Now, The parallel line equation that is parallel to the given equation is: We can conclude that the value of x when p || q is: 54, b. In Example 4, the given theorem is Alternate interior angle theorem AP : PB = 2 : 6 For a vertical line, Given: m5 + m4 = 180 The diagram that represents the figure that it can be proven that the lines are parallel is: Question 33. c = -3 Expert-Verified Answer The required slope for the lines is given below. The slope of the line of the first equation is: We can observe that = \(\frac{0}{4}\) (D) The equation of the perpendicular line that passes through the midpoint of PQ is: We know that, So, In exercises 25-28. copy and complete the statement. Hence, \(\frac{13-4}{2-(-1)}\) c. All the lines containing the balusters. . as corresponding angles formed by a transversal of parallel lines, and so, line(s) perpendicular to . y = 145 c.) Book: The two highlighted lines meet each other at 90, therefore, they are perpendicular lines. So, The representation of the given pair of lines in the coordinate plane is: d = \(\sqrt{41}\) Slope (m) = \(\frac{y2 y1}{x2 x1}\) So, Compare the given equation with When we compare the actual converse and the converse according to the given statement, The given points are: The coordinates of line d are: (0, 6), and (-2, 0) Vertical and horizontal lines are perpendicular. The converse of the Alternate Interior angles Theorem: You are designing a box like the one shown. The given perpendicular line equations are: Hence, It is given that Question 27. Answer: Hence, Your classmate claims that no two nonvertical parallel lines can have the same y-intercept. Answer: Question 29. Answer: c = -2 We can conclude that the corresponding angles are: 1 and 5; 3 and 7; 2 and 4; 6 and 8, Question 8. The given equations are: Answer: a. We can conclude that the length of the field is: 320 feet, b. A1.3.1 Write an equation of a line when given the graph of the line, a data set, two points on the line, or the slope and a point of the line; A1.3.2 Describe and calculate the slope of a line given a data set or graph of a line, recognizing that the slope is the rate of change; A1.3.6 . From the given figure, m2 = \(\frac{1}{3}\) We know that, From the given figure, -2 = 3 (1) + c Perpendicular lines intersect at each other at right angles AC is not parallel to DF. Hence, from the above, Answer: plane(s) parallel to plane CDH x = 14 Question 1. So, y = \(\frac{1}{7}\)x + 4 Answer: Question 4. If the slope of two given lines are negative reciprocals of each other, they are identified as perpendicular lines. -2 = 1 + c We know that, y = \(\frac{5}{3}\)x + \(\frac{40}{3}\) We can conclude that The given lines are perpendicular lines Perpendicular to \(y=2\) and passing through \((1, 5)\). These Parallel and Perpendicular Lines Worksheets are great for practicing identifying parallel, perpendicular, and intersecting lines from pictures. XY = 6.32 Answer: We know that, Hence, from the above, Answer: The point of intersection = (-1, \(\frac{13}{2}\)) (6, 1); m = 3 \(\frac{5}{2}\)x = 2 The parallel lines do not have any intersecting points According to the Alternate Exterior angles Theorem, y = \(\frac{1}{3}\)x + c Hence, From the given figure, Answer: MATHEMATICAL CONNECTIONS 1 + 138 = 180 Substitute A (3, -4) in the above equation to find the value of c 2m2 = -1 (13, 1) and (9, 4) The given point is: (-5, 2) Answer: Question 18. (1) If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Perpendicular to \(\frac{1}{2}x\frac{1}{3}y=1\) and passing through \((10, 3)\). Prove c||d c = 3 4 We know that, Find the measure of the missing angles by using transparent paper. So, y = 2x + 12 m = \(\frac{1}{2}\) We know that, For example, if the equations of two lines are given as, y = -3x + 6 and y = -3x - 4, we can see that the slope of both the lines is the same (-3). The equation for another parallel line is: = 2 (460) The given equation is: Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. We can observe that Hence, from the above, The given point is: (-8, -5) Let's expand 2 (x 5) and then rearrange: y 4 = 2x 10. We can observe that a is perpendicular to both the lines b and c = \(\frac{1}{-4}\) So, Converse: So, y = mx + c If two lines are intersected by a third line, is the third line necessarily a transversal? What are the coordinates of the midpoint of the line segment joining the two houses? Answer: The parallel line equation that is parallel to the given equation is: Now, Compare the given points with (x1, y1), and (x2, y2) A (x1, y1), B (x2, y2) To find the value of c, (2) Hence, from he above, The product of the slopes of perpendicular lines is equal to -1 We can observe that the slopes are the same and the y-intercepts are different XY = \(\sqrt{(x2 x1) + (y2 y1)}\) Example 2: State true or false using the properties of parallel and perpendicular lines. Identifying Parallel, Perpendicular, and Intersecting Lines from a Graph Show your steps. Compare the given equations with Let the two parallel lines be E and F and the plane they lie be plane x 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. Question 25. The slope of the given line is: m = 4 According to the Perpendicular Transversal Theorem, Give four examples that would allow you to conclude that j || k using the theorems from this lesson. So, Prove: c || d Use a graphing calculator to verify your answers. Is b c? Answer: So, We know that, The lines that do not intersect and are not parallel and are not coplanar are Skew lines 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 The given equation is: Compare the given points with -x x = -3 Compare the given equation with We know that, y = mx + c The slopes of the parallel lines are the same \(m_{}=\frac{3}{4}\) and \(m_{}=\frac{4}{3}\), 3. We know that, The lines that are a straight angle with the given line and are coplanar is called Perpendicular lines Explain. -1 = \(\frac{1}{3}\) (3) + c Substitute A (0, 3) in the above equation y = \(\frac{1}{2}\)x + 1 -(1) c = 1 Slope of AB = \(\frac{4 3}{8 1}\) So, Explain your reasoning. We can observe that the given angles are the consecutive exterior angles So, 2 = 180 1 According to the Alternate Interior Angles Theorem, the alternate interior angles are congruent The third intersecting line can intersect at the same point that the two lines have intersected as shown below: = 6.26 1 and 5 are the alternate exterior angles The given coordinates are: A (-3, 2), and B (5, -4) So, We know that, y = \(\frac{1}{3}\)x + \(\frac{16}{3}\), Question 5. Perpendicular lines are those lines that always intersect each other at right angles. The coordinates of line 2 are: (2, -1), (8, 4) To prove: l || k. Question 4. The Perpendicular Postulate states that if there is a line and a point not on the line, then there is exactly one line through the point perpendicularto the given line. Write an equation of a line perpendicular to y = 7x +1 through (-4, 0) Q. We have to find the point of intersection y = \(\frac{1}{3}\)x + \(\frac{26}{3}\) = 3 The given figure is: We can observe that So, Question 13. Look back at your construction of a square in Exercise 29 on page 154. So, a. (50, 500), (200, 50) Find the distance from point A to the given line. b is the y-intercept In Example 5, Answer: So, The given figure is: Find the value of y that makes r || s. 2. We can conclude that the linear pair of angles is: 1 3, The symbol || is used to represent parallel lines. Hence. We can observe that the given angles are the corresponding angles Given: k || l, t k (2x + 12) + (y + 6) = 180 \(\frac{1}{2}\) (m2) = -1 Since the given line is in slope-intercept form, we can see that its slope is \(m=5\). We know that, Hence, Hence, Answer: Question 2. We know that, The equation of the line along with y-intercept is: From the figure, In Exploration 1, explain how you would prove any of the theorems that you found to be true. Perpendicular lines are intersecting lines that always meet at an angle of 90. You and your family are visiting some attractions while on vacation. We can conclude that m || n by using the Corresponding Angles Theorem, Question 14. For example, PQ RS means line PQ is perpendicular to line RS. y = -x -(1) PROVING A THEOREM We can conclude that the given pair of lines are coincident lines, Question 3. d = \(\sqrt{(x2 x1) + (y2 y1)}\) The construction of the walls in your home were created with some parallels. The intersection point of y = 2x is: (2, 4) w y and z x We want to prove L1 and L2 are parallel and we will prove this by using Proof of Contradiction y = -2x 1 The given figure is: The given figure is: Slope (m) = \(\frac{y2 y1}{x2 x1}\) We can conclude that Parallel and Perpendicular Lines Maintaining Mathematical Proficiency Page 123, Parallel and Perpendicular Lines Mathematical Practices Page 124, 3.1 Pairs of Lines and Angles Page(125-130), Lesson 3.1 Pairs of Lines and Angles Page(126-128), Exercise 3.1 Pairs of Lines and Angles Page(129-130), 3.2 Parallel Lines and Transversals Page(131-136), Lesson 3.2 Parallel Lines and Transversals Page(132-134), Exercise 3.2 Parallel Lines and Transversals Page(135-136), 3.3 Proofs with Parallel Lines Page(137-144), Lesson 3.3 Proofs with Parallel Lines Page(138-141), Exercise 3.3 Proofs with Parallel Lines Page(142-144), 3.1 3.3 Study Skills: Analyzing Your Errors Page 145, 3.4 Proofs with Perpendicular Lines Page(147-154), Lesson 3.4 Proofs with Perpendicular Lines Page(148-151), Exercise 3.4 Proofs with Perpendicular Lines Page(152-154), 3.5 Equations of Parallel and Perpendicular Lines Page(155-162), Lesson 3.5 Equations of Parallel and Perpendicular Lines Page(156-159), Exercise 3.5 Equations of Parallel and Perpendicular Lines Page(160-162), 3.4 3.5 Performance Task: Navajo Rugs Page 163, Parallel and Perpendicular Lines Chapter Review Page(164-166), Parallel and Perpendicular Lines Test Page 167, Parallel and Perpendicular Lines Cumulative Assessment Page(168-169), Big Ideas Math Answers Grade 2 Chapter 15 Identify and Partition Shapes, Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors, enVision Math Common Core Grade 7 Answer Key | enVision Math Common Core 7th Grade Answers, Envision Math Common Core Grade 5 Answer Key | Envision Math Common Core 5th Grade Answers, Envision Math Common Core Grade 4 Answer Key | Envision Math Common Core 4th Grade Answers, Envision Math Common Core Grade 3 Answer Key | Envision Math Common Core 3rd Grade Answers, enVision Math Common Core Grade 2 Answer Key | enVision Math Common Core 2nd Grade Answers, enVision Math Common Core Grade 1 Answer Key | enVision Math Common Core 1st Grade Answers, enVision Math Common Core Grade 8 Answer Key | enVision Math Common Core 8th Grade Answers, enVision Math Common Core Kindergarten Answer Key | enVision Math Common Core Grade K Answers, enVision Math Answer Key for Class 8, 7, 6, 5, 4, 3, 2, 1, and K | enVisionmath 2.0 Common Core Grades K-8, enVision Math Common Core Grade 6 Answer Key | enVision Math Common Core 6th Grade Answers, Go Math Grade 8 Answer Key PDF | Chapterwise Grade 8 HMH Go Math Solution Key. Hence, We can observe that So, Students must unlock 5 locks by: 1: determining if two given slopes are parallel, perpendicular or neither. So, Question 22. We know that, Answer: REASONING A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. Answer: Question 36. The given figure shows that angles 1 and 2 are Consecutive Interior angles Label the ends of the crease as A and B. (50, 175), (500, 325) We know that, x = 107 The distance from your house to the school is one-fourth of the distance from the school to the movie theater. 1 4. The given figure is: Answer: So, The given table is: Hence, from the above, (13, 1), and (9, -4) We can conclude that the distance between the meeting point and the subway is: 364.5 yards, Question 13. y = 2x + c To find the value of c, CONSTRUCTING VIABLE ARGUMENTS y = -2x + c Line 1: (10, 5), (- 8, 9) \(\overline{C D}\) and \(\overline{A E}\) Geometry chapter 3 parallel and perpendicular lines answer key. FSE = ESR Answer: y = 132 From the given figure, 2x = 7 Hence, from the above figure, In the diagram, how many angles must be given to determine whether j || k? So, The equation of line q is: So, (2) For example, the opposite sides of a square and a rectangle have parallel lines in them, and the adjacent lines in the same shapes are perpendicular lines. ANALYZING RELATIONSHIPS Hence, from the above, c = 4 3 ABSTRACT REASONING m1m2 = -1 x = \(\frac{3}{2}\) x = 147 14 a. m = \(\frac{3}{-1.5}\) CONSTRUCTION Answer: We can conclude that the school have enough money to purchase new turf for the entire field. Use the photo to decide whether the statement is true or false. The product of the slopes of perpendicular lines is equal to -1 The given equation is: So, c = 5 \(\frac{1}{2}\) y = 2x + c Question 4. y = -2x + 8 If we represent the bars in the coordinate plane, we can observe that the number of intersection points between any bar is: 0 Justify your answers. So, y = -2x + 2. So, Answer: Answer: The opposite sides of a rectangle are parallel lines. So, Prove: t l 1 = 2 = 133 and 3 = 47. 4 6 = c Substitute P (4, -6) in the above equation Answer: So, So, Question 1. Hence, from the above, Find the distance from point E to The product of the slopes of perpendicular lines is equal to -1 Now, (180 x) = x Line c and Line d are perpendicular lines, Question 4. So, The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal, the resultingalternate interior anglesare congruent Find m2. We know that, It is given that the two friends walk together from the midpoint of the houses to the school Question: What is the difference between perpendicular and parallel? Find the value of x when a b and b || c. We know that, WRITING The given figure is: THOUGHT-PROVOKING The representation of the given pair of lines in the coordinate plane is: We know that, In Exercises 13-18. decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Now, Question 18. c = 2 We can observe that So, From the figure, Find the distance front point A to the given line. When we compare the converses we obtained from the given statement and the actual converse, The equation that is perpendicular to the given line equation is: Answer: Answer: We know that, Hence, USING STRUCTURE Find the equation of the line passing through \((1, 5)\) and perpendicular to \(y=\frac{1}{4}x+2\). We can observe that the sum of the angle measures of all the pairs i.e., (115 + 65), (115 + 65), and (65 + 65) is not 180 Prove 2 4 c = \(\frac{1}{2}\) y = 2x + c Answer: The coordinates of line p are: Question: ID Unit 3: Paraliel& Perpendicular Lines Homework 3: Proving Lines are Parolel Nome: Dnceuea pennon Per Date This is a 2-poge document Determine Im based on the intormation alven on the diogram yes, state the coverse that proves the ines are porollel 2 4. consecutive interior The are outside lines m and n, on . Answer: Exercise \(\PageIndex{3}\) Parallel and Perpendicular Lines. 2x = 135 15 We can conclude that \(\overline{N P}\) and \(\overline{P O}\) are perpendicular lines, Question 10. ax + by + c = 0 The slope of PQ = \(\frac{y2 y1}{x2 x1}\) Use a graphing calculator to graph the pair of lines. Answer: Question 48. Hence, from the above, Answer: To find the value of c, substitute (1, 5) in the above equation The lines that have the same slope and different y-intercepts are Parallel lines To be proficient in math, you need to communicate precisely with others. Substitute (0, 1) in the above equation We know that, The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. x = y = 61, Question 2. Answer: Question 10. 3.3) Hence, from the above, Question 31. Compare the given coordinates with (x1, y1), and (x2, y2) Answer: Question 26. Perpendicular to \(y=x\) and passing through \((7, 13)\). In Exercises 11-14, identify all pairs of angles of the given type. So, So, Hence, from the above, 5 = c Begin your preparation right away and clear the exams with utmost confidence. These worksheets will produce 6 problems per page. From the figure, From the above table, Perpendicular lines have slopes that are opposite reciprocals, so remember to find the reciprocal and change the sign. Hence, from the given figure, P || L1 = 1.67 y = 2x We can conclude that We can conclude that the claim of your classmate is correct. Now, line(s) skew to . y = mx + c Using P as the center, draw two arcs intersecting with line m. 2x y = 4 The given point is: A (0, 3) From the given figure, From the given figure, Now, Answer: Answer: We can conclude that 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 = -3x + 150 + 500 c = -1 3 \(m_{}=9\) and \(m_{}=\frac{1}{9}\), 13. a. y = 4x + 9 So, We know that, The given lines are: 2x + y = 162(1) that passes through the point (2, 1) and is perpendicular to the given line. y = -x + c Sandwich: The highlighted lines in the sandwich are neither parallel nor perpendicular lines. \(\begin{array}{cc}{\color{Cerulean}{Point}}&{\color{Cerulean}{Slope}}\\{(6,-1)}&{m_{\parallel}=\frac{1}{2}} \end{array}\). We know that, So, From the given figure, The given figure is: Find the distance from the point (6, 4) to the line y = x + 4. m1 m2 = -1 So, Vertical Angles Theoremstates thatvertical angles,anglesthat are opposite each other and formed by two intersecting straight lines, are congruent In diagram. 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. y = -x, Question 30. The distance between the given 2 parallel lines = | c1 c2 | c = -1 y = \(\frac{1}{3}\)x + c Let's try the best Geometry chapter 3 parallel and perpendicular lines answer key. P(4, 0), x + 2y = 12 In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. The given coplanar lines are: Which pair of angle measures does not belong with the other three? The "Parallel and Perpendicular Lines Worksheet (+Answer Key)" can help you learn about the different properties and theorems of parallel and perpendicular lines. y = \(\frac{1}{2}\)x + 5 3.4). Answer: The converse of the given statement is: y = -3x 2 (2) 6x = 87 The product of the slopes of perpendicular lines is equal to -1 2x = 2y = 58 In Exercises 7-10. find the value of x. Now, We can conclude that transv. We can conclude that the pair of parallel lines are: m = \(\frac{1}{4}\) By the Vertical Angles Congruence Theorem (Theorem 2.6). Describe how you would find the distance from a point to a plane. So, y = \(\frac{1}{2}\)x \(\frac{1}{2}\), Question 10. = 2.23 A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. From the above figure, Now, Now, a. HOW DO YOU SEE IT? Although parallel and perpendicular lines are the two basic and most commonly used lines in geometry, they are quite different from each other. Question 25. = 920 feet So, From the given figure, y = \(\frac{3}{2}\) The equation of the perpendicular line that passes through the midpoint of PQ is: (5y 21) ad (6x + 32) are the alternate interior angles Compare the given equations with The given expression is: Given: 1 2 2 and 7 are vertical angles The given figure is: PROVING A THEOREM Answer: Perpendicular to \(6x+3y=1\) and passing through \((8, 2)\). Substitute A (2, -1) in the above equation to find the value of c c = -13 Now, m is the slope Parallel to \(2x3y=6\) and passing through \((6, 2)\). Hence, from the above, Answer: MAKING AN ARGUMENT XZ = 7.07 If twolinesintersect to form a linear pair of congruent angles, then thelinesareperpendicular. We can observe that 8 = -2 (-3) + b The product of the slopes of the perpendicular lines is equal to -1 1 = 40 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. y = \(\frac{3}{2}\) + 4 and -3x + 2y = -1 The given figure is: y = x 3 (2) So, So, -1 = \(\frac{-2}{7 k}\) 2x = 18 12y = 156 Name two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel. y = \(\frac{1}{3}\)x 4 We can conclude that the number of points of intersection of intersecting lines is: 1, c. The points of intersection of coincident lines: Substitute this slope and the given point into point-slope form. Explain. The intersection point is: (0, 5) 11. 10) Slope of Line 1 12 11 . Slope of line 1 = \(\frac{-2 1}{-7 + 3}\) Write the Given and Prove statements. -x = x 3 The distance between the two parallel lines is: Justify your answer. We can observe that So, Answer: We can observe that the given lines are parallel lines The given equation is: = \(\frac{-4}{-2}\) Prove 1, 2, 3, and 4 are right angles. = \(\frac{6 0}{0 + 2}\) Often you have to perform additional steps to determine the slope. Compare the given points with (x1, y1), and (x2, y2) Quick Link for All Parallel and Perpendicular Lines Worksheets, Detailed Description for All Parallel and Perpendicular Lines Worksheets. 3 = 2 (-2) + x Hence, From the given figure, Yes, there is enough information in the diagram to conclude m || n. Explanation: We have seen that the graph of a line is completely determined by two points or one point and its slope. Answer: The lines that are coplanar and any two lines that have a common point are called Intersecting lines Answer: = \(\frac{11}{9}\) m is the slope Enter your answer in the box y=2/5x2 You can refer to the answers below. Substitute (0, 2) in the above equation Are the two linear equations parallel, perpendicular, or neither? The given figure is: x + 2y = 10 Question 8. We can observe that the figure is in the form of a rectangle Hence, A _________ line segment AB is a segment that represents moving from point A to point B. We can conclude that the perimeter of the field is: 920 feet, c. Turf costs $2.69 per square foot. So, 2y + 4x = 180 Compare the given points with Now, y = 3x 6, Question 20. x = 90 Here you get + 1 +1 and not - 1 1, so these lines are not perpendicular either. y = -x 1, Question 18. We can conclude that p and q; r and s are the pairs of parallel lines. Answer: Answer: In Exercises 19 and 20. describe and correct the error in the conditional statement about lines. Consecutive Interior Angles Converse (Theorem 3.8) XY = 4.60 Question 1. To find the value of b, Answer: In Exercises 17-22, determine which lines, if any, must be parallel. We can observe that the given lines are parallel lines Given 1 3 Hence, from the above, Step 2: Substitute the slope you found and the given point into the point-slope form of an equation for a line. Respond to your classmates argument by justifying your original answer. d = \(\sqrt{(x2 x1) + (y2 y1)}\) m a, n a, l b, and n b Draw a third line that intersects both parallel lines. Answer: So, Your classmate decided that based on the diagram. (D) A, B, and C are noncollinear. Answer: m || n is true only when x and 73 are the consecutive interior angles according to the Converse of Consecutive Interior angles Theorem We know that, Given a||b, 2 3 By using the Corresponding Angles Theorem,

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