(m_{}=\frac{4}{3}\), 3. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Answer: Question 20. \(\frac{1}{3}\)x + 3x = -2 + 2 So, Find an equation of the line representing the bike path. Answer: d. AB||CD // Converse of the Corresponding Angles Theorem According to the above theorem, 12y 18 = 138 Answer: So, c = -2 b. m1 + m4 = 180 // Linear pair of angles are supplementary y = \(\frac{3}{2}\) + 4 and -3x + 2y = -1 The given point is: (1, 5) Answer: m is the slope ANALYZING RELATIONSHIPS We can conclude that the number of points of intersection of intersecting lines is: 1, c. The points of intersection of coincident lines: For perpendicular lines, Hence, From the given figure, Lines Perpendicular to a Transversal Theorem (Theorem 3.12): In a plane. The given perpendicular line equations are: Compare the given points with We can observe that Find the slope of the line perpendicular to \(15x+5y=20\). A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. The equation that is perpendicular to the given line equation is: Hence, In geometry, there are three different types of lines, namely, parallel lines, perpendicular lines, and intersecting lines. To find the value of c, The diagram of the control bar of the kite shows the angles formed between the Control bar and the kite lines. We have to find the point of intersection (50, 500), (200, 50) If the slope of two given lines are negative reciprocals of each other, they are identified as ______ lines. Find the equation of the line perpendicular to \(x3y=9\) and passing through \((\frac{1}{2}, 2)\). = (\(\frac{-5 + 3}{2}\), \(\frac{-5 + 3}{2}\)) Possible answer: 2 and 7 c. Possible answer: 1 and 8 d. Possible answer: 2 and 3 3. Using the properties of parallel and perpendicular lines, we can answer the given questions. The parallel line needs to have the same slope of 2. y = mx + c 6 + 4 = 180, Question 9. Determine which lines, if any, must be parallel. The corresponding angles are: and 5; 4 and 8, b. alternate interior angles = 4 The given equation is: Now, From the given figure, In a square, there are two pairs of parallel lines and four pairs of perpendicular lines. (7x 11) = (4x + 58) Now, According to the Corresponding Angles Theorem, the corresponding angles are congruent So, = 44,800 square feet Now, We can conclude that the value of x is: 14. y = \(\frac{7}{2}\) 3 CONSTRUCTING VIABLE ARGUMENTS We know that, 5 7 Is your friend correct? Consecutive Interior Angles Theorem (Thm. Answer: Answer: Answer: Compare the given equation with y = 2x and y = 2x + 5 The coordinates of line a are: (0, 2), and (-2, -2) The angles that have the same corner are called Adjacent angles Hence, from the above, The slope of perpendicular lines is: -1 Parallel to \(10x\frac{5}{7}y=12\) and passing through \((1, \frac{1}{2})\). d = \(\sqrt{(x2 x1) + (y2 y1)}\) So, Compare the given points with Line 2: (7, 0), (3, 6) 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. b is the y-intercept The given equation is: 6-3 Write Equations of Parallel and Perpendicular Lines Worksheet. The given figure is: We can conclude that 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. The given point is: A (2, -1) 2 = 122, Question 16. Alternate Exterior angle Theorem: The sum of the angle measures are not supplementary, according to the Consecutive Exterior Angles Converse, y = -9 We can say that any intersecting line do intersect at 1 point Answer: So, Converse: From the given figure, . So, Parallel to \(\frac{1}{5}x\frac{1}{3}y=2\) and passing through \((15, 6)\). We can observe that So, From the given figure, line(s) skew to Each unit in the coordinate plane corresponds to 10 feet. y = mx + c Answer: Draw \(\overline{P Z}\), CONSTRUCTION To find the coordinates of P, add slope to AP and PB The representation of the parallel lines in the coordinate plane is: Question 16. y = 3x + 2 m || n is true only when 147 and (x + 14) are the corresponding angles by using the Converse of the Alternate Exterior Angles Theorem Question 15. Answer: Hence, from the above, Use a square viewing window. We can conclude that the value of the given expression is: 2, Question 36. Use a graphing calculator to verify your answer. A(1, 6), B(- 2, 3); 5 to 1 From the given figure, Verify your answer. So, Compare the given points with (x1, y1), (x2, y2) CONSTRUCTION We know that, a. m5 + m4 = 180 //From the given statement x = 6 The equation of the line that is perpendicular to the given line equation is: 2x = 18 m2 = \(\frac{1}{2}\), b2 = 1 y = \(\frac{3}{5}\)x \(\frac{6}{5}\) We have to find the point of intersection Given: k || l, t k 68 + (2x + 4) = 180 Consider the 2 lines L1 and L2 intersected by a transversal line L3 creating 2 corresponding angles 1 and 2 which are congruent We can conclude that From the figure, For a parallel line, there will be no intersecting point So, d = \(\sqrt{(x2 x1) + (y2 y1)}\) The points of intersection of intersecting lines: In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. Question 42. c = 1 Likewise, parallel lines become perpendicular when one line is rotated 90. Prove the statement: If two lines are vertical. Question 27. d = \(\sqrt{(x2 x1) + (y2 y1)}\) We know that, We know that, Answer: c = 5 3 The given figure is: It can be observed that So, Now, Is b || a? So, Answer Keys - These are for all the unlocked materials above. According to the Perpendicular Transversal Theorem, The slopes of the parallel lines are the same Expert-Verified Answer The required slope for the lines is given below. We can observe that Hence, The point of intersection = (\(\frac{3}{2}\), \(\frac{3}{2}\)) This contradiction means our assumption (L1 is not parallel to L2) is false, and so L1 must be parallel to L2. c. m5=m1 // (1), (2), transitive property of equality -5 8 = c So, So, Answer: m2 = -2 = 0 The slopes of the parallel lines are the same x = 0 y = \(\frac{1}{6}\)x 8 Question 4. So, We know that, Definition of Parallel and Perpendicular Parallel lines are lines in the same plane that never intersect. Now, So, Answer: Question 10. Parallel lines do not intersect each other y = \(\frac{1}{4}\)x 7, Question 9. The coordinates of line 2 are: (2, -4), (11, -6) (C) Alternate Exterior Angles Converse (Thm 3.7) Answer: Label the ends of the crease as A and B. Compare the given points with (x1, y1), and (x2, y2) Observe the following figure and the properties of parallel and perpendicular lines to identify them and differentiate between them. From the given figure, x = 4 and y = 2 Answer: In Exercises 19 and 20, describe and correct the error in the reasoning. Answer: Question 24. To find the coordinates of P, add slope to AP and PB The completed proof of the Alternate Interior Angles Converse using the diagram in Example 2 is: Now, We can observe that the given angles are the consecutive exterior angles